Background: We endeavor to reproduce historical observations and to identify and remedy the cause of any disparate predictions before using models to inform public policy-making. We have no finely age- and time-stratified observations from historical pandemics, but prior exposure of older adults to a related strain is among the more compelling hypotheses for the w-shaped age-specific mortality characterizing the 1918 pandemic, blurring the distinction between annual and pandemic influenza.
Methods: We are attempting to reproduce patterns in annual influenza morbidity and mortality via a cross-classified compartmental model whose age class sojourns approximate the longevity of clusters of closely-related strains. In this population model, we represent effective inter-personal contacts via a generalization of Hethcote's formulation of mixing as a convex combination of contacts within and between age groups. Information about mixing has been sought in face-to-face conversations, a surrogate for contacts by which respiratory diseases might be transmitted, but could also be obtained from household and community transmission studies. We reanalyzed observations from several such studies to learn about age-specific preferences, proportions of contacts with others the same age. And we obtained age-specific forces of infection from proportions reporting illness in a prospective study of household transmission during the 1957 influenza pandemic, which we gamma distributed to correct for misclassification. Then we fit our model to weekly age-specific hospitalizations from Taiwan's National Health Insurance Program, 2000-07, by adjusting a) age-specific coefficients of harmonic functions by which we model seasonality and b) probabilities of hospitalization given influenza.
Results: While our model accounts for only 30% of the temporal variation in hospitalizations, estimated conditional probabilities resemble official health resource utilization statistics. Moreover, younger and older people are most likely to be hospitalized and elderly ones to die of influenza, with modeled deaths 10.6% of encoded influenza or pneumonia mortality.
Conclusions: Having satisfactorily reproduced recent patterns in influenza morbidity and mortality in Taiwan via a deterministic model, we will switch to a discrete event-time simulator and - possibly with different initial conditions and selected parameters - evaluate the sufficiency of projected pandemic vaccine production.
Joint work with Denis Taneri, and Jen-Hsiang Chuang
Refreshments will be served at 4PM in Skiles 236.
SPECIAL TIME AND LOCATION FOR THIS WEEK ONLY
PLEASE NOTE UNUSUAL TIME
Consider a class of multidimensional degenerate diffusion processes of the following form
X_t = x+\int_0^t (X_s) ds+\int_0^t \sigma(X_s) dW_s,
Y_t = y+\int_0^t F(X_s)ds,
where b,\sigma, F are assumed to be smooth and b,\sigma bounded. Suppose now that \sigma\sigma^* is uniformly elliptic and that \nabla F does not degenerate. These assumptions guarantee that only one Poisson bracket is needed to span the whole space. We obtain a parametrix representation of Mc Kean-Singer type for the density of (X_t,Y_t) from which we derive some explicit Gaussian controls that characterize the additional singularity induced by the degeneracy. This particular representation then allows to give a local limit theorem with the usual convergence rate for an associated Markov chain approximation. The "weak" degeneracy allows to use the local limit Theorem in Gaussian regime but also induces some difficulty to define the suitable approximating process. In particular two time scales appear. Another difficulty w.r.t. the standard literature on the topic, see e.g. Konakov and Mammen (2000), is the unboundedness of F.
Solvation process is of fundamental importance to other complex biological processes, such signal transduction, gene regulation, etc. Solvation models can be roughly divided into two classes: explicit solvent models that treat the solvent in molecular or atomic detail while implicit solvent models take a multiscale approach that generally replaces the explicit solvent with a dielectric continuum. Because of their fewer degrees of freedom, implicit solvent methods have become popular for many applications in molecular simulation with applications in the calculations of biomolecular titration states, folding energies, binding affinities, mutational effects, surface properties, and many other problems in chemical and biomedical research. In this talk, we introduce a geometric flow based multiscale solvation model that marries a microscopic discrete description of biomolecules with a macroscopic continuum treatment of the solvent. The free energy functional is minimized by coupled geometric and potential flows. The geometric flow is driven not only by intrinsic forces, such as mean curvatures, but also by extrinsic potential forces, such as those from electrostatic potentials. The potential flow is driven mainly by a Poisson-Boltzmann like operator. Efficient computational methods, namely the matched interface and boundary (MIB) method, is developed for to solve the Poisson- Boltzmann equation with discontinuous interface. A Dirichlet- to-Neumann mapping (DTN) approach is developed to regularize singular charges from biomolecules.
RECEPTION TO FOLLOW
Note change in time.
In this talk we will consider three different numerical methods for solving nonlinear PDEs:
All of the above methods are based on high-order approximations of the corresponding nonlinear PDE and respect a weak form of an entropy condition. Theoretical results and numerical examples for the performance of each of the three methods will be presented.
Note time change.
<p>You are cordially invited to attend a reception that will follow the seminar to chat informally with faculty and students. Refreshments will be provided.</p>
Please note this course runs from 3-5.
<p>(Please note this course runs from 3-5 pm.)</p>
Joint meeting at Emory
Joint meeting at Emory
These are two hour talks.
Note special day.
These are two hour lectures.
Note special time
Note special day
These are two hour lectures.
In this contribution we study the asymptotic behaviour of polynomials orthogonal with respect to a Sobolev-Type inner product
\langle p, q\rangle_S = \int^\infty_0 p(x)q(x)x^\alpha e^{-x} dx + IP(0)^t AQ(0), \alpha > -1,
where p and q are polynomials with real coefficients,
A = \pmatrix{M_0 & \lambda\\ \lambda & M_1},
IP(0) = \pmatrix{p(0)\\ p'(0)}, Q(0) = \pmatrix{q(0)\\ q'(0)},
and A is a positive semidefinite matrix.
First, we analyze some algebraic properties of these polynomials. More precisely, the connection relations between the polynomials orthogonal with respect to the above inner product and the standard Laguerre polynomials are deduced. On the other hand, the symmetry of the multiplication operator by x^2 yields a five term recurrence relation that such polynomials satisfy.
Second, we focus the attention on their outer relative asymptotics with respect to the standard Laguerre polynomials as well as on an analog of the Mehler-Heine formula for the rescaled polynomials.
Third, we find the raising and lowering operators associated with these orthogonal polynomials. As a consequence, we deduce the holonomic equation that they satisfy. Finally, some open problems will be considered.
We study several parameters of cubic graphs with large girth. In particular, we prove that every n-vertex cubic graph with sufficiently large girth satisfies the following:
The presentation is based on results obtained jointly with Tomas Kaiser, Andrew King, Petr Skoda and Jan Volec.
Pre-reception at 2:30 in Room N201. If you would like to meet with Prof. Ashtekar while he is on campus (at the Center for Relativistic Astrophysics - Boggs building), please contact <A class="moz-txt-link-abbreviated" href="mailto:lori.federico@physics.gatech.edu">lori.federico@physics.gatech.edu</a>.
Given a random word of size n whose letters are drawn independently<br />
from an ordered alphabet of size m, the fluctuations of the shape of<br />
the corresponding random RSK Young tableaux are investigated, when both<br />
n and m converge together to infinity. If m does not grow too fast and<br />
if the draws are uniform, the limiting shape is the same as the<br />
limiting spectrum of the GUE. In the non-uniform case, a control of<br />
both highest probabilities will ensure the convergence of the first row<br />
of the tableau, i.e., of the length of the longest increasing<br />
subsequence of the random word, towards the Tracy-Widom distribution.
(joint work with Csaba Biro, Dave Howard, Mitch Keller and Stephen Young. Biro and Young finished their Ph.D.'s at Georgia Tech in 2008. Howard and Keller will graduate in spring 2010)
(This is a 2 hour lecture.)
This talk continues from last week's colloquium.
This is a 2-hour talk.
Tea and light refreshments 1:30 in Room 2222. Organizer: Santosh Vempala
This is a 2 hour talk.
Preceded with a reception at 4:10pm.
This lecture is more for the general audience. Reception following lecture. Organizers: Chongchun Zeng and Hao Min Zhou
This lecture will be more for the mathematical audience
Organizer: Daniel Dadush, ACO Student, ISyE
A density functional theory of Ohta and Kawasaki gives rise to nonlocal perturbations of the well-studied Cahn-Hilliard and isoperimetric variational problems. In this talk, I will discuss these simple but rich variational problems in the context of diblock copolymers. Via a combination of rigorous analysis and numerical simulations, I will attempt to characterize minimizers without any preassigned bias for their geometry.
Tea and light refreshments 1:30 in Room 2222. Organizer: Santosh Vempala
This is joint work with Dr. Yi Zhao.
Stability methods are often used in extremal graph theory, Ramsey theory and similar areas, where an extremal problem is to be solved and
Of course, stability methods can also be used in other cases, but we restrict ourselves to the above two areas.
In my lecture I will give an introduction to the applications of the stability methods in extremal graph theory, describe cases in extremal graph theory, extremal hypergraph theory, in the Erdos-Frankl-Rold (= generalized Erdos-Kleitman-Rothschild theory) ...
In the second part of my lecture I shall describe the application of this method to the Erdos-Sos conjecture. This is part of our work with Ajtai, Komlos and Szemeredi.
Host: Meghan Duffy (School of Biology, Georgia Tech)
Linkage involves finding a set of internally disjoint paths in a graph with specified endpoints. Given graphs G and H, we say G is H-linked if for every injective mapping f:V(H) -> V(G) we can find a subgraph H' of G which is a subdivision of H, with f(v) being the vertex of H' corresponding to each vertex v of H. We describe two results on H-linkage for small graphs H.
(1) Goddard showed that 4-connected planar triangulations are 4-ordered, or in other words C_4-linked. We strengthen this by showing that 4-connected planar triangulations are (K_4-e)-linked.
(2) Xingxing Yu characterized certain graphs related to P_4-linkage. We use his characterization to show that every 7-connected graph is P_4-linked, and to construct 6-connected graphs that are not P_4-linked.
This is joint work with Michael D. Plummer and Gexin Yu.
Leo Chen: The Shape and Stability of a Flexible Sheet in a von Karman Vortex Street
Michelle Delcourt: Dessin and Manturov bracket shuffles
In this talk we will explore the connections between knot theory and combinatorics. Links are related to Grothendieck's dessins d'enfants. Cartographic one-vertex dessins can be represented by chord diagrams. The diagrams can be recorded as "words" using a finite alphabet (k-bracket parenthesis system). Many combinatorial objects are related to these Manturov bracket structures.
The Southeast Geometry Seminar is a series of semiannual one-day events focusing on geometric analysis. These events are hosted in rotation by the following institutions:
The following five speakers will give presentations on topics that include geometric analysis, and related fields, such as partial differential equations, general relativity, and geometric topology.
A poster session will be hosted. There will also be an evening public lecture by plenary speaker Sergiu Klainerman entitled The Mathematical Magic of Black Holes.
Refreshments at 4:00PM in Skiles 236
Hosted by: Huy Huynh and Yao Li
Hosted by: Huy Huynh and Yao Li
Refreshments at 4PM in Skiles 236
Refreshments in Room 2222, Klaus Building from 2-3 PM.
Hosted by: Huy Huynh and Yao Li
Hosted by: Huy Huynh and Yao Li
This is part 1 of a two part talk. The second part will continue next week.
Hosted by: Huy Huynh and Yao Li
Hosted by: Huy Huynh and Yao Li
Hosted by Academic Affairs Honors Program in collaboration with the College of Sciences.
For more information, see the <A href="/~rohrs/FranklinColloquium.pdf">flyer</a>.
Hosted by: Huy Huynh and Yao Li
Hosted by: Huy Huynh and Yao Li
Light refreshments will be available in Room 236 at 10:30 am.
***Refreshments at 4PM in Skiles 236.***
Hosted by: Huy and Yao
Hosted by: Huy Huynh and Yao Li
Hosted by: Huy Huynh and Yao Li
The speaker is visiting Georgia Tech for the full week. His office will be Skiles 133A.
Hosted by: Huy Huynh and Yao Li
Hosted by: Huy Huynh and Yao Li
Hosted by Vijay Vazirani
Anton Leykin is an invited speaker presenting "Certified numerical solving of systems of polynomial equations"
Please note the location: Last minute room change to Skiles 270.
This talk should be non-technical except the last few slides. The talk is<br />
based on a work done in collaboration with Denis Charles, Max Chickering,<br />
Nikhil Devanur, and Manan Sanghi, all from Microsoft.
I will propose two numerical approaches for minimizing the MFF. Approach<br />
I is good for high-dimensional systems and fixed endpoints. It is <br />
based on temperature relaxation strategy and Broyden's method. Approach<br />
II is good for low-dimensional systems and only one fixed endpoint. It<br />
is based on Sethian's Fast Marching Method.I will show the <br />
application of Approaches I and II to the problems of rearrangement of<br />
Lennard-Jones cluster of 38 atoms and of CO escape from the Myoglobin protein<br />
respectively.
Hosted by: Yao Li and Ricardo Restrepo
This talk is part of the oral exam for the speaker. Please note the special time, place. Also the talk itself will be 45 min long.
Hosted by Yao Li and Ricardo Restrepo.
Hosts: Yao Li and Ricardo Restrepo
This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will follow at 5.
This is the second talk in the Emory-Ga Tech-UGA joint seminar. The first talk will begin at 3:45.
Hosts: Yao Li and Ricardo Restrepo
Hosts: Yao Li and Ricardo Restrepo
Note this is a 2 hour talk (with a short break in the middle).
Hosted by Renato DC Monteiro, ISyE.
Hosts: Yao and Ricardo
Note this is a 2 hour talk.
Hosts: Yao Li and Ricardo Restrepo
Hosted by Christian Houdre and Liang Peng.
This talk will be the oral examination for Meredith Casey.
Hosts: Yao Li and Ricardo Restrepo
Note this is a 2 hour talk.
Hosts: Yao Li and Ricardo Restrepo
The talk is 1.5-2 hours long, and although some knowledge of HeegaardFloer homology and contact manifolds is useful I will spend some time inthe begining to review the basic notions. So the talk should be accessibleto everyone.
Hosts: Yao Li and Ricardo Restrepo.
Note this is a 2 hour talk.
**PLEASE NOTE SPECIAL TIME**
This is the first talk in the Emory-Ga Tech-UGA joint seminar. The second talk will begin at 5:00. (NOTE: These talks are on the UGA campus.)
This lecture is more for the general audience. <br />
Reception to follow in Klaus Atrium.
Mathematics lecture
Hosts: Yao Li and Ricardo Restrepo
This will be a 2 hour talk.
Hosts: Yao Li and Ricardo Restrepo
Note this is a two hour seminar.
Hosts: Amey Kaloti and Ricardo Restrepo
Hosts: Amey Kaloti and Ricardo Restrepo
Note the unusual time and room
If you wish to drive your own car and park, the closest parking deck<br />
is attached to the Oxford Rd Building. There will be a charge for<br />
parking, which is $6 for 2-3 hours. Once you have parked, exit the<br />
parking garage into the building and there will be an elevator to your<br />
right. Take the elevator to level 3. You should take a left out of<br />
the elevator and proceed through the glass doors into the courtyard<br />
area. The Mathematics and Science Center will be the building to your<br />
left.
If you wish to drive your own car and park, the closest parking deck is attached<br />
to the Oxford Rd Building. There will be a charge for parking, which is $6 for<br />
2-3 hours. Once you have parked, exit the parking garage into the building and<br />
there will be an elevator to your right. Take the elevator to level 3. You<br />
should take a left out of the elevator and proceed through the glass doors into<br />
the courtyard area. The Mathematics and Science Center will be the building to<br />
your left.
Hosted by Christian Houdre and Liang Peng
Hosted by Christian Houdre and Liang Peng
Tea and light refreshments 2:30 p.m. in Room 2222
Hosts: Amey Kaloti and Ricardo Restrepo
<a href="http://www.nimbios.org/press/MaoFeature" title="http://www.nimbios.org/press/MaoFeature">http://www.nimbios.org/press/Ma...
Hosts: Amey Kaloti and Ricardo Restrepo.
Hosts: Amey Kaloti and Ricardo Restrepo
Host: Predrag Cvitanovic, School of Physics
Robert J. Lang is recognized as one of the foremost origami artists in the world as well as a pioneer in computational origami and the development of formal design algorithms for folding. With a Ph.D. in Applied Physics from Caltech, he has, during the course of work at NASA/Jet Propulsion Laboratory, Spectra Diode Laboratories, and JDS Uniphase, authored or co-authored over 80 papers and 45 patents in lasers and optoelectronics as well as authoring, co-authoring, or editing 9 books and a CD-ROM on origami. He is a full-time artist and consultant on origami and its applications to engineering problems but moonlights in physics: from 2007-2010 as the Editor-in-Chief of the IEEE Journal of Quantum Electronics.
Refreshments will be served at 3:30.
The actual talk will be 40 minutes. Note the unusual time.
Hosts: Amey Kaloti and Ricardo Restrepo
Hosted by Predrag Cvitanović, School of Physics, Georgia Tech.
Hosted by Christian Houdre and Liang Peng.
Hosted also by Ben Webb
Other organizers include: Ruoting Gong, <br />
Huy Huynh, <br />
Jinyong Ma, <br />
Ruodu Wang, and<br />
Linwei Xin.
Advisor Chongchun Zeng
Recall this is a two hour seminar (running from 2-4).
Recall this is a two hour seminar (2-4).
Recall this is a 2 hour seminar (2-4).
Hosted by Christian Houdre and Liang Peng
Hosted by Predrag Cvitanovic, School of Physics
Recall this is a 2 hour seminar.
[Note unusual day and time!]
Hosted by Christian Houdre and Liang Peng
Recall this is a 2 hour seminar.
Refreshments at 10:30am in the atrium outside Skiles 006
Hosted by Christian Houdre and Liang Peng
Hosted by Christian Houdre and Liang Peng
Graduate Advisor: Eberhard Voit
Joint colloquium between the School of Physics & the School of Earth and Atmospheric Sciences<br />
hosted by Predrag Cvitanovi. <br />
<a href="https://docs.google.com/spreadsheet/ccc?key=0Avrez5uyvwE7dERQQkV1eElNRUd... />
To schedule a meeting with the speaker</a>.
Note that this talk is on the UGA campus.
Note that this talk is on the UGA campus.
There will be a reception in the Atrium of the Klaus building at 4PM.
Please contact Guantao Chen, <a href="mailto:gchen@gsu.edu">gchen@gsu.edu</a> if you are interested in participating this mini-conference.
This is a joint ARC-SoM colloquium, and is in conjunction with the ARC Theory Day on November 11, 2011
Advisor: Liang Peng
Hosts are Ernie Croot and Dan Margalit.
There will be a tea 30 minutes before the colloquium.
Hosts: Christian Houdre and Liang Peng.
Vladimir Koltchinskii's talk has been rescheduled for next Tuesday, February 14
Host: Carlos Sa de Melo, School of Physics
(Refreshments in the lounge outside Skiles 005 at 4:05pm)
Hosted by Dan Goldman, School of Physics
Reception in the Atrium of the Klaus building at 4PM.
Booksigning to follow.
Note nonstandard day and time.
Note this is a 2 hour talk
An undergraduate-accessible talk.
Note this is a 2 hour talk.
Hosts: Michael Schatz and Predrag Cvitanovic, School of Physics
Host: Turgay Uzer, School of Physics
Host: Daniel Goldman, School of Physics
Note this is a 2 hour talk.
Hosts Christian Houdre and Liang Peng
The general public lecture will be presented by Jason Cantarella (University of Georgia) entitled<br />
The Square Peg Theorems or What does it mean to solve simultaneous equations? to take place in Klaus 1116 at 5:00PM
General audience lecture
Mathematics lecture
Refreshments at 4PM in Lobby of Weber SST building
Host: Predrag Cvitanovic
Hosts Christian Houdre and Liang Peng
Joint with Applied and Computational Mathematics Seminar
Host: David Hu. Refreshments will be served.<br />
<a href="http://www2.me.gatech.edu/www/calendar/view_seminar.asp?speaker=Evelyn%2... target="_blank">Speaker's Bio</a>
<a href="http://www2.me.gatech.edu/www/calendar/view_seminar.asp?speaker=Zi%20Che... target="_blank">Speaker's Bio</a>. <br />
Host: David Hu, School of Mechanical Engineering
Hosted by Christian Houdre and Liang Peng
Given a set of tiles on a square grid (think polyominoes) and a region, can we tile the region by copies of the tiles? In general this decision problem is undecidable for infinite regions and NP-complete for finite regions. In the case of simply connected finite regions, the problem can be solved in polynomial time for some simple sets of tiles using combinatorial group theory; whereas the NP-completeness proofs rely heavily on the regions having lots of holes. We construct a fixed set of rectangular tiles whose tileability problem is NP-complete even for simply connected regions.This is joint work with Igor Pak.
In this talk I will briefly survey results on Vertex Sparsification and some of our results on Mimicking network(or Exact Cut Sparsifier). Ankur Moitra introduced the notion of vertex sparsification to construct a smaller graph which preserves the properties of a huge network that are relevant to the terminals. Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K \subset V$, a vertex cut sparsifier is a smaller graph $H=(V_H,E_H)$ that approximately(quality f>=1) preserves all the minimum cuts between the terminals. Mimicking networks are the best quality vertex cut sparsifiers i.e, with quality 1. We improve both the previous upper($2^{2^{k}}$ ) and lower bounds($k+1$) for mimicking network reducing the doubly-exponential gap between them to a single-exponential gap. 1. Given a graph $G$, we exhibit a construction of mimicking network with at most $k$'th Hosten-Morris number ($\approx 2^{{(k-1)} \choose {\lfloor {{(k-1)}/2} \rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {\itrestricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2. There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{\frac{k-1}{2}}$. 3. We also exhibit constructions of better mimicking networks for trees($\lfloor(\frac{3k}{2})-1\rfloor$), outerplanar graphs($5k-9$) and graphs of bounded($t$) tree-width($k 2^{(2t+1) \choose {(2t+1)/2}}$). The talk will be self-contained and with no prerequisite.
Hosts: Christian Houdre and Liang Peng
Kickoff of the Tech Topology Conference from December 7-9, 2012.
From the publisher's website: "... The goal of these lectures is to introduce newcomers from the different <br />
camps to algebraic statistics. The introduction will be centered around <br />
the following three observations: many important statistical models <br />
correspond to algebraic or semi-algebraic sets of parameters; the <br />
geometry of these parameter spaces determines the behaviour of widely <br />
used statistical inference procedures; computational algebraic geometry <br />
can be used to study parameter spaces and other features of statistical <br />
models... "
This is the first of 4 or 5, 1.5 hour talks.
Mr. Crawford grew up near Philadelphia and has a B.S. in Applied Mathematics<br />
from Georgia Tech. He served as an Air Force officer, retiring as a colonel in<br />
1996. In addition to being a member of Georgia Battlefields Association and the<br />
Civil War Round Table of Atlanta, Charlie is a life member of the Civil War<br />
Trust.
School of Computational Science and Engineering job candidate talk
Hosted by the School of Computational Science and Engineering
Hosted by the College of Computing<br />
Light refreshments served at 4:30 PM
Note: this is a 40 minute talk.
Host: Turgay Uzer, School of Physics
Note different time and day.
References<br />
[1] S. Arlot and P. Massart. Data-driven calibration of penalties for least-squares regression. J. Mach. Learn.<br />
Res., 10:245.279 (electronic), 2009.<br />
[2] L. Birgé and P. Massart. Minimal penalties for Gaussian model selection. Probab. Theory Related Fields,<br />
138(1-2):33.73, 2007.<br />
[3] Vladimir Koltchinskii. Oracle inequalities in empirical risk minimization and sparse recovery problems,<br />
volume 2033 of Lecture Notes in Mathematics. Springer, Heidelberg, 2011. Lectures from the 38th Prob-<br />
ability Summer School held in Saint-Flour, 2008, École d.Été de Probabilités de Saint-Flour. [Saint-Flour<br />
Probability Summer School].<br />
[4] Pascal Massart. Concentration inequalities and model selection, volume 1896 of Lecture Notes in Math-<br />
ematics. Springer, Berlin, 2007. Lectures from the 33rd Summer School on Probability Theory held in<br />
Saint-Flour, July 6.23, 2003, With a foreword by Jean Picard.
Hosts: Christian Houdre and Liang Peng
Mathematics Audience Lecture
Hosts: Christian Houdre and Liang Peng
El Soufi will be visiting Harrell for the week leading up to this seminar
General Audience Lecture. Reception to follow in Klaus Atrium.
Hosts Christian Houdre and Liang Peng
Advisor: Dr. Matthew Baker
Hosted by School of Computer Science.
Note this is a 1 hour seminar (not the usual 2 hours).
Note this is a 1 hour seminar (not the usual 2 hours).
Joint work with Saugata Basu sbasu@math.purdue.edu On a real analogue of Bezout inequality and the number of connected components of sign conditions. <a href="http://arxiv.org/abs/1303.1577" title="http://arxiv.org/abs/1303.1577">http://arxiv.org/abs/1303.1577</a>
Refreshements served at 4:00pm
This talk assumes no familiarity with directed topology, flow-cut dualities, or sheaf (co)homology.
Host: Dan Goldman, Physics
Kickoff of the Tech Topology Conference from December 6-8, 2013. For complete details see<br />
ttc.gatech.edu
Christian Sadel is a Mathematical Physicists with broad spectrum of competences, who has been working in different areas, Random Matrix Theory (with H. Schulz-Baldes), discrete Schrödinger operators and tree graphs (with A. Klein), cocycle theory (with S. Jitomirskaya & A. Avila), SLE and spectral theory (with B. Virag), application to Mott transports in semiconductors (with J. Bellissard).
SHORT BIO:<br />
Alexander Schrijver is Professor of Mathematics at the<br />
University of Amsterdam and researcher at the Center for<br />
Mathematics and Computer Science (CWI) in Amsterdam.<br />
His research focuses on discrete mathematics and optimization,<br />
in particular on applying methods from fundamental mathematics.<br />
He is the author of four books, including 'Theory of Linear and<br />
Integer Programming' and 'Combinatorial Optimization - Polyhedra<br />
and Efficiency'.<br />
<br />
He received Fulkerson Prizes in 1982 and 2003, Lanchester Prizes in<br />
1987 and 2004, a Dantzig Prize in 2003, a Spinoza Prize in 2005,<br />
a Von Neumann Theory Prize in 2006, and an Edelman Award in 2008.<br />
He is a member of the Royal Netherlands Academy of Arts and Sciences<br />
since 1995 and of three foreign academies, received honorary<br />
doctorates from the Universities of Waterloo and Budapest, and was<br />
knighted by the Dutch Queen in 2005.
This is a joint Seminar School of Mathematics and Center of Relativistic Astrophysics, Georgia Tech
Joint DOS-ACO Seminar. Food and refreshments will be provided.
Contact Yuliya Babenko, <a href="mailto:ybabenko@kennesaw.edu">ybabenko@kennesaw.edu</a>
Speaker is visiting the School of Biology, Georgia Tech
Note: This is a special time for Research Horizons.
This is a final project for Dr. Etnyre's Differential Geometry class.
Host: Shina Tan, School of Physics, Georgia Tech
Bio: Georgios Piliouras is a postdoc at Caltech, Center for Mathematics and<br />
Computation. He received his PhD in Computer Science from Cornell<br />
University and has been a Georgia Tech postdoc at the EE department.
Hosted by Dana Randall
This is a project for Prof. Margalit's course on Low-dimensional Topology and Hyperbolic Geometry.
This is a project for Prof. Margalit's course on Low-dimensional Topology and Hyperbolic Geometry.
After the lecture, there will be a reception and time to chat with Engle and other guests.
This is the third of several talks discussing embeddings of manifolds. I will discuss some general results for smooth manifolds, but focus on embeddings of contact manifolds into other contact manifolds. Particular attention will be paid to embeddings of contact 3-manifolds in contact 5-manifolds. I will discuss two approaches to this last problem that are being developed jointly with Yanki Lekili.
After the talk there will be a reception and time for visitors to chat with Donelan and each other.
Biography: Michael Levitt is an American-British-Israeli biophysicist and professor of structural biology in<br />
the Stanford University School of Medicine and a winner of the 2013 Nobel Prize in Chemistry. Born<br />
in South Africa in 1947, Levitt earned his Bachelor of Science in Physics from Kings College<br />
London and his Ph.D. in biophysics from Cambridge University. His research involves multi-scale<br />
approaches to molecular modeling: Coarse-grained models that merge atoms to allow folding<br />
simulation and hybrid models that combine classical and quantum mechanics to explain how enzymes<br />
works by electrostatic strain. Levitt's diverse interests have included RNA and DNA modeling,<br />
protein folding simulation, classification of protein folds and protein geometry, antibody<br />
modeling, x-ray refinement, antibody humanization, side-chain geometry, torsional normal mode,<br />
molecular dynamics in solution, secondary structure prediction, aromatic hydrogen bonds, structure<br />
databases, and mass spectrometry. His Stanford research team currently works on protein evolution,<br />
the crystallographic phase problem and Cryo-EM refinement. He is a member of both the Royal<br />
Society of London and the U.S. National Academy of Science. Levitt also remains an active computer<br />
programmer--"a craft skill of which I am particularly proud," he says.
Host: College of Sciences, Georgia Tech
Anna Vershynina is a job candidate. She is a Mathematical Physicist working on the rigorous mathematical theory of N-body problem and its relation with quantum information.
Colm Mulcahy is a professor of mathematics at Spelman College, in Atlanta, where he has<br />
taught since 1988. He's currently on leave in the DC area. Over the last decade, he has<br />
been at the forefront of publishing new mathemagical principles and effects for cards,<br />
particularly in his long-running bi-monthly Card Colm for the MAA. Some of his puzzles<br />
have been featured in the New York Times. His book <br />
<a href="http://www.crcpress.com/product/isbn/9781466509764" target="_blank">Mathematical Card Magic: Fifty-Two New Effects</a> was published by AK Peters/CRC Press in 2013.<br />
Colm is a recipient of MAA's Allendoerfer Award for excellence in expository writing, for<br />
an article on image compression using wavelets.
This is a project for Prof. Margalit's course on Low-dimensional Topology and Hyperbolic Geometry.
Predrag Cvitanovic, School of Physics
Kick-off of the <a href="http://ttc.gatech.edu">Tech Topology Conference</a>, December 5-7, 2014
Joint ARC colloquium/ACO student seminar
Please note the delayed start for this week only.
Reference[1] Moody T. Chu<br />
<br />
<br />
<br />
<br />
, Nonnegative Inverse Eigenvalue and Singular Value Problems, SIAM J. Numer. Anal (1992).[2] Wei Ma and Zheng-J. Bai, A regularized directional derivative-based Newton method for inverse singular value problems, Inverse Problems (2012).
Dr. Hurth is a recent graduate of the Georgia Tech School of Mathematics. After his talk, the AMS Graduate Chapter is taking Dr. Hurth to dinner at Gordon Biersch. Graduate students and others interested in speaking to Dr. Hurth are invited to join us. If interested, please RSVP to JD Walsh (in person or at <a href="mailto:walsh@math.gatech.edu">walsh@math.gatech.edu</a>).
This is a project for Prof. Wickelgren's course on Stable Homotopy Theory.
Hosted by GT Honors Program and College of Sciences
A reception will follow the talk and giving time for visitors to chat with Ellenberg and each other.
Useful background:The paper I’m discussing: <a href="http://arxiv.org/abs/1502.03736" title="http://arxiv.org/abs/1502.03736">http://arxiv.org/abs/1502.03736</a>Terry Tao’s blog post on Dvir’s theorem: <a href="https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel... title="https://terrytao.wordpress.com/2008/03/24/dvirs-proof-of-the-finite-fiel... earlier paper with Terry and Richard Oberlin about Kakeya restriction over finite fields: <a href="http://arxiv.org/abs/0903.1879" title="http://arxiv.org/abs/0903.1879">http://arxiv.org/abs/0903.1879</a>
Hosted by Predrag Cvitanovic, School of Physics
Karl Liechty is the<br />
winner of the 2015 Szego prize in orthogonal polynomials and special functions.
Speaker’s Biography:Michael Malisoff received his PhD in 2000 from <br />
the Department of Mathematics at Rutgers University in New Brunswick, <br />
NJ. In 2001, he joined the faculty of the Department of Mathematics at <br />
Louisiana State University in Baton Rouge (LSU), where he is now the Roy<br />
Paul Daniels Professor #3 in theLSU College of Science. His main <br />
research has been on controller design and analysis for nonlinear <br />
control systems with time delays and uncertainty and their applications <br />
in engineering. One of his projects is joint with the Georgia Tech <br />
Savannah Robotics team, and helped develop marine robotic methods to <br />
help understand the environmental impacts of oil spills. His more than <br />
100 publications include a Springer monograph on constructive Lyapunov <br />
methods. His awards include the First Place Student Best Paper Award at <br />
the 1999 IEEE Conference on Decision and Control, two three-year <br />
NationalScience Foundation Mathematical Sciences Priority Area <br />
grants, and 9 Best Presentation awards in American Control Conference <br />
sessions. He is an associate editor for IEEE Transactions on Automatic <br />
Control and for SIAM Journal on Control and Optimization.
For Prof. Wickelgren's Stable Homotopy Theory class
This is the 3rd Jorge Ize Memorial lecture, at IIMAS, Mexico City. We will join a videoconference of the event.
For Prof. Wickelgren's Stable Homotopy Theory class
Special time.
Refreshments will be served in the atrium immediately following the talk. Please join us to welcome the new class of ACO students.
Food and Drinks will be provided after the seminar.
Food and Drinks will be provided before the seminar.
Light refreshments at 6:30pm
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
Review of a recent paper by Chatterjee et al. (Arxiv 1406.5291)
Food and Drinks will be provided before the seminar
Joint with School of Math Colloquium. Special time (colloquium time).
Advisor: Dr. Federico Bonetto
Food and Drinks will be provided before the seminar.
Refreshments will be served in the atrium after the talk.
Joint work with Vladimir Koltchinskii.
Refreshments will be served in the atrium after the talk.
First featured lecture in the Atlanta Lecture Series in Combinatorics and Graph Theory mini-conference
Second featured lecture in the Atlanta Lecture Series in Combinatorics and Graph Theory mini-conference
This talk should interest people in Algebra, Dynamical Systems and Mathematical Physics in addition to Geometry and Topology. Volodia Nekrashevych will visit Atlanta from Sunday November 15th evening until Tuesday November 17th afternoon. He will be available for private talks on Monday November 14th after noon or on Tueasday morning before 10AM. Contact him directly by email or contact <a href="mailto:jeanbel@math.gatech.edu">jeanbel@math.gatech.edu</a> to schedule a meeting or to have a dinner with him.
Food and Drinks will be provided before the seminar.
Hosted by Roman Grigoriev, School of Physics
Please not non-standard day for seminar.
Joint work with Shachar Lovett.
Kick-off of the <a href="http://ttc.gatech.edu/">Tech Topology Conference</a>, December 4-6, 2015
Food and Drinks will be provided before the seminar.
Paper available on arXiv:1412.3661
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
Joint work with Yinon Spinka.
Please note different day and time for the seminar
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
When a disease outbreak occurs, mathematical models are used to<br />
estimate the potential severity of the epidemic. The average number of<br />
secondary infections resulting from the initial infection or reproduction<br />
number, R_0, quantifies this severity. R_0 is estimated from the models by<br />
leveraging observed case data and understanding of disease epidemiology.<br />
However, the leveraged data is not perfect. How confident should we be<br />
about measurements of R_0 given noisy data? I begin my talk by introducing<br />
techniques used to model epidemics. I show how to adapt standard models to<br />
specific diseases by using the 2014-2015 Ebola outbreak in West Africa as<br />
an example throughout the talk. Nest, I introduce the inverse problem:<br />
given real data tracking the infected population how does one estimate the<br />
severity of the outbreak. Through a novel method I show how to account for<br />
both inherent noise arising from discrete interactions between individuals<br />
(demographic stochasticity) and from uncertainty in epidemiological<br />
parameters. By applying this, I argue that the first estimates of R_0<br />
during the Ebola outbreak were overconfident because demographic<br />
stochasticity was ignored.<br />
This talk will be accessible to undergraduates.
In this talk we first introduce a new "singularity-free" approach to the proof of Seidel's long exact sequence, including the fixed-point version. This conveniently generalizes to Dehn twists along Lagrangian submanifolds which are rank one symmetric spaces and their covers, including RPn and CPn, matching a mirror prediction due to Huybrechts and Thomas. The idea of the proof can be interpreted as a "mirror" of the construction in algebraic geometry, realized by a new surgery and cobordism construction. This is a joint work with Cheuk-Yu Mak.
Link to the Stelson Lecture announcement <a href="http://www.math.gatech.edu/news/stelson-lecture-dr-g-rard-ben-arous" title="http://www.math.gatech.edu/news/stelson-lecture-dr-g-rard-ben-arous">htt...
Joint work with Will Perkins and Prasad Tetali.
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
Food and Drinks will be provided before the seminar.
Note the unusual time.
Bio: Tomas Zegard is a postdoctoral fellow in the School of Civil and Environmental Engineering at Georgia Tech. He received a PhD in Structural Engineering from the University of Illinois at Urbana-Champaign in 2014. Afterwards, he took a position at SOM LLP in Chicago, an Architecture + Engineering firm specializing in skyscrapers. He has made significant contributions to the field of topology optimization through research papers and free open-source tools. Xiaojia Zhang is a doctoral candidate in the School of Civil and Environmental Engineering at Georgia Tech. She received her bachelor’s and master’s degrees in structural engineering from the University of Illinois at Urbana-Champaign. Her major research interests are structural topology optimization with material and geometric nonlinearity, stochastic programming, and additive manufacturing.
Many varieties of interest in algebraic geometry and applications<br />
are given as images of regular maps, i.e. via a parametrization.<br />
Implicitization is the process of converting a parametric description of a<br />
variety into an intrinsic (i.e. implicit) one. Theoretically,<br />
implicitization is done by computing (a Grobner basis for) the kernel of a<br />
ring map, but this can be extremely time-consuming -- even so, one would<br />
often like to know basic information about the image variety. The purpose<br />
of the NumericalImplicitization package is to allow for user-friendly<br />
computation of the basic numerical invariants of a parametrized variety,<br />
such as dimension, degree, and Hilbert function values, especially when<br />
Grobner basis methods take prohibitively long.
Food and Drinks will be provided before the seminar.
Refreshments will be provided before the seminar.
Refreshments will be provided before the seminar.
Joint work with Micha Sharir (Tel-Aviv University).
Talk by Shuozhi Xu, <br />
<br />
Title: Algorithms and Implementation for the Concurrent Atomistic-Continuum Method. <br />
<br />
Abstract: Unlikemany other multiscale methods, the concurrent atomistic-continuum<br />
(CAC) method admits the migration of dislocations and intrinsic<br />
stacking faults through a lattice while employing an underlying<br />
interatomic potential as the only constitutive relation. Here, we<br />
build algorithms and develop a new CAC code which runs in parallel<br />
using MPI with a domain decomposition algorithm. New features of the<br />
code include, but are not limited to: (i) both dynamic and<br />
quasistatic CAC simulations are available, (ii) mesh refinement<br />
schemes for both dynamic fracture and curved dislocation migration<br />
are implemented, and (iii) integration points in individual finite<br />
elements are shared among multiple processors to minimize the amount<br />
of data communication. The CAC program is then employed to study a<br />
series of metal plasticity problems in which both dislocation core<br />
effects at the nanoscale and the long range stress field of<br />
dislocations at the submicron scales are preserved. Applications<br />
using the new code include dislocation multiplication from Frank-Read<br />
sources, dislocation/void interactions, and dislocation/grain<br />
boundary interactions.
Tentative schedule: 9-12: mini-presentations, informal discussion, Q&A, led by Jose Rodriguez (numerical decomposition), Elizabeth Gross (reaction networks), Dan Bates (numerical AG for sciences and engineering); 12-1: lunch; 1pm+: catch flights, continue talking in groups.
This lecture is part of ACO25, a conference celebrating the 25th anniversary of the ACO Program. For more details about the conference please visit <a href="http://aco25.gatech.edu/" title="http://aco25.gatech.edu/">http://aco25.gatech.edu/</a>
This lecture is part of ACO25, a conference celebrating the 25th anniversary of the ACO Program. For more details about the conference please visit <a title="http://aco25.gatech.edu/" href="http://aco25.gatech.edu/">http://aco25.gatech.edu/</a>
Note the semianr scheduled for 1.5 hours. (We might take a short break in the middle and then go slightly longer.)
Note the semianr scheduled for 1.5 hours. (We might take a short break in the middle and then go slightly longer.)
Note todays seminar is just form 2 to 3 to accomodate a seminar at 3.
Please note the special time! This is Stochastic & Analysis seminars joint.
This will be a 1.5 hour seminar.
This will be a 1.5 hour (maybe slightly longer) seminar.
Dissertation advisor: Luca Dieci
This workshop is sponsored by College of Science, School of Mathematics, GT-MAP and NSF.
NOTE: This is the first in a forthcoming series of colloquia in quantum mathematical physics that will take place this semester. The series is a spin-off of last year's QMath conference, and is intended to be of broad interest to people wanting to know the state of the art of current topics in mathematical physics.
postponed from September 18
CORRECTED DATE. NOTE: This is the first in a forthcoming series of colloquia in quantum mathematical physics that will take place this semester. The series is a spin-off of last year's QMath conference, and is intended to be of broad interest to people wanting to know the state of the art of current topics in mathematical physics.
Lunch will be provided. The talk will be the first 25 minutes of the hour and then will be followed by discussion.
Note this talk is only 1 hour (to allow for the GT MAP seminar at 3.
Notice the seminar is back to 1.5 hours this week.
This is part of the 2017 Quolloquium series.
This is part of the 2017 Quolloquium series.
Bio: Hyenkyun Woo is an assistant professor at KOREATECH (Korea University of Technology and Education). He got a Ph.D at Yonsei university. and was a post-doc at Georgia Tech and Korea Institute of Advanced Study and others.
Planar contact manifolds have been intensively studied to understand several aspects of 3-dimensional contact geometry. In this talk, we define "iterated planar contact manifolds", a higher-dimensional analog of planar contact manifolds, by using topological tools such as "open book decompositions" and "Lefschetz fibrations”. We provide some history on existing low-dimensional results regarding Reeb dynamics, symplectic fillings/caps of contact manifolds and explain some generalization of those results to higher dimensions via iterated planar structure. This is partly based on joint work in progress with J. Etnyre and B. Ozbagci.
[CV: Prof. Oded Margalit, PhD in Computer Science from Tel-Aviv University under the <br />
supervision of Prof. Zvi Galil has worked at IBM's Haifa research lab on<br />
machine learning, constraint satisfaction, verification and more. Currently he is the CTO <br />
of the IBM Cyber security center of excellence at Ben Gurion University <br />
of the Negev. Oded participates in organising several computer science <br />
competitions (like the international IEEEXtreme and the national CodeGuru). He loves riddles and authors the monthly <br />
challenge corner of IBM research: "Ponder-This".]
[CV: Prof. Oded Margalit has a PhD in computer science from Tel Aviv University under the supervision of Prof. Zvi Galil. He has worked at IBM Research – Haifa in the areas of machine learning, constraint satisfaction, verification, and more. Currently, he is the CTO of the IBM Cybersecurity Center of Excellence in Beer Sheva, Israel. Oded helps organize several computer science competitions, like the international IEEEXtreme and the Israeli national CodeGuru competition. He loves riddles and authors the IBM Research monthly challenge corner Ponder This.]
Transition State Theory describes how a reactive system crosses an energy barrier that is marked by a saddle point of the potential energy. The transition from the reactant to the product side of the barrier is regulated by a system of invariant manifolds that separate trajectories with qualitatively different behaviour. <br />
<br />
The situation becomes more complex if there are more than two reaction channels, or possible outcomes of the reaction. Indeed, the monkey saddle potential, with three channels, is known to exhibit chaotic dynamics at any energy. We investigate the boundaries between initial conditions with different outcomes in an attempt to obtain a qualitative and quantitative description of the relevant invariant structures.
This will be a 90 minute seminar
This theorem is one of earliest instance of the h-principle, and there will be a series of talks on it this semester.
We show that there is a symmetric n-dimensional convex set whose Banach--Mazur distance to the cube is bounded below by n^{5/9}/polylog(n). This improves previously know estimate due to S.Szarek, and confirms a conjecture of A.Naor. The proof is based on probabilistic arguments.
The concentration of Lipschitz functions around their expectation is a classical topic and continues to be very active. In these talks, we will discuss some recent progress in detail, including: A tight log-Sobolev inequality for isotropic logconcave densities A unified and improved large deviation inequality for convex bodies An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. (Warning: the talk will involve elementary calculus on the board, sometimes at an excruciatingly slow pace). Joint work with Yin Tat Lee.
Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs
is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.
In this talk we describe some new controllability and identifability results for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more
regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems
for such systems more diffcult.
We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of thestrings, lengths of the edges, attached masses, and the topology of the graph. The proofs are based on the boundary control and leaf peeling methods developed in our previous papers. The boundary control method is a powerful
method in inverse theory which uses deep connections between controllability and identifability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.
The concentration of Lipschitz functions around their expectation is a classical topic and continues to be very active. In these talks, we will discuss some recent progress in detail, including: A tight log-Sobolev inequality for isotropic logconcave densities A unified and improved large deviation inequality for convex bodies An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. (Warning: the talk will involve elementary calculus on the board, sometimes at an excruciatingly slow pace). Joint work with Yin Tat Lee.
This is a joint seminar by College of Engineering and School of Math.
It is natural to question whether the center of mass of a convex body $K\subset \mathbb{R}^n$ lies in its John ellipsoid $B_K$, i.e., in the maximal volume ellipsoid contained in $K$. This question is relevant to the efficiency of many algorithms for convex bodies. We obtain an unexpected negative result. There exists a convex body $K\subset \mathbb{R}^n$ such that its center of mass does not lie in the John ellipsoid $B_K$ inflated $(1-o(1))n$ times about the center of $B_K$. (Yet, if one inflate $B_K$ by a factor $n$, it contains $K$.)Moreover, there exists a polytope $P \subset \mathbb{R}^n$ with $O(n^2)$ facets whose center of mass is not contained in the John ellipsoid $B_P$ inflated $O(\frac{n}{\log(n)})$ times about the center of $B_P$.
I shall tell about some background and known results in regards to the celebrated and fascinating Log-Brunn-Minkowski inequality, setting the stage for Xingyu to discuss connections with elliptiic operators a week later.
The n-dimensional L^p Brunn-Minkowski inequality for p<1 , in particular the log-Brunn-Minkowski inequality, is proposed by Boroczky-Lutwak-Yang-Zhang in 2013, based on previous work of Firey and Lutwak . When it came out, it promptly became the major problem in convex geometry. Although some partial results on some specific convex sets are shown to be true, the general case stays wide open. In this talk I will present a breakthrough on this conjecture due to E. Milman and A Kolesnikov, where we can obeserve a beautiful interaction of PDE, operator theory, Riemannian geometry and all sorts of best constant estimates. They showed the validity of the local version of this inequality for orgin-symmtric convex sets with a C^{2} smooth boundary and strictly postive mean curvature, and for p between 1-c/(n^{3/2}) and 1. Their infinitesimal formulation of this inequality reveals the deep connection with the poincare-type inequalities. It turns out, after a sophisticated transformation, the desired inequality follows from an estimate of the universal constant in Poincare inequality on convex sets.
It has been known that when an equiangular tight frame (ETF) of size |Φ|=N exists, Φ ⊂ Fd (real or complex), for p > 2 the p-frame potential ∑i ≠ j | < φj, φk > |p achieves its minimum value on an ETF over all N sized collections of vectors. We are interested in minimizing a related quantity: 1/ N2 ∑i, j=1 | < φj, φk > |p . In particular we ask when there exists a configuration of vectors for which this quantity is minimized over all sized subsets of the real or complex sphere of a fixed dimension. Also of interest is the structure of minimizers over all unit vector subsets of Fd of size N. We shall present some results for p in (2, 4) along with numerical results and conjectures. Portions of this talk are based on recent work of D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.
Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.
We already know that the Euclidean unit ball is at the center of the Banach-Mazur compactum, however its structure is still being explored to this day. In 1987, Szarek and Talagrand proved that the maximum distance $R_{\infty} ^n$ between an arbitrary $n$-dimensional normed space and $\ell _{\infty} ^n$, or equivalently the maximum distance between a symmetric convex body in $\mathbb{R} ^n$ and the $n$-dimensional unit cube is bounded above by $c n^{7/8}$. In this talk, we will discuss a related paper by A. Giannopoulos, "A note to the Banach-Mazur distance to the cube", where he proves that $R_{\infty} ^n < c n^{5/6}$.
Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.
We will discuss several open problems concerning unique determination of convex bodies in the n-dimensional Euclidean space given some information about their projections or sectionson all sub-spaces of dimension n-1. We will also present some related results.
Thanks are due to our colleague, Vladimir Koltchinskii, for arranging this visit. Please write to Vladimir if you would like to meet with Professor Gabor Lugosi during his visit, or for additional information.
Alesker has introduced the notion of a smooth valuation on a smooth manifold M. This is a special kind of set function, defined on sufficiently regular compact subsets A of M, extending the corresponding idea from classical convexity theory. Formally, a smooth valuation is a kind of curvature integral; informally, it is a sum of Euler characteristics of intersections of A with a collection of objects B. Smooth valuations admit a natural multiplication, again due to Alesker. I will aim to explain the rather abstruse formal definition of this multiplication, and its relation to the ridiculously simple informal counterpart given by intersections of the objects B.
This talk is organized by the Association for Women in Math (AWM). Everyone is welcome to attend.
High-dimensional data arise in many fields of contemporary science and introduce new challenges in statistical learning and data recovery. Many datasets in image analysis and signal processing are in a high-dimensional space but exhibit a low-dimensional structure. We are interested in building efficient representations of these data for the purpose of compression and inference, and giving performance guarantees depending on the intrinsic dimension of data. I will present two sets of problems: one is related with manifold learning; the other arises from imaging and signal processing where we want to recover a high-dimensional, sparse vector from few linear measurements. In the first problem, we model a data set in $R^D$ as samples from a probability measure concentrated on or near an unknown $d$-dimensional manifold with $d$ much smaller than $D$. We develop a multiscale adaptive scheme to build low-dimensional geometric approximations of the manifold, as well as approximating functions on the manifold. The second problem arises from source localization in signal processing where a uniform array of sensors is set to collect propagating waves from a small number of sources. I will present some theory and algorithms for the recovery of the point sources with high precision.
The following is a well-known and difficult problem in rare event simulation: given a set and a Gaussian distribution, estimate the probability that a sample from the Gaussian distribution falls outside the set. Previous approaches to this question are generally inefficient in high dimensions. One key challenge with this problem is that the probability of interest is normally extremely small. I'll discuss a new, provably efficient method to solve this problem for a general polytope and general Gaussian distribution. Moreover, in practice, the algorithm seems to substantially outperform our theoretical guarantees and we conjecture that our analysis is not tight. Proving the desired efficiency relies on a careful analysis of (highly) correlated functions of a Gaussian random vector.Joint work with Ton Dieker.
please note special time!
We consider one or more volumes of a liquid or semi-molten material sitting on a substrate, while the vapor above is assumed to have the same medium in suspension. There may be both evaporation and condensation to move mass from one cell to another. We explore possible equilibrium states of such configurations. Our examples include a single sessile drop (or cell) on the plate, connected clusters of cells of the material on the plate, as well as a periodic configuration of connected cells on the plate. The shape of the configurations will depend on the type of energy that we take into consideration, and in settings with a vertical gravitational potential energy the clusters are shown to exhibit a preferred granular scale. The majority of our results are in a lower dimensional setting, however, some results will be presented in 3-D.
This should be unpublished. Againx3
Recently there has been an outburst of parallelization techniques to speed up optimization algorithms, particularly in applications in statistical learning and structured linear programs. Motivated by these developments, we seek for theoretical explanations of provable improvements (or the lack thereof) in performance obtained by parallelizing optimization algorithms. In 1994, Nemirovski proved that for low-dimensional optimization problems there is a very limited improvement that could be obtained by parallelization, and furthermore conjectured that no acceleration should be achievable by these means. In this talk, I will present new results showing that in high-dimensional settings no acceleration can be obtained by parallelization, providing strong evidence towards Nemirovski's conjecture. This is joint work with Jelena Diakonikolas (UC Berkeley).
This is a part of GT MAP seminar. See gtmap.gatech.edu for more information.
In this talk I will describe those linear subspaces of $\mathbf{R}^d$ which can be formed by taking the linear span of lattice points in a half-open parallelepiped. I will draw some connections between this problem and Keith Ball's cube slicing theorem, which states that the volume of any slice of the unit cube $[0,1]^d$ by a codimension-$k$ subspace is at most $2^{k/2}$.
Oral Comprehensive Exam
The purpose of this work is approximation of generic Hamiltonian dynamical systems by those with a finite number of islands. In this work, we will consider a Lemon billiard as our Hamiltonian dynamical system apparently with an infinitely many islands. Then, we try to construct a Hamiltonian dynamical system by deforming the boundary of our lemon billiard to have a finite number of islands which are the same or sub-islands of our original system. Moreover, we want to show elsewhere in the phase space of the constructed billiard is a chaotic sea. In this way, we will have a dynamical system which preserves some properties of our lemon billiards while it has much simpler structure.
Note the special time!
In joint work with J. Martinez-Garcia we study the classification problem of asymptotically log del Pezzo surfaces in algebraic geometry. This turns out to be equivalent to understanding when certain convex bodies in high-dimensions intersect the cube non-trivially. Beyond its intrinsic interest in algebraic geometry this classification is relevant to differential geometery and existence of new canonical metricsin dimension 4.
The talk will discuss a paper by Gompf and Miyazaki of the same name. This paper introduces the notion of dualisable patterns, a technique which is widely used in knot theory to produce knots with similar properties. The primary objective of the paper is to first find a knot which is not obviously ribbon, and then show that it is. It then goes on to construct a related knot which is not ribbon. The talk will be aimed at trying to unwrap the basic definitions and techniques used in this paper, without going too much into the heavy technical details.
Given a Hamiltonian system, normally hyperbolic invariant manifolds and their stable and unstable manifolds are important landmarks that organize the long term behaviour.
When the stable and unstable manifolds of a normally hyperbolic invarriant manifold intersect transversaly, there are homoclinic orbits that converge to the manifold both in the future and in the past. Actually, the orbits are asymptotic both in the future and in the past.
One can construct approximate orbits of the system by chainging several of these homoclinic excursions.
A recent result with M. Gidea and T. M.-Seara shows that if we consider long enough such excursions, there is a true orbit that follows it. This can be considered as an extension of the classical shadowing theorem, that allows to handle some non-hyperbolic directions
For each n, let M be an n by n random matrix with independent ±1 entries. We show that the probability that M is not invertable equals (1/2 + o(1/n))^n, which settles an old problem. Some generalizations are considered.
It is anticipated that chaotic regimes (characterized by, e.g., sensitivity with respect to initial conditions and loss of memory) arise in a wide variety of dynamical systems, including those arising from the study of ensembles of gas particles and fluid mechanics. However, in most cases the problem of rigorously verifying asymptotic chaotic regimes is notoriously difficult. For volume-preserving systems (e.g., incompressible fluid flow or Hamiltonian systems), these issues are exemplified by coexistence phenomena: even in quite simple models which should be chaotic, e.g. the Chirikov standard map, completely opposite dynamical regimes (elliptic islands vs. hyperbolic sets) can be tangled together in phase space in a convoluted way.
Recent developments have indicated, however, that verifying chaos is tractable for systems subjected to a small amount of noise— from the perspective of modeling, this is not so unnatural, as the real world is inherently noisy. In this talk, I will discuss two recent results: (1) a large positive Lyapunov exponent for (extremely small) random perturbations of the Chirikov standard map, and (2) a positive Lyapunov exponent for the Lagrangian flow corresponding to various incompressible stochastic fluids models, including stochastic 2D Navier-Stokes and 3D hyperviscous Navier-Stokes on the periodic box. The work in this talk is joint with Jacob Bedrossian, Samuel Punshon-Smith, Jinxin Xue and Lai-Sang Young.
The celebrated Hadwiger's theorem says that linear combinations of intrinsic volumes on convex sets are the only isometry invariant continuous valuations(i.e. finitely additive measures). On the other hand H. Weyl has extended intrinsic volumes beyond convexity, to Riemannian manifolds. We try to understand the continuity properties of this extension under theGromov-Hausdorff convergence (literally, there is no such continuityin general). First, we describe a new conjectural compactification of the set of all closed Riemannian manifolds with given upper bounds on dimensionand diameter and lower bound on sectional curvature. Points of thiscompactification are pairs: an Alexandrov space and a constructible(in the Perelman-Petrunin sense) function on it. Second, conjecturally all intrinsic volumes extend by continuity to this compactification. No preliminary knowledge of Alexandrov spaces will be assumed, though it will be useful.
We shall survey a variety of results, some recent, some going back a long time, where combinatorial methods are used to prove or disprove the existence of orthogonal exponential bases and Gabor bases. The classical Erdos distance problem and the Erdos Integer Distance Principle play a key role in our discussion.
This is a SCMB MathBioSys Seminar posted on behalf of Melissa Kemp (GT BME)
Constriction of blood vessels in the extremities due to traumatic injury to halt excessive blood loss or resulting from pathologic occlusion can cause considerable damage to the surrounding tissues with significant morbidity and mortality. Optimal healing of damaged tissue relies on the precise balance of pro-inflammatory and pro-healing processes of innate inflammation. In this talk, we will present a discrete multiscale mathematical model that spans the tissue and intracellular scales, and captures the consequences of targeting various regulatory components. We take advantage of the canalization properties of some of the functions, which is a type of hierarchical clustering of the inputs, and use it as control to steer the system away from a faulty attractor and understand better the regulatory relations that govern the system dynamics.EDIT: CANCELLED
We discuss a problem of asymptotically efficient (that is, asymptotically normal with minimax optimal limit variance) estimation of functionals of the form $\langle f(\Sigma), B\rangle$ of unknown covariance $\Sigma$ based on i.i.d.mean zero Gaussian observations $X_1,\dots, X_n\in {\mathbb R}^d$ with covariance $$\Sigma$. Under the assumptions that the dimension $d\leq n^{\alpha}$ for some $\alpha\in (0,1)$ and $f:{\mathbb R}\mapsto {\mathbb R}$ is of smoothness $s>\frac{1}{1-\alpha},$ we show how to construct an asymptotically efficient estimator of such functionals (the smoothness threshold $\frac{1}{1-\alpha}$ is known to be optimal for a simpler problem of estimation of smooth functionals of unknown mean of normal distribution).
The proof of this result relies on a variety of probabilistic and analytic tools including Gaussian concentration, bounds on the remainders of Taylor expansions of operator functions and bounds on finite differences of smooth functions along certain Markov chains in the spaces of positively semi-definite matrices.
(The talk will be at 1-2pm, then it follows by a discussion session from 2 pm to 2:45 pm.)
Powerful AI systems, which are driven by machine learning, are increasingly controlling various aspects of modern society: from social interactions (e.g., Facebook, Twitter, Google, YouTube), economics (e.g., Uber, Airbnb, Banking), learning (e.g., Wikipedia, MOOCs), governance (Judgements, Policing, Voting), to autonomous vehicles and weapons. These systems have a tremendous potential to change our lives for the better, but, via the ability to mimic and nudge human behavior, they also have the potential to be discriminatory, reinforce societal prejudices, and polarize opinions. Moreover, recent studies have demonstrated that these systems can be quite brittle and generally lack the required robustness to be deployed in various civil/military situations. The reason being that considerations such as fairness, robustness, stability, explainability, accountability etc. have largely been an afterthought in the development of AI systems. In this talk, I will discuss the opportunities that lie ahead in a principled and thoughtful development of AI systems.
BioNisheeth Vishnoi is a Professor of Computer Science at Yale University. He received a B.Tech in Computer Science and Engineering from IIT Bombay in 1999 and a Ph.D. in Algorithms, Combinatorics and Optimization from Georgia Tech in 2004. His research spans several areas of theoretical computer science: from approximability of NP-hard problems, to combinatorial, convex and non-convex optimization, to tackling algorithmic questions involving dynamical systems, stochastic processes and polynomials. He is also broadly interested in understanding and addressing some of the key questions that arise in nature and society from the viewpoint of theoretical computer science. Here, his current focus is on natural algorithms, emergence of intelligence, and questions at the interface of AI, ethics, and society. He was the recipient of the Best Paper Award at FOCS in 2005, the IBM Research Pat Goldberg Memorial Award in 2006, the Indian National Science Academy Young Scientist Award in 2011, and the IIT Bombay Young Alumni Achievers Award in 2016.
In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary.
GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.
In the design of complex engineering systems like aircraft/rotorcraft/spacecraft, computer experiments offer a cheaper alternative to physical experiments due to high-fidelity(HF) models. However, such models are still not cheap enough for application to Global Optimization(GO) and Uncertainty Quantification(UQ) to find the best possible design alternative. In such cases, surrogate models of HF models become necessary. The construction of surrogate models requires an offline database of the system response generated by running the expensive model several times. In general, the training sample size and distribution for a given problem is unknown apriori and can be over/under predicted, which leads to wastage of resources and poor decision-making. An adaptive model building approach eliminates this problem by sequentially sampling points based on information gained in the previous step. However, an approach that works for highly non-stationary response is still lacking in the literature. Here, we use Gaussian Process(GP) models as surrogate model. We employ a novel process-convolution approach to generate parameterized non-stationary
GPs that offer control on the process smoothness. We show that our approach outperforms existing methods, particularly for responses that have localized non-smoothness. This leads to better performance in terms of GO, UQ and mean-squared-prediction-errors for a given budget of HF function calls.
We study delocalization properties of null vectors and eigenvectors of matrices with i.i.d. subgaussian entries. Such properties describe quantitatively how "flat" is a vector and confirm one of the universality conjectures stating that distributions of eigenvectors of many classes of random matrices are close to the uniform distribution on the unit sphere. In particular, we get lower bounds on the smallest coordinates of eigenvectors, which are optimal as the case of Gaussian matrices shows. The talk is based on the joint work with Konstantin Tikhomirov.
Strong edge coloring of a graph $G$ is a coloring of the edges of the graph such that each color class is an induced subgraph. The strong chromatic index of $G$ is the smallest number $k$ such that $G$ has a $k$-strong edge coloring. Erdős and Nešetřil conjecture that the strong chromatic index of a graph of max degree $\Delta$ is at most $5\Delta^2/4$ if $\Delta$ is even and $(5\Delta^2-2\Delta + 1)/4$ if $\Delta$ is odd. It is known for $\Delta=3$ that the conjecture holds, and in this talk I will present part of Anderson's proof that the strong chromatic index of a subcubic planar graph is at most $10$
A major challenge in clinical and biomedical research is on translating in-vitro and in- vivo model findings to humans. Translation success rate of all new compounds going through different clinical trial phases is generally about 10%. (i) This field is challenged by a lack of robust methods that can be used to translate model findings to humans (or interpret preclinical finds to accurately design successful patient regimens), hence providing a platform to evaluate a plethora of agents before they are channeled in clinical trials. Using set theory principles of mapping morphisms, we recently developed a novel translational framework that can faithfully map experimental results to clinical patient results. This talk will demonstrate how this method was used to predict outcomes of anti-TB drug clinical trials. (ii) Translation failure is deeply rooted in the dissimilarities between humans and experimental models used; wide pathogen isolates variation, patient population genetic diversities and geographic heterogeneities. In TB, bacteria phenotypic heterogeneity shapes differential antibiotic susceptibility patterns in patients. This talk will also demonstrate the application of dynamical systems in Systems Biology to model (a) gene regulatory networks and how gene programs influence Mycobacterium tuberculosis bacteria metabolic/phenotypic plasticity. (b) And then illustrate how different bacteria phenotypic subpopulations influence treatment outcomes and the translation of preclinical TB therapeutic regimens. In general, this talk will strongly showcase how mathematical modeling can be used to critically analyze experimental and patient data.
We will go to lunch together after the talk with the graduate students.
The popularity of machine learning is increasingly growing in transportation, with applications ranging from traffic engineering to travel demand forecasting and pavement material modeling, to name just a few. Researchers often find that machine learning achieves higher predictive accuracy compared to traditional methods. However, many machine-learning methods are often viewed as “black-box” models, lacking interpretability for decision making. As a result, increased attention is being devoted to the interpretability of machine-learning results.
In this talk, I introduce the application of machine learning to study travel behavior, covering both mode prediction and behavioral interpretation. I first discuss the key differences between machine learning and logit models in modeling travel mode choice, focusing on model development, evaluation, and interpretation. Next, I apply the existing machine-learning interpretation tools and also propose two new model-agnostic interpretation tools to examine behavioral heterogeneity. Lastly, I show the potential of using machine learning as an exploratory tool to tune the utility functions of logit models.
I illustrate these ideas by examining stated-preference travel survey data for a new mobility-on-demand transit system that integrates fixed-route buses and on-demand shuttles. The results show that the best-performing machine-learning classifier results in higher predictive accuracy than logit models as well as comparable behavioral outputs. In addition, results obtained from model-agnostic interpretation tools show that certain machine-learning models (e.g. boosting trees) can readily account for individual heterogeneity and generate valuable behavioral insights on different population segments. Moreover, I show that interpretable machine learning can be applied to tune the utility functions of logit models (e.g. specifying nonlinearities) and to enhance their model performance. In turn, these findings can be used to inform the design of new mobility services and transportation policies.
Abstract: Reiher, Rödl, Ruciński, Schacht, and Szemerédi proved, via a modification of the absorbing method, that every 3-uniform $n$-vertex hypergraph, $n$ large, with minimum vertex degree at least $(5/9+\alpha)n^2/2$ contains a tight Hamiltonian cycle. Recently, owing to a further modification of the method, the same group of authors joined by Bjarne Schuelke, extended this result to 4-uniform hypergraphs with minimum pair degree at least, again, $(5/9+\alpha)n^2/2$. In my talk I will outline these proofs and point to the crucial ideas behind both modifications of the absorbing method.
Abstract: In this talk, we consider the Cauchy problem of the N-dimensional incompressible viscoelastic fluids with N ≥ 2. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group e∓itΛ. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations. The proof relies heavily on the dispersive estimates for the system of acoustics, and a careful study of the nonlinear terms. And we also obtain the similar result for the isentropic compressible Navier-Stokes equations. Here, the initial velocity with arbitrary B⋅N 2 −1 2,1 norm of potential part P⊥u0 and large highly oscillating are allowed in our results. (Joint works with Daoyuan Fang and Ruizhao Zi)
Moment problem is a classical question in real analysis, which asks whether a set of moments can be realized as integration of corresponding monomials with respect to a Borel measure. Truncated moment problem asks the same question given a finite set of moments. I will explain how some of the fundamental results in the truncated moment problem can be proved (in a very general setting) using elementary convex geometry. No familiarity with moment problems will be assumed. This is joint work with Larry Fialkow.
The Georgia Scientific Computing Symposium is a forum for professors, postdocs, graduate students and other researchers in Georgia to meet in an informal setting, to exchange ideas, and to highlight local scientific computing research. The symposium has been held every year since 2009 and is open to the entire research community.
This year, the symposium will be held on Saturday, February 16, 2019, at Georgia Institute of Technology. Please see
http://gtmap.gatech.edu/events/2019-georgia-scientific-computing-symposium
for more information
We will discuss the regularity of the conjugacy between an Anosov automorphism L of a torus and its small perturbation. We assume that L has no more than two eigenvalues of the same modulus and that L^4 is irreducible over rationals. We consider a volume-preserving C^1-small perturbation f of L. We show that if the Lyapunov exponents of f with respect to the volume are the same as the Lyapunov exponents of L, then f is C^1+ conjugate to L. Further, we establish a similar result for irreducible partially hyperbolic automorphisms with two-dimensional center bundle. This is joint work with Andrey Gogolev and Victoria Sadovskaya
A single soap bubble has a spherical shape since it minimizes its surface area subject to a fixed enclosed volume of air. When two soap bubbles collide, they form a “double-bubble” composed of three spherical caps. The double-bubble minimizes total surface area among all sets enclosing two fixed volumes. This was proven mathematically in a landmark result by Hutchings-Morgan-Ritore-Ros and Reichardt using the calculus of variations in the early 2000s. The analogous case of three or more Euclidean sets is considered difficult if not impossible. However, if we replace Lebesgue measure in these problems with the Gaussian measure, then recent work of myself (for 3 sets) and of Milman-Neeman (for any number of sets) can actually solve these problems. We also use the calculus of variations. Time permitting, we will discuss an improvement to the Milman-Neeman result and applications to optimal clustering of data and to designing elections that are resilient to hacking. http://arxiv.org/abs/1901.03934
Wiener-Hopf factorization (WHf) encompasses several important results in probability and stochastic processes, as well as in operator theory. The importance of the WHf stems not only from its theoretical appeal, manifested, in part, through probabilistic interpretation of analytical results, but also from its practical applications in a wide range of fields, such as fluctuation theory, insurance and finance. The various existing forms of the WHf for Markov chains, strong Markov processes, Levy processes, and Markov additive process, have been obtained only in the time-homogeneous case. However, there are abundant real life dynamical systems that are modeled in terms of time-inhomogenous processes, and yet the corresponding Wiener-Hopf factorization theory is not available for this important class of models. In this talk, I will first provide a survey on the development of Wiener-Hopf factorization for time-homogeneous Markov chains, Levy processes, and Markov additive processes. Then, I will discuss our recent work on WHf for time-inhomogensous Markov chains. To the best of our knowledge, this study is the first attempt to investigate the WHf for time-inhomogeneous Markov processes.
Inference of evolutionary dynamics of heterogeneous cancer and viral populations Abstract: Genetic diversity of cancer cell populations and intra-host viral populations is one of the major factors influencing disease progression and treatment outcome. However, evolutionary dynamics of such populations remain poorly understood. Quantification of selection is a key step to understanding evolutionary mechanisms driving cancer and viral diseases. We will introduce a mathematical model and an algorithmic framework for inference of fitness landscapes of heterogeneous populations from genomic data. It is based on a maximal likelihood approach, whose objective is to estimate a vector of clone/strain fitnesses which better fits the observed tumor phylogeny, observed population structure and the dynamical system describing evolution of the population as a branching process. We will discuss our approach to solve the problem by transforming the original continuous maximum likelihood problem into a discrete optimization problem, which could be considered as a variant of scheduling problem with precedent constraints and with non-linear cumulative cost function.
Linear Schur multipliers, which act on matrices by entrywisemultiplications, as well as their generalizations have been studiedfor over a century and successfully applied in perturbation theory (asdemonstrated in the previous talk). In this talk, we will discussestimates for finite dimensional multilinear Schur multipliersunderlying these applications.
Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Minimum s-t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization.
Here, we are given a graph with edges labeled + or - and the goal is to produce a clustering that agrees with the labels as much as possible: + edges within clusters and - edges across clusters.
The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective, e.g., minimizing the total number of disagreements or maximizing the total number of agreements.
We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective.
This naturally gives rise to a family of basic min-max graph cut problems.
A prototypical representative is Min-Max s-t Cut: find an s-t cut minimizing the largest number of cut edges incident on any node.
In this talk we will give a short introduction of Correlation Clustering and discuss the following results:
Joint work with Moses Charikar and Neha Gupta.
First, we introduce a new field theoretical interpretation of quantum mechanical wave functions, by postulating that the wave function is the common wave function for all particles in the same class determined by the external potential V, of the modulus of the wave function represents the distribution density of the particles, and the gradient of phase of the wave function provides the velocity field of the particles. Second, we show that the key for condensation of bosonic particles is that their interaction is sufficiently weak to ensure that a large collection of boson particles are in a state governed by the same condensation wave function field under the same bounding potential V. For superconductivity, the formation of superconductivity comes down to conditions for the formation of electron-pairs, and for the electron-pairs to share a common wave function. Thanks to the recently developed PID interaction potential of electrons and the average-energy level formula of temperature, these conditions for superconductivity are explicitly derived. Furthermore, we obtain both microscopic and macroscopic formulas for the critical temperature. Third, we derive the field and topological phase transition equations for condensates, and make connections to the quantum phase transition, as a topological phase transition. This is joint work with Tian Ma.
We identify principal component analysis (PCA) as an empirical risk minimization problem with respect to the reconstruction error and prove non-asymptotic upper bounds for the corresponding excess risk. These bounds unify and improve existing upper bounds from the literature. In particular, they give oracle inequalities under mild eigenvalue conditions. We also discuss how our results can be transferred to the subspace distance and, for instance, how our approach leads to a sharp $\sin \Theta$ theorem for empirical covariance operators. The proof is based on a novel contraction property, contrasting previous spectral perturbation approaches. This talk is based on joint works with Markus Reiß and Moritz Jirak.
If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of f? Stein conjectured in the 1960s that for any $p>3$, $\|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}$.
We make a small progress toward this conjecture and show that it holds for $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao.
Let $\nu$ denote the maximum size of a packing of edge-disjoint triangles in a graph $G$. We can clearly make $G$ triangle-free by deleting $3\nu$ edges. Tuza conjectured in 1981 that $2\nu$ edges suffice, and proved it for planar graphs. The best known general bound is $(3-\frac{3}{23})\nu$ proven by Haxell in 1997. We will discuss this proof and some related results.
Two recent extensions of optimal mass transport theory will be covered. In the first part of the talk, we will discuss measure-valued spline, which generalizes the notion of cubic spline to the space of distributions. It addresses the problem to smoothly interpolate (empirical) probability measures. Potential applications include time sequence interpolation or regression of images, histograms or aggregated datas. In the second part of the talk, we will introduce matrix-valued optimal transport. It extends the optimal transport theory to handle matrix-valued densities. Several instances are quantum states, color images, diffusion tensor images and multi-variate power spectra. The new tool is expected to have applications in these domains. We will focus on theoretical side of the stories in both parts of the talk.
Mathapalooza! is simultaneously a Julia Robinson Mathematics Festival and an event of the Atlanta Science Festival. There will be puzzles and games, a magic show by Matt Baker, mathematically themed courtroom skits by GT Club Math, a presentation about math and dance by Manuela Manetta, a presentation about math and music by David Borthwick, and a gallery of mathematical art curated by Elisabetta Matsumoto. It is free, and we anticipate engaging hundreds of members of the public in the wonders of mathematics. More info at https://mathematics-in-motion.org/about/Be there or B^2 !
Let X be a degree d curve in the projective space P^r.
A general hyperplane H intersects X at d distinct points; varying H defines a monodromy action on X∩H. The resulting permutation group G is the sectional monodromy group of X. When the ground field has characteristic zero the group G is known to be the full symmetric group.
By work of Harris, if G contains the alternating group, then X satisfies a strengthened Castelnuovo's inequality (relating the degree and the genus of X).
The talk is concerned with sectional monodromy groups in positive characteristic. I will describe all non-strange non-degenerate curves in projective spaces of dimension r>2 for which G is not symmetric or alternating. For a particular family of plane curves, I will compute the sectional monodromy groups and thus answer an old question on Galois groups of generic trinomials.
We will try to address a few universality questions for the behavior of large random matrices over finite fields, and then present a minimal progress on one of these questions.
Hadwiger (Hajos and Gerards and Seymour, respectively) conjectured that the vertices of every graph with no K_{t+1} minor (topological minor and odd minor, respectively) can be colored with t colors such that any pair of adjacent vertices receive different colors. These conjectures are stronger than the Four Color Theorem and are either wide open or false in general. A weakening of these conjectures is to consider clustered coloring which only requires every monochromatic component to have bounded size instead of size 1. It is known that t colors are still necessary for the clustered coloring version of those three conjectures. Joint with David Wood, we prove a series of tight results about clustered coloring on graphs with no subgraph isomorphic to a fixed complete bipartite graph. These results have a number of applications. In particular, they imply that the clustered coloring version of Hajos' conjecture is true for bounded treewidth graphs in a stronger sense: K_{t+1} topological minor free graphs of bounded treewidth are clustered t-list-colorable. They also lead to the first linear upper bound for the clustered coloring version of Hajos' conjecture and the currently best upper bound for the clustered coloring version of the Gerards-Seymour conjecture.
If a finite group $G$ acts on a Cohen-Macaulay ring $A$, and the order of $G$ is a unit in $A$, then the invariant ring $A^G$ is Cohen-Macaulay as well, by the Hochster-Eagon theorem. On the other hand, if the order of $G$ is not a unit in $A$ then the Cohen-Macaulayness of $A^G$ is a delicate question that has attracted research attention over the last several decades, with answers in several special cases but little general theory. In this talk we show that the statement that $A^G$ is Cohen-Macaulay is equivalent to a statement quantified over the inertia groups for the action of G$ on $A$ acting on strict henselizations of appropriate localizations of $A$. In a case of long-standing interest—a permutation group acting on a polynomial ring—we show how this can be applied to find an obstruction to Cohen-Macaulayness that allows us to completely characterize the permutation groups whose invariant ring is Cohen-Macaulay regardless of the ground field. This is joint work with Sophie Marques.
Many problems of spherical discrete and metric geometry may be reformulated as energy minimization problems and require techniques that stem from harmonic analysis, potential theory, optimization etc. We shall discuss several such problems as well of applications of these ideas to combinatorial geometry, discrepancy theory, signal processing etc.
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One of the most famous methods for solving large-scale over-determined linear systems is Kaczmarz algorithm, which iteratively projects the previous approximation x_k onto the solution spaces of the next equation in the system. An elegant proof of the exponential convergence of this method using correct randomization of the process is due to Strohmer and Vershynin (2009). Many extensions and generalizations of the method were proposed since then, including the works of Needell, Tropp, Ward, Srebro, Tan and many others. An interesting unifying view on a number of iterative solvers (including several versions of the Kaczmarz algorithm) was proposed by Gower and Richtarik in 2016. The main idea of their sketch-and-project framework is the following: one can observe that the random selection of a row (or a row block) can be represented as a sketch, that is, left multiplication by a random vector (or a matrix), thereby pre-processing every iteration of the method, which is represented by a projection onto the image of the sketch.
I will give an overview of some of these methods, and talk about the role that random matrix theory plays in the showing their convergence. I will also discuss our new results with Deanna Needell on the block Gaussian sketch and project method.
One of the most famous methods for solving large-scale over-determined linear systems is Kaczmarz algorithm, which iteratively projects the previous approximation x_k onto the solution spaces of the next equation in the system. An elegant proof of the exponential convergence of this method using correct randomization of the process is due to Strohmer and Vershynin (2009). Many extensions and generalizations of the method were proposed since then, including the works of Needell, Tropp, Ward, Srebro, Tan and many others. An interesting unifying view on a number of iterative solvers (including several versions of the Kaczmarz algorithm) was proposed by Gower and Richtarik in 2016. The main idea of their sketch-and-project framework is the following: one can observe that the random selection of a row (or a row block) can be represented as a sketch, that is, left multiplication by a random vector (or a matrix), thereby pre-processing every iteration of the method, which is represented by a projection onto the image of the sketch.
I will give an overview of some of these methods, and talk about the role that random matrix theory plays in the showing their convergence. I will also discuss our new results with Deanna Needell on the block Gaussian sketch and project method.
I will talk about the structure of large square random matrices with centered i.i.d. heavy-tailed entries (only two finite moments are assumed). In our previous work with R. Vershynin we have shown that the operator norm of such matrix A can be reduced to the optimal sqrt(n)-order with high probability by zeroing out a small submatrix of A, but did not describe the structure of this "bad" submatrix, nor provide a constructive way to find it. Now we can give a very simple description of this small "bad" subset: it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms to bring its operator norm to the almost optimal sqrt(loglog(n)*n)-order, under additional assumption that the entries of A are symmetrically distributed. As a corollary, one can also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same regularization.
Im am planning to discuss some details of the proof, the main component of which is the development of techniques that extend constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.
Consider a uniformly chosen proper coloring with q colors of a domain in Z^d (a graph homomorphism to a clique). We show that when the dimension is much higher than the number of colors, the model admits a staggered long-range order, in which one bipartite class of the domain is predominantly colored by half of the q colors and the other bipartite class by the other half. In the q=3 case, this was previously shown by Galvin-Kahn-Randall-Sorkin and independently by Peled. The result further extends to homomorphisms to other graphs (covering for instance the cases of the hard-core model and the Widom-Rowlinson model), allowing also vertex and edge weights (positive temperature models). Joint work with Ron Peled.
This is a two day conference (March 30-31) to be held at Georgia Tech on algebraic geometry and related areas. We will have talks by Sam Payne, Eric Larson, Angelica Cueto, Rohini Ramadas, and Jennifer Balakrishnan. See https://sites.google.com/view/gattaca/home for more information.
One of the characteristics observed in real networks is that, as a network's topology evolves so does the network's ability to perform various complex tasks. To explain this, it has also been observed that as a network grows certain subnetworks begin to specialize the function(s) they perform. We introduce a model of network growth based on this notion of specialization and show that as a network is specialized its topology becomes increasingly modular, hierarchical, and sparser, each of which are properties observed in real networks. This model is also highly flexible in that a network can be specialized over any subset of its components. By selecting these components in various ways we find that a network's topology acquires some of the most well-known properties of real networks including the small-world property, disassortativity, power-law like degree distributions and clustering coefficients. This growth model also maintains the basic spectral properties of a network, i.e. the eigenvalues and eigenvectors associated with the network's adjacency network. This allows us in turn to show that a network maintains certain dynamic properties as the network's topology becomes increasingly complex due to specialization.
A link in the 3-sphere is doubly slice if it is the cross-section of an unknotted 2-sphere in the 4-sphere. The double branched cover of a doubly slice link is a 3-manifold which embeds in the 4-sphere. For doubly slice Montesinos links, this produces embeddings of Seifert fibered spaces in S^4. In this pre-talk, I'll discuss a construction and an obstruction to being doubly slice.
The classical statement that there are 27 lines on every smooth cubic surface in $\mathbb{P}^3$ fails to hold under tropicalization: a tropical cubic surface in $\mathbb{TP}^3$ often contains infinitely many tropical lines. This pathology can be corrected by reembedding the cubic surface in $\mathbb{P}^{44}$ via the anticanonical bundle.
Under this tropicalization, the 27 classical lines become an arrangement of metric trees in the boundary of the tropical cubic surface in $\mathbb{TP}^{44}$. A remarkable fact is that this arrangement completely determines the combinatorial structure of the corresponding tropical cubic surface. In this talk, we will describe their metric and topological type as we move along the moduli space of tropical cubic surfaces. Time permitting, we will discuss the matroid that emerges from their tropical convex hull.
This is joint work with Anand Deopurkar.
Which 3-manifolds smoothly embed in the 4-sphere? This seemingly simple question turns out to be rather subtle. Using Donaldson's theorem, we derive strong restrictions to embedding a Seifert fibered space over an orientable base surface, which in particular gives a complete classification when e > k/2, where k is the number of exceptional fibers and e is the normalized central weight. Our results point towards a couple of interesting conjectures which I'll discuss. This is joint work with Duncan McCoy.
In a joint work with Sameer Iyer, the validity of steady Prandtl layer expansion is established in a channel. Our result covers the celebrated Blasius boundary layer profile, which is based on uniform quotient estimates for the derivative Navier-Stokes equations, as well as a positivity estimate at the flow entrance.
It is anticipated that the invariant statistics of many of smooth dynamical systems with a `chaotic’ asymptotic character are given by invariant measures with the SRB property- a geometric property of invariant measures which, roughly, means that the invariant measure is smooth along unstable directions. However, actually verifying the existence of SRB measures for concrete systems is extremely challenging: indeed, SRB measures need not exist, even for systems exhibiting asymptotic hyperbolicity (e.g., the figure eight attractor).
The study of asymptotic properties for dynamical systems in the presence of noise is considerably simpler. One manifestation of this principle is the theorem of Ledrappier and Young ’89, where it was proved that under very mild conditions, stationary measures for a random dynamical system with a positive Lyapunov exponent are automatically random SRB measures (that is, satisfy the random analogue of the SRB property). I will talk today about a new proof of this result in a joint work with Lai-Sang Young. This new proof has the benefit of being (1) conceptually lucid and to-the-point (the original proof is somewhat indirect) and (2) potentially easily adapted to more general settings, e.g., to appropriate infinite-dimensional random dynamics, such as time-t solutions to certain classes SPDE (this generalization is an ongoing work, joint with LSY).
Mathematical billiards naturally arise in mechanics, optics, acoustics, etc. They also form the most visual class of dynamical systems with evolution covering all the possible spectrum of behaviours from integrable (extremely regular) to strongly chaotic. Billiard is a (deterministic) dynamical system generated by an uniform (by inertia) motion of a point particle within a domain with piecewise smooth walls ("a billiard table"). I will introduce all needed notions on simple examples and outline some open problems. This talk is also a preparatory talk to a Mathematical Physics seminar (on Monday April 8) where a new direction of research will be discussed which consider physical billiards where instead of a point (mathematical) particle a real physical hard sphere moves. To a complete surprise of mathematicians and PHYSICISTS evolution of a billiard may completely change (and in different ways) in transition from mathematical to physical billiards. It a rare example when mathematicians surprise physicists. Some striking results with physicists are also already obtained. I will (again visually) explain at the end of RH why it is surprising that there could be difference between Math and Phys billiards.
In the setup of classical knot theory---the study of embeddings of the circle into S^3---we recall two examples of classical knot invariants: the Alexander polynomial and the Seifert form.
We then introduce notions from knot-concordance theory, which is concerned with the study of slice surfaces of a knot K---surfaces embedded in the 4-ball B^4 with boundary the knot K. We will comment on the difference between the smooth and topological theory with a focus on a surprising feature of the topological theory: classical invariants govern the existence of slice surfaces of low genus in a way that is not the case in the smooth theory. This can be understood as an analogue of a dichotomy in the study of smooth and topological 4-manifolds.
In this talk we will discuss some some extremal problems for polynomials. Applications to the problems in discrete dynamical systems as well as in the geometric complex analysis will be suggested.
Following an idea of Hugelmeyer, we give a knot theory reproof of a theorem of Schnirelman: Every smooth Jordan curve in the Euclidian plane has an inscribed square. We will comment on possible generalizations to more general Jordan curves.
Our main knot theory result is that the torus knot T(2n,1) in S^1xS^2 does not arise as the boundary of a locally-flat Moebius band in S^1xB^3 for square-free integers n>1. For context, we note that for n>2 and the smooth setting, this result follows from a result of Batson about the non-orientable 4-genus of certain torus knots. However, we show that Batson's result does not hold in the locally flat category: the smooth and topological non-orientable 4-genus differ for the T(9,10) torus knot in S^3.
Based on joint work with Marco Golla.
Computing the eigenvalues and eigenvectors of a large matrix is a basic task in high dimensional data analysis with many applications in computer science and statistics. In practice, however, data is often perturbed by noise. A natural question is the following: How much does a small perturbation to the matrix change the eigenvalues and eigenvectors? In this talk, I will consider the case where the perturbation is random. I will discuss perturbation results for the eigenvalues and eigenvectors as well as for the singular values and singular vectors. This talk is based on joint work with Van Vu, Ke Wang, and Philip Matchett Wood.
In an optimal design problem, we are given a set of linear experiments v1,...,vn \in R^d and k >= d, and our goal is to select a set or a multiset S subseteq [n] of size k such that Phi((\sum_{i \in [n]} v_i v_i^T )^{-1}) is minimized. When Phi(M) = det(M)^{1/d}, the problem is known as the D-optimal design problem, and when Phi(M) = tr(M), it is known as the A-optimal design problem. One of the most common heuristics used in practice to solve these problems is the local search heuristic, also known as the Fedorov's exchange method. This is due to its simplicity and its empirical performance. However, despite its wide usage no theoretical bound has been proven for this algorithm. In this paper, we bridge this gap and prove approximation guarantees for the local search algorithms for D-optimal design and A-optimal design problems. We show that the local search algorithms are asymptotically optimal when $\frac{k}{d}$ is large. In addition to this, we also prove similar approximation guarantees for the greedy algorithms for D-optimal design and A-optimal design problems when k/d is large.
In this talk I will discuss a particular fast-slow system, and describe an averaging theorem. I will also explain how this particular slow-fast system arises in a certain problem of energy transport in an open system of interacting hard-spheres. The technical aspect involved in this is how to deal with singularities present and the fact that the dynamics is fully coupled.
Unusual time.
In standard (mathematical) billiards a point particle moves uniformly in a billiard table with elastic reflections off the boundary. We show that in transition from mathematical billiards to physical billiards, where a finite size hard sphere moves in the same billiard table, virtually anything may happen. Namely a non-chaotic billiard may become chaotic and vice versa. Moreover, both these transitions may occur softly, i.e. for any (arbitrarily small) positive value of the radius of a physical particle, as well as by a ”hard” transition when radius of the physical particle must exceed some critical strictly positive value. Such transitions may change a phase portrait of a mathematical billiard locally as well as completely (globally). These results are somewhat unexpected because for all standard examples of billiards their dynamics remains absolutely the same after transition from a point particle to a finite size (”physical”) particle. Moreover we show that a character of dynamics may change several times when the size of the particle is increasing. This approach already demonstrated a sensational result that quantum system could be more chaotic than its classical counterpart.
We are going talk about three topics. First of all, Principal Components Analysis (PCA) as a dimension reduction technique. We investigate how useful it is for real life problems. The problem is that, often times the spectrum of the covariance matrix is wrongly estimated due to the ratio between sample space dimension over feature space dimension not being large enough. We show how to reconstruct the spectrum of the ground truth covariance matrix, given the spectrum of the estimated covariance for multivariate normal vectors. We then present an algorithm for reconstruction the spectrum in the case of sparse matrices related to text classification.
In the second part, we concentrate on schemes of PCA estimators. Consider the problem of finding the least eigenvalue and eigenvector of ground truth covariance matrix, a famous classical estimator are due to Krasulina. We state the convergence proof of Krasulina for the least eigenvalue and corresponding eigenvector, and then find their convergence rate.
In the last part, we consider the application problem, text classification, in the supervised view with traditional Naive-Bayes method. We find out an updated Naive-Bayes method with a new loss function, which loses the unbiased property of traditional Naive-Bayes method, but obtains a smaller variance of the estimator.
Committee: Heinrich Matzinger (Advisor); Karim Lounici (Advisor); Ionel Popescu (school of math); Federico Bonetto (school of math); Xiaoming Huo (school of ISYE);
In this talk, we will study Seifert fibered three-manifolds. While simple to define, they comprise 6 of the 8 Thurston geometries, and are an important testing ground for many questions and invariants. We will present several constructions/definitions of these manifolds and learn how to work with them explicitly.
In this talk we discuss the following problem due to Peskine and Kollar: Let E be a flat family of rank two bundles on P^n parametrized by a smooth variety T. If E_t is isomorphic to O(a)+O(b) for general t in T, does it mean E_0 is isomorphic to O(a)+O(b) for a special point 0 in T? We construct counter-examples in over P^1 and P^2, and discuss the problem in P^3 and higher P^n.
Inference (aka predictive modeling) is in the core of many data science problems. Traditional approaches could be either statistically or computationally efficient, however not necessarily both. The existing principles in deriving these models - such as the maximal likelihood estimation principle - may have been developed decades ago, and do not take into account the new aspects of the data, such as their large volume, variety, velocity and veracity. On the other hand, many existing empirical algorithms are doing extremely well in a wide spectrum of applications, such as the deep learning framework; however they do not have the theoretical guarantee like these classical methods. We aim to develop new algorithms that are both computationally efficient and statistically optimal. Such a work is fundamental in nature, however will have significant impacts in all data science problems that one may encounter in the society. Following the aforementioned spirit, I will describe a set of my past and current projects including L1-based relaxation, fast nonlinear correlation, optimality of detectability, and nonconvex regularization. All of them integrates statistical and computational considerations to develop data analysis tools.
We will use Heegaard Floer homology to analyze maps between a certain family of three-manifolds akin to the Gromov norm/hyperbolic volume. Along the way, we will study the Heegaard Floer homology of splices. This is joint work with Cagri Karakurt and Eamonn Tweedy.
Unusual time.
Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times for every two revolutions it makes around the Sun. It is generally accepted that this is due to the large value of Mercury's eccentricity. However, the mathematical model commonly used to study the problem -- sometimes called the spin-orbit model -- proved not to be entirely convincing, because of the expression used for the tidal torque. Only recently, a different model for the tidal torque has been proposed, with the advantage of both being more realistic and providing a higher probability of capture into the 3:2 resonance with respect to the previous models. On the other hand, a drawback of the model is that the function describing the tidal torque is not smooth and appears as a superposition of peaks, so that both analytical and numerical computations turn out to be rather delicate. We shall present numerical and analytical results about the nature of the librations of Mercury's spin in the 3:2 resonance, as predicted by the realistic model. In particular we shall provide evidence that the librations are quasi-periodic in time, so that the very concept of resonance should be revisited. The analytical results are mainly based on perturbation theory and leave several open problems, that we shall discuss.
We define the notion of a knot type having Legendrian large cables and
show that having this property implies that the knot type is not uniformly thick.
Moreover, there are solid tori in this knot type that do not thicken to a solid torus
with integer sloped boundary torus, and that exhibit new phenomena; specifically,
they have virtually overtwisted contact structures. We then show that there exists
an infinite family of ribbon knots that have Legendrian large cables. These knots fail
to be uniformly thick in several ways not previously seen. We also give a general
construction of ribbon knots, and show when they give similar such examples.
When equiangular tight frames (ETF's), a type of structured optimal packing of lines, exist and are of size $|\Phi|=N$, $\Phi\subset\mathbb{F}^d$ (where $\mathbb{F}=\mathbb{R}$, $\mathbb{C}$, or $\mathbb{H}$), for $p > 2$ the so-called $p$-frame energy $E_p(\Phi)=\sum\limits_{i\neq j} |\langle \varphi_{i}, \varphi_{j} \rangle|^p$ achieves its minimum value on an ETF over all sized $N$ collections of unit vectors. These energies have potential functions which are not positive definite when $p$ is not even. For these cases the apparent complexity of the problem of describing minimizers of these energies presents itself. While there are several open questions about the structure of these sets for fixed $N$ and fixed $p$, we focus on another question:
What structural properties are expressed by minimizing probability measures for the quantity $I_{p}(\mu)=\int\limits_{\mathbb{S}_{\mathbb{F}}^{d-1}}\int\limits_{\mathbb{S}_{\mathbb{F}}^{d-1}} |\langle x, y \rangle|^p d\mu(x) d\mu(y)$?
We collect a number of surprising observations. Whenever a tight spherical or projective $t$-design exists for the sphere $\mathbb{S}_{\mathbb{F}}^d$, equally distributing mass over it gives a minimizer of the quantity $I_{p}$ for a range of $p$ between consecutive even integers associated with the strength $t$. We show existence of discrete minimizers for several related potential functions, along with conditions which guarantee emptiness of the interior of the support of minimizers for these energies.
This talk is based on joint work with D. Bilyk, A. Glazyrin, R. Matzke, and O. Vlasiuk.
Casson invariant is defined for the class of oriented integral homology 3-spheres. It satisfies certain properties, and reduce to Rohlin invariant after mod 2. We will define Casson invariant as half of the algebraic intersection number of irreducible representation spaces (space consists of representations of fundamental group to SU(2)), and then prove this definition satisfies the expected properties.
We discuss a general approach to a problem of estimation of a smooth function $f(\theta)$ of a high-dimensional parameter $\theta$<br />
of statistical models. In particular, in the case of $n$ i.i.d. Gaussian observations $X_1,\doot, X_n$ with mean $\mu$ and covariance <br />
matrix $\Sigma,$ the unknown parameter is $\theta = (\mu, \Sigma)$ and our approach yields an estimator of $f(\theta)$ <br />
for a function $f$ of smoothness $s>0$ with mean squared error of the order $(\frac{1}{n} \vee (\frac{d}{n})^s) \wedge 1$ <br />
(provided that the Euclidean norm of $\mu$ and operator norms of $\Sigma,\Sigma^{-1}$ are uniformly bounded),<br />
with the error rate being minimax optimal up to a log factor (joint result with Mayya Zhilova). The construction of optimal estimators <br />
crucially relies on a new bias reduction method in high-dimensional problems<br />
and the bounds on the mean squared error are based on controlling finite differences of smooth functions along certain Markov chains<br />
in high-dimensional parameter spaces as well as on concentration inequalities.
In a fractional coloring, vertices of a graph are assigned subsets of the [0, 1]-interval such that adjacent vertices receive disjoint subsets. The fractional chromatic number of a graph is at most k if it admits a fractional coloring in which the amount of "color" assigned to each vertex is at least 1/k. We investigate fractional colorings where vertices "demand" different amounts of color, determined by local parameters such as the degree of a vertex. Many well-known results concerning the fractional chromatic number and independence number have natural generalizations in this new paradigm. We discuss several such results as well as open problems. In particular, we will sketch a proof of a "local demands" version of Brooks' Theorem that considerably generalizes the Caro-Wei Theorem and implies new bounds on the independence number. Joint work with Luke Postle.
Milnor K-theory is a field invariant that originated as an attempt to study algebraic K-theory. Instead, Milnor K-theory has proved to have many other applications, including Galois cohomology computations, Voevodsky's proof of the Bloch-Kato conjecture, and Kato's higher class field theory. In this talk, we will go over the basic definitions and theorems of Milnor K-theory. We will also discuss some of these applications.
Stability is a multivariate generalization for real-rootedness in univariate polynomials. Within the past ten years, the theory of stable polynomials has contributed to breakthroughs in combinatorics, convex optimization, and operator theory. I will introduce a generalization of stability, called complete log-concavity, that satisfies many of the same desirable properties. These polynomials were inspired by work of Adiprasito, Huh, and Katz on combinatorial Hodge theory, but can be defined and understood in elementary terms. The structure of these polynomials is closely tied with notions of discrete convexity, including matroids, submodular functions, and generalized permutohedra. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and applications to matroid theory, including a proof of Mason’s conjecture on numbers of independent sets. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.
(*Refreshments available at 2:30pm before the colloquium.*)
In this talk we will follow the paper titled "Aubry-Mather theory for homeomorphisms", in which it is developed a variational approach to study the dynamics of a homeomorphism on a compact metric space. In particular, they are described orbits along which any Lipschitz Lyapunov function has to be constant via a non-negative Lipschitz semidistance. This is work of Albert Fathi and Pierre Pageault.
Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T-periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement. The stability conditions are developed in non-dimensional parameter space through two basic nonlinear maps based on switching manifolds that correspond to impacts with the top and bottom membranes of the energy harvesting device. The range for stable simple T-periodic behavior is reduced with increasing angle of incline β, since the influence of gravity increases the asymmetry of dynamics following impacts at the bottom and top. These asymmetric T-periodic solutions lose stability to period doubling solutions for β ≥ 0, which appear through increased asymmetry. The period doubling, symmetric and asymmetric periodic motion are illustrated by bifurcation diagrams, phase portraits and velocity time series.
I will give a brief survey of concordance in high-dimensional knot theory and how slice results have classically been obtained in this setting with the aid of surgery theory. Time permitting, I will then discuss an example of how some non-abelian slice obstructions come into the picture for 1-knots, as intuition for the seminar talk about L^2 invariants.
Tropical geometry provides a new set of purely combinatorial tools, which has been used to approach classical problems. In tropical geometry most algebraic computations are done on the classical side - using the algebra of the original variety. The theory developed so far has explored the geometric aspect of tropical varieties as opposed to the underlying (semiring) algebra and there are still many commutative algebra tools and notions without a tropical analogue. In the recent years, there has been a lot of effort dedicated to developing the necessary tools for commutative algebra using different frameworks, among which prime congruences, tropical ideals, tropical schemes. These approaches allows for the exploration of the properties of tropicalized spaces without tying them up to the original varieties and working with geometric structures inherently defined in characteristic one (that is, additively idempotent) semifields. In this talk we explore the relationship between tropical ideals and congruences to conclude that the variety of a prime (tropical) ideal is either empty or consists of a single point. This is joint work with D. Joó.
The question of which high-dimensional knots are slice was entirely solved by Kervaire and Levine. Compared to this, the question of which knots are doubly slice in high-dimensions is still a largely open problem. Ruberman proved that in every dimension, some version of the Casson-Gordon invariants can be applied to obtain algebraically doubly slice knots that are not doubly slice. I will show how to use L^2 signatures to recover the result of Ruberman for (4k-3)-dimensional knots, and discuss how the derived series of the knot group might be used to organise the high-dimensional doubly slice problem.
I this talk I will summerize some of our contributions in the analysis of parabolic elliptic Keller-Segel system, a typical model in chemotaxis. For the case of linear diffusion, after introducing the critical mass in two dimension, I will show our result for blow-up conditions for higher dimension. The second part of the talk is concentrated in the critical exponent for Keller-Segel system with porus media type diffusion. In the end, motivated from the result on nonlocal Fisher-KPP equation, we show that the nonlocal reaction will also help in preventing the blow-up of the solutions.
Optimal transport is a thoroughly studied field in mathematics and introduces the concept of Wasserstein distance, which has been widely used in various applications in computational mathematics, machine learning as well as many areas in engineering. Meanwhile, control theory and path planning is an active branch in mathematics and robotics, focusing on algorithms that calculates feasible or optimal paths for robotic systems. In this defense, we use the properties of the gradient flows in Wasserstein metric to design algorithms to handle different types of path planning and control problems as well as the K-means problems defined on graphs.
We will see some instances of swindles in mathematics, primarily focusing on some in geometric topology due to Barry Mazur.
We discuss the asymptotic value of the maximal perimeter of a convex set in an n-dimensional space with respect to certain classes of measures. Firstly, we derive a lower bound for this quantity for a large class of probability distributions; the lower bound depends on the moments only. This lower bound is sharp in the case of the Gaussian measure (as was shown by Nazarov in 2001), and, more generally, in the case of rotation invariant log-concave measures (as was shown by myself in 2014). We discuss another class of measures for which this bound is sharp. For isotropic log-concave measures, the value of the lower bound is at least n^{1/8}.
In addition, we show a uniform upper bound of Cn||f||^{1/n}_{\infty} for all log-concave measures in a special position, which is attained for the uniform distribution on the cube. We further estimate the maximal perimeter of isotropic log-concave measures by n^2.
The well known Erdos-Hajnal Conjecture states that every graph has the Erdos-Hajnal (EH) property. That is, for every $H$, there exists a $c=c(H)>0$ such that every graph $G$ with no induced copy of $H$ has the property $hom(G):=max\{\alpha(G),\omega(G)\}\geq |V(G)|^{c}$. Let $H,J$ be subdivisions of caterpillar graphs. Liebenau, Pilipczuk, Seymour and Spirkl proved that the EH property holds if we forbid both $H$ and $\overline{J}.$ We will discuss the proof of this result.
I will talk about a conjecture that in Gibbs states of one-dimensional spin chains with short-ranged gapped Hamiltonians the quantum conditional mutual information (QCMI) between the parts of the chain decays exponentially with the length of separation between said parts. The smallness of QCMI enables efficient representation of these states as tensor networks, which allows their efficient construction and fast computation of global quantities, such as entropy. I will present the known partial results on the way of proving of the conjecture and discuss the probable approaches to the proof and the obstacles that are encountered.
This talk will be about polynomial decompositions that are relevant in machine learning. I will start with the well-known low-rank symmetric tensor decomposition, and present a simple new algorithm with local convergence guarantees, which seems to handily outperform the state-of-the-art in experiments. Next I will consider a particular generalization of symmetric tensor decomposition, and apply this to estimate subspace arrangements from very many, very noisy samples (a regime in which current subspace clustering algorithms break down). Finally I will switch gears and discuss representability of polynomials by deep neural networks with polynomial activations. The various polynomial decompositions in this talk motivate questions in commutative algebra, computational algebraic geometry and optimization. The first part of this talk is joint with Emmanuel Abbe, Tamir Bendory, Joao Pereira and Amit Singer, while the latter part is joint with Matthew Trager.
The talk presents an extension for high dimensions of an idea from a recent result concerning near optimal adaptive finite element methods (AFEM). The usual adaptive strategy for finding conforming partitions in AFEM is ”mark → subdivide → complete”. In this strategy any element can be marked for subdivision but since the resulting partition often contains hanging nodes, additional elements have to be subdivided in the completion step to get a conforming partition. This process is very well understood for triangulations received via newest vertex bisection procedure. In particular, it is proven that the number of elements in the final partition is limited by constant times the number of marked cells. This motivated us [B., Fierro, Veeser, in preparation] to design a marking procedure that is limited only to cells of the partition whose subdivision will result in a conforming partition and therefore no completion step is necessary. We also proved that this procedure is near best in terms of both error of approximation and complexity. This result is formulated in terms of tree approximations and opens the possibility to design similar algorithms in high dimensions using sparse occupancy trees introduced in [B., Dahmen, Lamby, 2011]. The talk describes the framework of approximating high dimensional data using conforming sparse occupancy trees.
One can regard a (trained) feedforward neural network as a particular type of function , where is a (typically high-dimensional) Euclidean space parameterizing some data set, and the value of the function on a data point is the probability that the answer to a particular yes/no question is "yes." It is a classical result in the subject that a sufficiently complex neural network can approximate any function on a bounded set. Last year, J. Johnson proved that universality results of this kind depend on the architecture of the neural network (the number and dimensions of its hidden layers). His argument was novel in that it provided an explicit topological obstruction to representability of a function by a neural network, subject to certain simple constraints on its architecture. I will tell you just enough about neural networks to understand how Johnson's result follows from some very simple ideas in piecewise linear geometry. Time permitting, I will also describe some joint work in progress with K. Lindsey aimed at developing a general theory of how the architecture of a neural network constrains its topological expressiveness.
We study whether all stationary solutions of 2D Euler equation must be radially symmetric, if the vorticity is compactly supported or has some decay at infinity. Our main results are the following:
(1) On the one hand, we are able to show that for any non-negative smooth stationary vorticity that is compactly supported (or has certain decay as |x|->infty), it must be radially symmetric up to a translation.
(2) On the other hand, if we allow vorticity to change sign, then by applying bifurcation arguments to sign-changing radial patches, we are able to show that there exists a compactly-supported, sign-changing smooth stationary vorticity that is non-radial.
We have also obtained some symmetry results for uniformly-rotating solutions for 2D Euler equation, as well as stationary/rotating solutions for the SQG equation. The symmetry results are mainly obtained by calculus of variations and elliptic equation techniques. This is a joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.
First talk at 4:00 by by Ananth Shankar (MIT http://math.mit.edu/~ananths/)
Exceptional splitting of abelian surfaces over global function fields.
Let A denote a non-constant ordinary abelian surface over a global function field (of characteristic p > 2) with good reduction everywhere. Suppose that $A$ does not have real multiplication by any real quadratic field with discriminant a multiple of $p$. Then we prove that there are infinitely many places modulo which $A$ is isogenous to the product of two elliptic curves. If time permits, I will also talk about applications of our results to the p-adic monodromy of such abelian surfaces. This is joint work with Davesh Maulik and Yunqing Tang.
Second talk at 5:15 Jordan Ellenberg (University of Wisconsin http://www.math.wisc.edu/~ellenber/)
What is the tropical Ceresa class and what should it be?
This is a highly preliminary talk about joint work with Daniel Corey and Wanlin Li. The Ceresa cycle is an algebraic cycle canonically attached to a curve C, which appears in an intriguing variety of contexts; its height can sometimes be interpreted as a special value, the vanishing of its cycle class is related to the Galois action on the nilpotent fundamental group, it vanishes on hyperelliptic curves, etc. In practice it is not easy to compute, and we do not in fact know an explicit non-hyperelliptic curve whose Ceresa class vanishes. We will discuss a definition of the Ceresa class for a tropical curve, explain how to compute it in certain simple cases, and describe progress towards understanding whether it is possible for the Ceresa class of a non-hyperelliptic tropical curve to vanish. (The answer is: "sort of”.) The tropical Ceresa class sits at the interface of algebraic geometry, graph theory (because a tropical curve is more or less a metric graph), and topology: for we can also frame the tropical Ceresa class as an entity governed by the mapping class group, and in particular by the question of when a product of commuting Dehn twists can commute with a hyperelliptic involution in the quotient of the mapping class group by the Johnson kernel.
During the last 30 years there has been much interest in random graph processes, i.e., random graphs which grow by adding edges (or vertices) step-by-step in some random way. Part of the motivation stems from more realistic modeling, since many real world networks such as Facebook evolve over time. Further motivation stems from extremal combinatorics, where these processes lead to some of the best known bounds in Ramsey and Turan Theory (that go beyond textbook applications of the probabilistic method). I will review several random graph processes of interest, and (if time permits) illustrate one of the main proof techniques using a simple toy example.
It is known that non-negative homogeneous polynomials(forms) over $\mathbb{R}$ are same as sums of squares if it is bivariate, quadratic forms, or ternary quartic by Hilbert. Once we know a form is a sum of squares, next natural question would be how many forms are needed to represent it as sums of squares. We denote the minimal number of summands in the sums of squares by rank (of the sum of squares). Ranks of some class of forms are known. For example, any bivariate forms (allowing all monomials) can be written as sum of $2$ squares.(i.e. its rank is $2$) and every nonnegative ternary quartic can be written as a sum of $3$ squares.(i.e. its rank is $3$). Our question is that "if we do not allow some monomials in a bivariate form, how its rank will be?". In the talk, we will introduce this problem in algebraic geometry flavor and provide some notions and tools to deal with.
I will talk about the structure of large square random matrices with centered i.i.d. heavy-tailed entries (only two finite moments are assumed). In our previous work with R. Vershynin we have shown that the operator norm of such matrix A can be reduced to the optimal sqrt(n)-order with high probability by zeroing out a small submatrix of A, but did not describe the structure of this "bad" submatrix, nor provide a constructive way to find it. Now we can give a very simple description of this small "bad" subset: it is enough to zero out a small fraction of the rows and columns of A with largest L2 norms to bring its operator norm to the almost optimal sqrt(loglog(n)*n)-order, under additional assumption that the entries of A are symmetrically distributed. As a corollary, one can also obtain a constructive procedure to find a small submatrix of A that one can zero out to achieve the same regularization.
I am planning to discuss some details of the proof, the main component of which is the development of techniques that extend constructive regularization approaches known for the Bernoulli matrices (from the works of Feige and Ofek, and Le, Levina and Vershynin) to the considerably broader class of heavy-tailed random matrices.
The unusual day
New and proposed missions for approaching moons, and particularly icy moons, increasingly require the design of trajectories within challenging multi-body environments that stress or exceed the capabilities of the two-body design methodologies typically used over the last several decades. These current methods encounter difficulties because they often require appreciable user interaction, result in trajectories that require significant amounts of propellant, or miss potential mission-enabling options. The use of dynamical systems methods applied to three-body and multi-body models provides a pathway to obtain a fuller theoretical understanding of the problem that can then result in significant improvements to trajectory design in each of these areas. The search for approach trajectories within highly nonlinear, chaotic regimes where multi-body effects dominate becomes increasingly complex, especially when landing, orbiting, or flyby scenarios must be considered in the analysis. In the case of icy moons, approach trajectories must also be tied into the broader tour which includes flybys of other moons. The tour endgame typically includes the last several flybys, or resonances, before the final approach to the moon, and these resonances further constrain the type of approach that may be used.
In this seminar, new methods for approaching moons by traversing the chaotic regions near the Lagrange point gateways will be discussed for several examples. The emphasis will be on landing trajectories approaching Europa including a global analysis of trajectories approaching any point on the surface and analyses for specific landing scenarios across a range of different energies. The constraints on the approach from the tour within the context of the endgame strategy will be given for a variety of different moons and scenarios. Specific approaches using quasiperiodic or Lissajous orbits will be shown, and general landing and orbit insertion trajectories will be placed into context relative to the invariant manifolds of unstable periodic and quasiperiodic orbits. These methods will be discussed and applied for the specific example of the Europa Lander mission concept. The Europa Lander mission concept is particularly challenging in that it requires the redesign of the approach scenario after the spacecraft has launched to accommodate landing at a wide range of potential locations on the surface. The final location would be selected based on reconnaissance from the Europa Clipper data once Europa Lander is in route. Taken as a whole, these methods will provide avenues to find both fundamentally new approach pathways and reduce cost to enable new missions.
Multidimensional data is ubiquitous in the application, e.g., images and videos. I will introduce some of my previous and current works related to this topic.
1) Lattice metric space and its applications. Lattice and superlattice patterns are found in material sciences, nonlinear optics and sampling designs. We propose a lattice metric space based on modular group theory and
metric geometry, which provides a visually consistent measure of dissimilarity among lattice patterns. We apply this framework to superlattice separation and grain defect detection.
2) We briefly introduce two current projects. First, we propose new algorithms for automatic PDE modeling, which drastically improves the efficiency and the robustness against additive noise. Second, we introduce a new model for surface reconstruction from point cloud data (PCD) and provide an ADMM type fast algorithm.
In this talk we study master equations arising from mean field game
problems, under the crucial monotonicity conditions.
Classical solutions of such equations require very strong technical
conditions. Moreover, unlike the master equations arising from mean
field control problems, the mean field game master equations are
non-local and even classical solutions typically do not satisfy the
comparison principle, so the standard viscosity solution approach seems
infeasible. We shall propose a notion of weak solution for such
equations and establish its wellposedness. Our approach relies on a new
smooth mollifier for functions of measures, which unfortunately does not
keep the monotonicity property, and the stability result of master
equations. The talk is based on a joint work with Jianfeng Zhang.
An electron interacting with the vibrational modes of a polar crystal is called a polaron. Polarons are the simplest Quantum Field Theory models, yet their most basic features such as the effective mass, ground-state energy and wave function cannot be evaluated explicitly. And while several successful theories have been proposed over the years to approximate the energy and effective mass of various polarons, they are built entirely on unjustified, even questionable, Ansätze for the wave function.
In this talk I shall provide the first explicit description of the ground-state wave function of a polaron in an asymptotic regime: For the Fröhlich polaron localized in a Coulomb potential and exposed to a homogeneous magnetic field of strength $B$ it will be shown that the ground-state electron density in the direction of the magnetic field converges pointwise and in a weak sense as $B\rightarrow\infty$ to the square of a hyperbolic secant function--a sharp contrast to the Gaussian wave functions suggested in the physics literature.
In independent bond percolation with parameter p, if one removes the vertices of the infinite cluster (and incident edges), for which values of p does the remaining graph contain an infinite cluster? Grimmett-Holroyd-Kozma used the triangle condition to show that for d > 18, the set of such p contains values strictly larger than the percolation threshold pc. With the work of Fitzner-van der Hofstad, this has been reduced to d > 10. We reprove this result by showing that for d > 10 and some p>pc, there are infinite paths consisting of "shielded"' vertices --- vertices all whose adjacent edges are closed --- which must be in the complement of the infinite cluster. Using numerical values of pc, this bound can be reduced to d > 7. Our methods are elementary and do not require the triangle condition.
Invasion percolation is a stochastic growth model that follows a greedy algorithm. After assigning i.i.d. uniform random variables (weights) to all edges of d-dimensional space, the growth starts at the origin. At each step, we adjoin to the current cluster the edge of minimal weight from its boundary. In '85, Chayes-Chayes-Newman studied the "acceptance profile"' of the invasion: for a given p in [0,1], it is the ratio of the expected number of invaded edges until time n with weight in [p,p+dp] to the expected number of observed edges (those in the cluster or its boundary) with weight in the same interval. They showed that in all dimensions, the acceptance profile an(p) converges to one for ppc. In this paper, we consider an(p) at the critical point p=pc in two dimensions and show that it is bounded away from zero and one as n goes to infinity.
For a first order (deterministic) mean-field game with non-local running and initial couplings, a classical solution is constructed for the associated, so-called master equation, a partial differential equation in infinite-dimensional space with a non-local term, assuming the time horizon is sufficiently small and the coefficients are smooth enough, without convexity conditions on the Hamiltonian.
In network routing users often tradeoff different objectives in selecting their best route. An example is transportation networks, where due to uncertainty of travel times, drivers may tradeoff the average travel time versus the variance of a route. Or they might tradeoff time and cost, such as the cost paid in tolls.
We wish to understand the effect of two conflicting criteria in route selection, by studying the resulting traffic assignment (equilibrium) in the network. We investigate two perspectives of this topic: (1) How does the equilibrium cost of a risk-averse population compare to that of a risk-neutral population? (i.e., how much longer do we spend in traffic due to being risk-averse) (2) How does the equilibrium cost of a heterogeneous population compare to that of a comparable homogeneous user population?
We provide characterizations to both questions above.
Based on joint work with Richard Cole, Thanasis Lianeas and Nicolas Stier-Moses.
At the end I will mention current work of my research group on algorithms and mechanism design for power systems.
Biography: Evdokia Nikolova is an Assistant Professor in the Department of Electrical and Computer Engineering at the University of Texas at Austin, where she is a member of the Wireless Networking & Communications Group. Previously she was an Assistant Professor in Computer Science and Engineering at Texas A&M University. She graduated with a BA in Applied Mathematics with Economics from Harvard University, MS in Mathematics from Cambridge University, U.K. and Ph.D. in Computer Science from MIT.
Nikolova's research aims to improve the design and efficiency of complex systems (such as networks and electronic markets), by integrating stochastic, dynamic and economic analysis. Her recent work examines how human risk aversion transforms traditional computational models and solutions. One of her algorithms has been adapted in the MIT CarTel project for traffic-aware routing. She currently focuses on developing algorithms for risk mitigation in networks, with applications to transportation and energy. She is a recipient of an NSF CAREER award and a Google Faculty Research Award. Her research group has been recognized with a best student paper award and a best paper award runner-up. She currently serves on the editorial board of the journal Mathematics of Operations Research.
One of the classical problems in scissors congruence is
this: given two polytopes in $n$-dimensional Euclidean space, when is
it possible to decompose them into finitely many pieces which are
pairwise congruent via translations? A complete set of invariants is
provided by the Hadwiger invariants, which measure "how much area is
pointing in each direction." Proving that these give a complete set
of invariants is relatively straightforward, but determining the
relations between them is much more difficult. This was done by
Dupont, in a 1982 paper. Unfortunately, this result is difficult to
describe and work with: it uses group homological techniques which
produce a highly opaque formula involving twisted coefficients and
relations in terms of uncountable sums. In this talk we will discuss
a new perspective on Dupont's proof which, together with more
topological simplicial techniques, simplifies and clarifies the
classical results. This talk is partially intended to be an
advertisement for simplicial techniques, and will be suitable for
graduate students and others unfamiliar with the approach.
In this talk, I will discuss progress in our understanding of Legendrian surfaces. First, I will present a new construction of Legendrian surfaces and a direct description for their moduli space of microlocal sheaves. This Legendrian invariant will connect to classical incidence problems in algebraic geometry and the study of flag varieties, which we will study in detail. There will be several examples during the talk and, in the end, I will indicate the relation of this theory to the study of framed local systems on a surface. This talk is based on my work with E. Zaslow.
We prove that every rational homology cobordism class in the subgroup generated by lens spaces contains a unique connected sum of lens spaces whose first homology embeds in any other element in the same class. As a consequence we show that several natural maps to the rational homology cobordism group have infinite rank cokernels, and obtain a divisibility condition between the determinants of certain 2-bridge knots and other knots in the same concordance class. This is joint work with Daniele Celoria and JungHwan Park.
The KAM (Kolmogorov Arnold and Moser) theory studies
the persistence of quasi-periodic solutions under perturbations.
It started from a basic set of theorems and it has grown
into a systematic theory that settles many questions.
The basic theorem is rather surprising since it involves delicate
regularity properties of the functions considered, rather
subtle number theoretic properties of the frequency as well
as geometric properties of the dynamical systems considered.
In these lectures, we plan to cover a complete proof of
a particularly representative theorem in KAM theory.
In the first lecture we will cover all the prerequisites
(analysis, number theory and geometry). In the second lecture
we will present a complete proof of Moser's twist map theorem
(indeed a generalization to more dimensions).
The proof also lends itself to very efficient numerical algorithms.
If there is interest and energy, we will devote a third lecture
to numerical implementations.
he KAM (Kolmogorov Arnold and Moser) theory studies
the persistence of quasi-periodic solutions under perturbations.
It started from a basic set of theorems and it has grown
into a systematic theory that settles many questions.
The basic theorem is rather surprising since it involves delicate
regularity properties of the functions considered, rather
subtle number theoretic properties of the frequency as well
as geometric properties of the dynamical systems considered.
In these lectures, we plan to cover a complete proof of
a particularly representative theorem in KAM theory.
In the first lecture we will cover all the prerequisites
(analysis, number theory and geometry). In the second lecture
we will present a complete proof of Moser's twist map theorem
(indeed a generalization to more dimensions).
The proof also lends itself to very efficient numerical algorithms.
If there is interest and energy, we will devote a third lecture
to numerical implementations.
Isospectral reductions is a network/graph reduction that preserves the
eigenvalues and the eigenvectors of the adjacency matrix. We analyze the
conditions under which the generalized eigenvectors would be preserved and
simplify the proof of the preservation of eigenvectors. Isospectral reductions
are associative and form a dynamical system on the set of all matrices/graphs.
We study the spectral equivalence relation defined by specific characteristics
of nodes under isospectral reductions and show some examples of the attractors.
Cooperation among antigens, cross-immunoreactivity (CR) has been observed in
various diseases. The complex viral population dynamics couldn't be explained
by traditional math models. A new math model was constructed recently with
promising numerical simulations. In particular, the numerical results recreated
local immunodeficiency (LI), the phenomenon where some viruses sacrifice
themselves while others are not attacked by the immune system. Here we analyze
small CR networks to find the minimal network with a stable LI. We also
demonstrate that you can build larger CR networks with stable LI using this
minimal network as a building block.
The pedestrian-induced lateral oscillation of London's Millennium bridge on the day it opened in 2000 has become a much cited paradigm of an instability caused by phase synchronization of coupled oscillators. However, a closer examination of subsequent theoretical studies and experimental observations have brought this interpretation into question.
To elucidate the true cause of instability, we study a model in which each pedestrian is represented by a simplified biomechanically-inspired two-legged inverted pendulum. The key finding is that synchronization between individual pedestrians is not a necessary ingredient of instability onset. Instead, the side-to-side pedestrian motion should on average lag that of the bridge oscillation by a fraction of a cycle. Using a multi-scale asymptotic analysis, we derive a mathematically rigorous general criterion for bridge instability based on the notion of effective negative damping. This criterion suggests that the initiation of wobbling is not accompanied by crowd synchrony and crowd synchrony is a consequence but not the cause of bridge instability.
The first half of this dissertation concerns the following problem: Given a lattice in $\mathbf{R}^d$ which refines the integer lattice $\mathbf{Z}^d$, what can be said about the distribution of the lattice points inside of the half-open unit cube $[0,1)^d$? This question is of interest in discrete geometry, especially integral polytopes and Ehrhart theory. We observe a combinatorial description of the linear span of these points, and give a formula for the dimension of this span. The proofs of these results use methods from classical multiplicative number theory.
In the second half of the dissertation, we investigate oriented matroids from the point of view of tropical geometry. Given an oriented matroid, we describe, in detail, a polyhedral complex which plays the role of the Bergman complex for ordinary matroids. We show how this complex can be used to give a new proof of the celebrated Bohne-Dress theorem on tilings of zonotopes by zonotopes with an approach which relies on a novel interpretation of the chirotope of an oriented matroid.
We say that trees with common root are (edge-)independent if, for any vertex in their intersection, the paths to the root induced by each tree are internally (edge-)disjoint. The relationship between graph (edge-)connectivity and the existence of (edge-)independent spanning trees is explored. The (Edge-)Independent Spanning Tree Conjecture states that every k-(edge-)connected graph has k-(edge-)independent spanning trees with arbitrary root.
We prove the case k=4 of the Edge-Independent Spanning Tree Conjecture using a graph decomposition similar to an ear decomposition, and give polynomial-time algorithms to construct the decomposition and the trees. We provide alternate geometric proofs for the cases k=3 of both the Independent Spanning Tree Conjecture and Edge-Independent Spanning Tree Conjecture by embedding the vertices or edges in a 2-simplex, and conjecture higher-dimension generalizations. We provide a partial result towards a generalization of the Independent Spanning Tree Conjecture, in which local connectivity between the root and a vertex set S implies the existence of trees whose independence properties hold only in S. Finally, we prove and generalize a theorem of Györi and Lovász on partitioning a k-connected graph, and give polynomial-time algorithms for the cases k=2,3,4 using the graph decompositions used to prove the corresponding cases of the Independent Spanning Tree Conjecture.
We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to quantum Teichmuller space, or filtering the algebra and obtaining an associated graded algebra that is a monomial subalgebra of a quantum torus. We utilize the former method to generalize the Chebyshev homomorphism of Bonahon and Wong between skein algebras of surfaces to a Chebyshev-Frobenius homomorphism between skein modules of marked 3-manifolds, in the course of which we define a surgery theory, and whose image we show is either transparent or (skew)-transparent. The latter method is used to show that skein algebras of surfaces are maximal orders, which implies a refined unicity theorem, shows that SL_2C-character varieties are normal, and suggests a conjecture on how this result may be utilized for topological quantum compiling.
In this talk we will review compactness results and singularity theorems related to harmonic maps. We first talk about maps from Riemann surfaces with tension fields bounded in a local Hardy space, then talk about stationary harmonic maps from higher dimensional manifolds, finally talk about heat flow of harmonic maps.
The study of LIn, the length of the longest increasing subsequences, and of LCIn, the length of the longest common and increasing subsequences in random words is classical in computer science and bioinformatics, and has been well explored over the last few decades. This dissertation studies a generalization of LCIn for two binary random words, namely, it analyzes the asymptotic behavior of LCbBn, the length of the longest common subsequences containing a fixed number, b, of blocks. We first prove that after proper centerings and scalings, LCbBn, for two sequences of i.i.d. Bernoulli random variables with possibly two different parameters, converges in law towards limits we identify. This dissertation also includes an alternative approach to the one-sequence LbBn problem, and Monte-Carlo simulations on the asymptotics of LCbBn and on the growth order of the limiting functional, as well as several extensions of the LCbBn problem to the Markov context and some connection with percolation theory.
Please note different day and room.
In this talk, I will describe joint work with Maximilien Péroux on understanding Koszul duality in ∞-topoi. An ∞-topos is a particularly well behaved higher category that behaves like the category of compactly generated spaces. Particularly interesting examples of ∞-topoi are categories of simplicial sheaves on Grothendieck topologies. The main theorem of this work is that given a group object G of an ∞-topos, there is an equivalence of categories between the category of G-modules in that topos and the category of BG-comodules, where BG is the classifying object for G-torsors. In particular, given any pointed space X, the space of loops on X, denoted ΩX, can be lifted to a group object of any ∞-topos, so if X is in addition a connected space, there is an equivalence between objects of any ∞-topos with an ΩX-action, and objects with an X-coaction (where X is a coalgebra via the usual diagonal map). This is a generalization of the classical equivalence between G-spaces and spaces over BG for G a topological group.
Residential crime is one of the toughest issues in modern society. A quantitative, informative, and applicable model of criminal behavior is needed to assist law enforcement. We have made progress to the pioneering statistical agent-based model of residential burglary (Short et al., Math. Models Methods Appl., 2008) in two ways. (1) In one space dimension, we assume that the movement patterns of the criminals involve truncated Lévy distributions for the jump length, other than classical random walks (Short et al., Math. Models Methods Appl., 2008) or Lévy flights without truncation (Chaturapruek et al., SIAM J. Appl. Math, 2013). This is the first time that truncated Lévy flights have been applied in crime modeling. Furthermore (2), in two space dimensions, we used the Poisson clocks to govern the time steps of the evolution of the model, rather than a discrete time Markov chain with deterministic time increments used in the previous works. Poisson clocks are particularly suitable to model the times at which arrivals enter a system. Introduction of the Poisson clock not only produces similar simulation output, but also brings in theoretically the mathematical framework of the Markov pure jump processes, e.g., a martingale approach. The martingale formula leads to a continuum equation that coincides with a well-known mean-field continuum limit. Moreover, the martingale formulation together with statistics quantifying the relevant pattern formation leads to a theoretical explanation of the finite size effects. Our conjecture is supported by numerical simulations.
Great News Everyone! The cartoon series Futurama is packed with science jokes. Adopting my Professor Farnsworth Alterego, I will explain some of these mathematical jokes with stills and clips from the series.
A brief meeting to discuss the plan for the semester, followed by an informal discussion over lunch (most likely at Ferst Place).
A random array indexed by the paths of an infinitely-branching rooted tree of finite depth is hierarchically exchangeable if its joint distribution is invariant under rearrangements that preserve the tree structure underlying the index set. Austin and Panchenko (2014) prove that such arrays have de Finetti-type representations, and moreover, that an array indexed by a finite collection of such trees has an Aldous-Hoover-type representation.
Motivated by problems in Bayesian nonparametrics and probabilistic programming discussed in Staton et al. (2018), we generalize hierarchical exchangeability to a new kind of partial exchangeability for random arrays which we call DAG-exchangeability. In our setting a random array is indexed by N^{|V|} for some DAG G=(V,E), and its exchangeability structure is governed by the edge set E. We prove a representation theorem for such arrays which generalizes the Aldous-Hoover representation theorem, and for which the Austin-Panchenko representation is a special case.
In this talk, we consider a quasi-periodically forced system arising from the problem of chemical reactions. For we demonstrate efficient algorithms to calculate the normally hyperbolic invariant manifolds and their stable/unstable manifolds using parameterization method. When a random noise is added, we use similar ideas to give a streamlined proof of the existence of the stochastic invariant manifolds.
The field of complex dynamics melds a number of disciplines, including complex analysis, geometry and topology. I will focus on the influences from the latter, introducing some important concepts and questions in complex dynamics, with an emphasis on how the concepts tie into and can be enhanced by a topological viewpoint.
In complex dynamics, the main objects of study are rational maps on the Riemann sphere. For some large subset of such maps, there is a way to associate to each map a marked torus. Moving around in the space of these maps, we can then track the associated tori and get induced mapping classes. In this talk, we will explore what sorts of mapping classes arise in this way and use this to say something about the topology of the original space of maps.
Soot particles are major pollutants emitted from propulsion and power generation systems. In turbulent combustion, soot evolution is heavily influenced by soot-turbulence-chemistry interaction. Specifically, soot is formed during combustion of fuel-rich mixtures and is rapidly oxidized before being transported by turbulence into fuel-lean mixtures. Furthermore, different soot evolution mechanisms are dominant over distinct regions of mixture fraction. For these reasons, a new subfilter Probability Density Function (PDF) model is proposed to account for this distribution of soot in mixture fraction space. At the same time, Direct Numerical Simulation (DNS) studies of turbulent nonpremixed jet flames have revealed that Polycyclic Aromatic Hydrocarbons (PAH), the gas-phase soot precursors, are confined to spatially intermittent regions of low scalar dissipation rates due to their slow formation chemistry. The length scales of these regions are on the order of the Kolmogorov scale (i.e., the smallest turbulence scale) or smaller, where molecular diffusion dominates over turbulent mixing irrespective of the large-scale turbulent Reynolds number. A strain-sensitivity parameter is developed to identify such species. A Strain-Sensitive Transport Approach (SSTA) is then developed to model the differential molecular transport in the nonpremixed “flamelet” equations. These two models are first validated a priori against a DNS database, and then implemented within a Large Eddy Simulation (LES) framework, applied to a series of turbulent nonpremixed sooting jet flames, and validated via comparisons with experimental measurements of soot volume fraction.
We will discuss how to solve algebraic equations using symbolic, numerical, and combinatorial methods.
Prym varieties are a class of abelian varieties that arise from double covers of tropical or algebraic curves. The talk will revolve around the Prym--Brill--Noether locus, a subvariety determined by divisors of a given rank. Using a connection to Young tableaux, we determine the dimensions of these loci for certain tropical curves, with applications to algebraic geometry. Furthermore, these loci are always pure dimensional and path connected. Finally, we compute the first homologies of the Prym--Brill--Noether loci under certain conditions.
special time
In asymptotic analysis and numerical approximation of highly-oscillatory evolution problems, it is commonly supposed that the oscillation frequency is either constant or, at least, bounded from below by a strictly positive constant uniformly in time. Allowing for the possibility that the frequency actually depends on time and vanishes at some instants introduces additional difficulties from both the asymptotic analysis and numerical simulation points of view. I will present a first step towards the resolution of these difficulties. In particular, we show that it is still possible in this situation to infer the asymptotic behavior of the solution at the price of more intricate computations and we derive a second order uniformly accurate numerical method.
Fine properties of spherical averages in the continuous setting include
$L^p$ improving estimates
and sparse bounds, interesting in the settings of a fixed radius, lacunary sets of radii, and the
full set of radii. There is a parallel theory in the setting of discrete spherical averages, as studied
by Elias Stein, Akos Magyar, and Stephen Wainger. We recall the continuous case, outline the
discrete case, and illustrate a unifying proof technique. Joint work with Robert Kesler, and
Dario Mena Arias.
We will consider the problem of estimating the singularity probability of sparse Bernoulli matrices, and a related question of anti-concentration of weighted sums of dependent Bernoulli(p) variables.
Based on joint work with Alexander Litvak.
special time
Our ambition is to derive asymptotic equations of the Vlasov-Poisson system in the strong magntic field regime. This work is thus an attempt to (re-)derive rigorously gyrokinetic equations and to design uniformly accurate methods for solving fast-oscillating kinetic equations, i.e. methods whose cost and accuracy do not depend the stiffness parameter. The main tools used to reach this objective are averaging and PDE techniques. In this talk, I will focus primarily on the first.
In first-passage percolation, we place i.i.d. nonnegative weights (t_e) on the edges of a graph and consider the induced weighted graph metric T(x,y). When the underlying graph is the two-dimensional square lattice, there is a phase transition in the model depending on the probability p that an edge weight equals zero: for p<1/2, the metric T(0,x) grows linearly in x, whereas for p>1/2, it remains stochastically bounded. The critical case occurs for p=1/2, where there are large but finite clusters of zero-weight edges. In this talk, I will review work with Wai-Kit Lam and Xuan Wang in which we determine the rate of growth for T(0,x) up to a constant factor for all critical distributions. Then I will explain recent work with Jack Hanson and Wai-Kit Lam in which we determine the "time constant" (leading order constant in the rate of growth) in the special case where the graph is the triangular lattice, and the weights are placed on the vertices. This is the only class of distributions known where this time constant is computable: we find that it is an explicit function of the infimum of the support of t_e intersected with (0,\infty).
Special time
I will start by giving a short overview of the history around stability and instability issues in gravitational systems driven by kinetic equations. Conservations properties and families of non-homogeneous steady states will be first presented. A well-know conjecture in both astrophysics and mathematics communities was that "all steady states of the gravitational Vlasov-Poisson system which are decreasing functions of the energy, are non linearly stable up to space translations". We explain why the traditional variational approaches are not sufficient to answer this conjecture. An alternative approach, inspired by astrophysics literature, will be then presented and quantitative stability inequalities will be shown, therefore solving the above conjecture for Vlasov-Poisson systems. This have been achieved by using a refined notion for the rearrangement of functions and Poincaré-like functional inequalities. For other systems like the so-called Hamiltonian Mean Field (HMF), the decreasing property of the steady states is no more sufficient to guarantee their stability. An additional explicit criteria is needed, under which their non-linear stability is proved. This criteria is sharp as non linear instabilities can be constructed if it is not satisfied.
In this talk we introduce some machine learning problems in the setting of undirected graphical models, also known as spin systems. We take proper colorings as a representative example of a hard-constraint graphical model. The classic problem of sampling a proper coloring uniformly at random of a given graph has been well-studied. Here we consider two inverse problems: Given random colorings of an unknown graph G, can we recover the underlying graph G exactly? If we are also given a candidate graph H, can we tell if G=H? The former problem is known as structure learning in the machine learning field and the latter is called identity testing. We show the complexity of these problems in different range of parameters and compare these results with the corresponding decision and sampling problems. Finally, we give some results of the analogous problems for the Ising model, a typical soft-constraint model. Based on joint work with Ivona Bezakova, Antonio Blanca, Daniel Stefankovic and Eric Vigoda.
A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points. Stable polynomials have close relations to matroids and hyperbolic programming. We will discuss a generalization of stability to algebraic varieties of codimension larger than one. They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.
I will provide an introduction to Lorentzian Polynomials in the sense of https://arxiv.org/abs/1902.03719
Understanding the folding of RNA sequences into three-dimensional structures is one of the fundamental challenges in molecular biology. For example, the branching of an RNA secondary structure is an important molecular characteristic yet difficult to predict correctly. However, recent results in geometric combinatorics (both theoretical and computational) yield new insights into the distribution of optimal branching configurations, and suggest new directions for improving prediction accuracy.
The systems of coupled NLS equations occur in some physical problems, in particular in nonlinear optics (coupling between two optical waveguides, pulses or polarized components...).
From the mathematical point of view, the coupling effects can lead to truly nonlinear behaviors, such as the beating effect (solutions with Fourier modes exchanging energy) of Grébert, Paturel and Thomann (2013). In this talk, I will use the coupling between two NLS equations on the 1D torus to construct a family of linearly unstable tori, and therefore unstable quasi-periodic solutions.
The idea is to take profit of the Hamiltonian structure of the system via the construction of a Birkhoff normal form and the application of a KAM theorem. In particular, we will see of this surprising behavior (this is the first example of unstable tori for a 1D PDE) is strongly related to the existence of beating solutions.
This is a work in collaboration with Benoît Grébert (Université de Nantes).
We will explore some of the basic notions in quantum topology. Our focus will be on introducing some of the foundations of diagrammatic algebra through the lens of the Temperley-Lieb algebra. We will attempt to show how these diagrammatic techniques can be applied to low dimensional topology. Every effort will be made to make this as self-contained as possible. If time permits we will also discuss some applications to topological quantum computing.
Mathematicians have long been trying to understand which domains admit an orthogonal (or, sometimes, not) basis of exponentials of the form , for some set of frequencies (this is the spectrum of the domain). It is well known that we can do so for the cube, for instance (just take ), but can we find such a basis for the ball? The answer is no, if we demand orthogonality, but this problem is still open when, instead of orthogonality, we demand just a Riesz basis of exponentials.
A 2-knot is a smooth embedding of S^2 in S^4, and a 0-concordance of 2-knots is a concordance with the property that every regular level set of the concordance is just a collection of S^2's. In his thesis, Paul Melvin proved that if two 2-knots are 0-concordant, then a Gluck twist along one will result in the same smooth 4-manifold as a Gluck twist on the other. He asked the following question: Are all 2-knots 0-slice (i.e. 0-concordant to the unknot)? I will explain all relevant definitions, and mostly follow the paper by Nathan Sunukjian on this topic.
In this talk I will briefly talk about coding theory and introduce a specific family of codes called Quasi-quadratic residue (QQR) codes. These codes have large minimum distances, which means they have good error-correcting capabilities. The weights of their codewords are directly related to the number of points on corresponding hyperelliptic curves. I will show a heuristic model to count the number of points on hyperelliptic curves using a coin-toss model, which in turn casts light on the relation between efficiency and the error-correcting capabilities of QQR codes. I will also show an interesting phenomenon we found about the weight enumerator of QQR codes. Lastly, using the bridge between QQR codes and hyperelliptic curves again, we derive the asymptotic behavior of point distribution of a family of hyperelliptic curves using results from coding theory.
We study the effect of sparsity on the appearance of outliers in the semi-circular law. As a corollary of our main results, we show that, for the Erdos-Renyi random graph with parameter p, the second largest eigenvalue is (asymptotically almost surely) detached from the bulk of the spectrum if and only if pn
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In 1999, Alon proved the “Combinatorial Nullstellensatz” which resembles Hilbert’s Nullstellensatz and gives combinatorial structure on the roots of a multivariate polynomial. This method has numerous applications, most notably in additive number theory, but also in many other areas of combinatorics. We will prove the Combinatorial Nullstellensatz and give some of its applications in graph theory.
For the first time in 2020, the US Census Bureau will apply a differentially private algorithm before publicly releasing decennial census data. Recently, the Bureau publicly released their code and end-to-end tests on the 1940 census data at various privacylevels. We will outline the DP algorithm (which is still being developed) and discuss the accuracy of these end-to-end tests. In particular, we focus on the bias and variance of the reported population counts. Finally, we discuss the choices the Bureau has yet to make that will affect the balance between privacy and accuracy. This talk is based on joint work with Abraham Flaxman.
We will define Newton polygons for polynomials over a valued field and prove a couple theorems using them. For example, relating the valuations of the roots of the polynomial to the slopes of the Newton polygon and proving the algebraic closure of the Puiseux series in characteristic 0.
This is a general audience Geometry-Topology talk where I will give a broad overview of my research interests and techniques I use in my work. My research concerns the study of link concordance using groups, both extracting concordance data from group theoretic invariants and determining the properties of group structures on links modulo concordance. Milnor's invariants are one of the more fundamental link concordance invariants; they are thought of as higher order linking numbers and can be computed using both Massey products (due to Turaev and Porter) and higher order intersections (due to Cochran). In my work, I have generalized Milnor's invariants to knots inside a closed, oriented 3-manifold M. I call this the Dwyer number of a knot and show methods to compute it for null-homologous knots inside a family of 3-manifolds with free fundamental group. I further show Dwyer number provides the weight of the first non-vanishing Massey product in the knot complement in the ambient manifold. Additionally, I proved the Dwyer number detects knots K in M bounding smoothly embedded disks in specific 4-manifolds with boundary M which are not concordant to the unknot in M x I. This result further motivates my definition of a new link concordance group in joint work with Matthew Hedden using the knotification construction of Ozsv'ath and Szab'o. Finally, I will briefly discuss my recent result that the string link concordance group modulo its pure braid subgroup is non-abelian.
Starting from Total Variation, this talk will overview some mathematical approaches for image processing, such as removing noise. We will also consider numerical application to data understanding. A few more application maybe presented.
I will show some operations that preserve Lorentzian property following Section 6 of https://arxiv.org/pdf/1902.03719.pdf
Phylogenetic trees are the fundamental mathematical representation of evolutionary processes in biology. As data objects, they are characterized by the challenges associated with "big data," as well as the complication that their discrete geometric structure results in a non-Euclidean phylogenetic tree space, which poses computational and statistical limitations.
In this talk, I will compare the geometric and statistical properties between a well-studied framework - the BHV space, and an alternative framework that we propose, which is based on tropical geometry. Our framework exhibits analytic, geometric, and topological properties that are desirable for theoretical studies in probability and statistics, as well as increased computational efficiency. I also demonstrate our approach on an example of seasonal influenza data.
Let $f$ be defined on $\mathbb{Z}$. Let $A_N f$ be the average of $f$ along the square integers.
Cutting a polyhedron along some spanning tree of its edges will yield an isometric immersion of the polyhedron into the plane. If this immersion is also injective, we call it an unfolding. In this talk I will give some general results about unfoldings of polyhedra. There is also a notion of pseudo-edge unfolding, which involves cutting on a pseudo edge graph, as opposed to an edge graph. A pseudo edge graph is a 3-connected graph on the surface of the polyhedron, whose vertices coincide with the vertices of the polyhedron, and whose edges are geodesics. I will explain part of the paper "Pseudo-Edge Unfoldings of Convex Polyhedra," a joint work of mine with Professor Ghomi, which proves the existence of a convex polyhedron with a pseudo edge graph along which it is not unfoldable. Finally, I will discuss some connections between pseudo edge graphs and edge graphs.
Using ideas from information theory, we establish lower bounds on the volume and the surface area of a geometric body using the size of its slices along different directions. In the first part of the talk, we derive volume bounds for convex bodies using generalized subadditivity properties of entropy combined with entropy bounds for log-concave random variables. In the second part, we investigate a new notion of Fisher information which we call the L1-Fisher information and show that certain superadditivity properties of the L1-Fisher information lead to lower bounds for the surface areas of polyconvex sets in terms of its slices.
A combinatorial structure is said to be quasirandom if it resembles a random structure in a certain robust sense. For example, it is well-known that a graph G with edge-density p is quasirandom if and only if the density of C_4 in G is p^4+o(p^4); this property is known to equivalent to several other properties that hold for truly random graphs. A similar phenomenon was established for permutations: a permutation is quasirandom if and only if the density of every 4-point pattern (subpermutation) is 1/4!+o(1). We strengthen this result by showing that a permutation is quasirandom if and only if the sum of the densities of eight specific 4-point patterns is 1/3+o(1). More generally, we classify all sets of 4-point patterns having such property.
The talk is based on joint work with Timothy F. N. Chan, Jonathan A. Noel, Yanitsa Pehova, Maryam Sharifzadeh and Jan Volec.
Graphs, which in their simplest forms are vertices connected by edges,
are widely used in high performance computing, machine learning and
network science. This talk will use recent progresses on two
well-studied algorithmic problems in static and dynamic graph,
max-flows and dynamic matchings, to discuss a methodology for
designing faster algorithm for large graphs. This approach is
motivated by a fundamental phenomenon in data structures: the
advantages of offline data structures over online ones.
I will start by describing how work on max-flows led to a focus on
finding short paths in residual graphs, and how investigating more
global notions of progress in residual graphs led to a more
sophisticated and general understanding of iterative methods and
preconditioning. I will then discuss a similar phenomenon in dynamic
graphs, where maintaining a large matching seems to require the online
detection of short augmenting paths, but can once again be
circumvented through the offline construction of smaller equivalent
graphs.
The structure of sums-of-squares representations of (nonnegative homogeneous) polynomials is one interesting subject in real algebraic geometry. The sum-of-squares representations of a given polynomial are parametrized by the convex body of positive semidefinite Gram matrices, called the Gram spectrahedron. In this talk, I will introduce Gram spectrahedron, connection to toric variety, a new result that if a variety $X$ is arithmetically Cohen-Macaulay and a linearly normal variety of almost minimal degree (i.e. $\deg(X)=\text{codim}(X)+2$), then every sum of squares on $X$ is a sum of $\dim(X)+2$ squares.
As countless examples show, sequences of complicated objects should be studied all at once via the formalism of generating functions. We apply this point of view to the homology and combinatorics of (orbit-)configuration spaces: using the notion of twisted commutative algebras, which categorify exponential generating functions. With this idea the configuration space “generating function” factors into an infinite product, whose terms are surprisingly easy to understand. Beyond the intrinsic aesthetic of this decomposition and its quantitative consequences, it also gives rise to representation stability - a notion of homological stability for sequences of representations of differing groups.
Continued fractions play a key role in number theory, especially in understanding how well we can approximate irrational numbers by rational numbers. They also play an important role in function theory, in understanding how well we can approximate analytic functions by rational functions. We discuss a few of the main achievements of the theory.
I will consider the isotropic XY chain with a transverse magnetic field acting on a single site, and analyze the long time behaviour of the time-dependent state of the system when a periodic perturbation drives the impurity. I will show that, under some conditions, the state approaches a periodic orbit synchronized with the forcing. Moreover I will provide the explicit rate of convergence to the asymptotics. This is a joint work with G. Genovese.
I will discuss a proof of the statement that the support of a Lorentzian polynomial is M-convex, based on sections 3-5 of the Brändén—Huh paper.
In this talk, I will discuss from a mathematical viewpoint some sufficient conditions that guarantee the energy equality for weak solutions. I will mainly focus on a fluid equation example, namely the inhomogeneous Euler equations. The main tools are the commutator Lemmas. This is a joint work with Ming Chen.
When hybridization plays a role in evolution, networks are necessary to describe species-level relationships. In this talk, we show that most topological features of a level-1 species network (networks with no interlocking cycles) are identifiable from gene tree topologies under the network multispecies coalescent model (NMSC). We also present the theory behind NANUQ, a new practical method for the inference of level-1 networks under the NMSC.
This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function. After a brief review of inverse spectral theory of one dimensional regular Schroedinger operators, we will discuss complex analytic methods for the following problem: Can one spectrum together with subsets of another spectrum and norming constants recover the potential?
I will give an introduction to surface bundles and will discuss several places where they arise naturally. A surface bundle is a fiber bundle where the fiber is a surface. A first example is the mapping torus construction for 3-manifolds, which is a surface bundle over the circle. Topics will include a construction of 4-manifolds as well as section problems related to surface bundles. The talk will be based on a forthcoming Notices survey article by Salter and Tshishiku.
We study the subject of approximation of convex bodies by polytopes in high dimension.
For a convex set K in R^n, we say that K can be approximated by a polytope of m facets by a distance R>1 if there exists a polytope of P m facets such that K contains P and RP contains K.
When K is symmetric, the maximal volume ellipsoid of K is used heavily on how to construct such polytope of poly(n) facets to approximate K. In this talk, we will discuss why the situation is entirely different for non-symmetric convex bodies.
In deep generative models, the latent variable is generated by a time-inhomogeneous Markov chain, where at each time step we pass the current state through a parametric nonlinear map, such as a feedforward neural net, and add a small independent Gaussian perturbation. In this talk, based on joint work with Belinda Tzen, I will discuss the diffusion limit of such models, where we increase the number of layers while sending the step size and the noise variance to zero. The resulting object is described by a stochastic differential equation in the sense of Ito. I will first show that sampling in such generative models can be phrased as a stochastic control problem (revisiting the classic results of Föllmer and Dai Pra) and then build on this formulation to quantify the expressive power of these models. Specifically, I will prove that one can efficiently sample from a wide class of terminal target distributions by choosing the drift of the latent diffusion from the class of multilayer feedforward neural nets, with the accuracy of sampling measured by the Kullback-Leibler divergence to the target distribution.
In this talk we introduce a refined alteration approach for constructing $H$-free graphs: we show that removing all edges in $H$-copies of the binomial random graph does not significantly change the independence number (for suitable edge-probabilities); previous alteration approaches of Erdös and Krivelevich remove only a subset of these edges. We present two applications to online graph Ramsey games of recent interest, deriving new bounds for Ramsey, Paper, Scissors games and online Ramsey numbers (each time extending recent results of Fox–He–Wigderson and Conlon–Fox–Grinshpun–He).
Based on joint work with Lutz Warnke.
In this informal chat, I will introduce the braid group and several equivalent topological perspectives from which to view it. In particular, we will discuss the role that complex polynomials play in this setting, along with a few classical results.
The Jacobian Conjecture is a famous open problem in commutative algebra and algebraic geometry. Suppose you have a polynomial function $f:\mathbb{C}^n\to\mathbb{C}^n$. The Jacobian Conjecture asserts that if the Jacobian of $f$ is a non-zero constant, then $f$ has a polynomial inverse. Because the conjecture is so easy to state, there have been many claimed proofs that turned out to be false. We will discuss some of these incorrect proofs, as well as several correct theorems relating to the Jacobian Conjecture.
Deep neural networks (DNNs) have revolutionized machine learning by gradually replacing the traditional model-based algorithms with data-driven methods. While DNNs have proved very successful when large training sets are available, they typically have two shortcomings: First, when the training data are scarce, DNNs tend to suffer from overfitting. Second, the generalization ability of overparameterized DNNs still remains a mystery. In this talk, I will discuss two recent works to “inject” the “modeling” flavor back into deep learning to improve the generalization performance and interpretability of the DNN model. This is accomplished by DNN regularization through applied differential geometry and harmonic analysis. In the first part of the talk, I will explain how to improve the regularity of the DNN representation by enforcing a low-dimensionality constraint on the data-feature concatenation manifold. In the second part, I will discuss how to impose scale-equivariance in network representation by conducting joint convolutions across the space and the scaling group. The stability of the equivariant representation to nuisance input deformation is also proved under mild assumptions on the Fourier-Bessel norm of filter expansion coefficients.
The space of degree-n complex polynomials with distinct roots appears frequently and naturally throughout mathematics, but its shape and structure could be better understood. In recent and ongoing joint work with Jon McCammond, we present a deformation retraction of this space onto a simplicial complex with rich structure given by the combinatorics of noncrossing partitions. In this talk, I will describe the deformation retraction and the resulting combinatorial data associated to each generic complex polynomial. We will also discuss a helpful comment from Daan Krammer which connects our work with the ideas of Bill Thurston and his collaborators.
A knot is a smooth embedding of a circle into R^3. Closely related are tangles, which are properly embedded arcs in a 3-ball. We will model certain tangles using jump ropes and relate this to Conway's classification of rational tangles.
For online optimization, the input instance is revealed in a sequence of steps and, after each step, the algorithm has to take an immediate and irrevocable decision based on the previous inputs. Online algorithms produce a sequence of decisions for such problems without the complete information of the future. In the worst-case analysis of online optimization problems, sometimes, it is impossible to achieve any bounded competitive ratio. An interesting way to bypass these impossibility results is to incorporate a stochastic component into the input model. In the random-order arrival model, the adversary specifies an input instance in advance but the input appears according to a random permutation. The knapsack problem is one of the classical problems in combinatorial optimization: Given a set of items, each specified by its size and profit, the goal is to find a maximum profit packing into a knapsack of bounded capacity. The generalized assignment problem (GAP) includes, besides the knapsack problem, several important problems related to scheduling and matching. In this talk, we will present improved competitive algorithms under random-order arrival for these two problems. This is joint work with Susanne Albers and Leon Ladewig.
A double occurrence word (DOW) is a word in which every symbol appears exactly twice; two DOWs are equivalent if one is a symbol-to-symbol image of the other. In the context of genomics, DOWs and operations on DOWs have been used in studies of DNA rearrangement. By modeling the DNA rearrangement process using DOWs, it was observed that over 95% of the scrambled genome of the ciliate Oxytricha trifallax could be described by iterative insertions of the ``repeat pattern'' and the ``return pattern''. These patterns generalize square and palindromic factors of DOWs, respectively. We introduce a notion of inserting repeat/return words into DOWs and study how two distinct insertions into the same word can produce equivalent DOWs. Given a DOW w, we characterize the structure of w which allows two distinct insertions to yield equivalent DOWs. This characterization depends on the locations of the insertions and on the length of the inserted repeat/return words and implies that when one inserted word is a repeat word and the other is a return word, then both words must be trivial (i.e., have only one symbol). The characterization also introduces a method to generate families of words recursively.
Back in the year 2000, Christ and Kiselev introduced a useful "maximal trick" in their study of spectral properties of Schro edinger operators.
The trick was completely abstract and only at the level of basic functional analysis and measure theory. Over the years it was reproven,
generalized, and reused by many authors. We will present its recent application in the theory of restriction of the Fourier transform to
surfaces in the Euclidean space.
I will talk about a connection between graph theory and sutured Floer homology. In fact, there is a one to one correspondence between hypergraphs of a planar bipartite graph and the dimension of sutured Floer homology of a complement of a neighborhood of special alternating link In a three sphere. This is based on the work of Juhas, Kalman and Rasmussen.
In the realm of Laplacians of Riemannian manifolds, nodal domains have been the subject of intensive research for well over a hundred years.
Given a Riemannian manifold M, let f be an eigenfunctions f of the Laplacian with respect to some boundary conditions. A nodal domain associated with f is the maximal connected subset of the domain M for which the f does not change sign.
Here we examine the discrete cases, namely we consider nodal domains for graphs. Dekel-Lee-Linial shows that for a Erdős–Rényi graph G(n, p), with high probability there are exactly two nodal domains for each eigenvector corresponding to a non-leading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately equal to each other.
As a direct corollary of this purely combinatorial result, the sensitivity and degree of every boolean function are polynomially related. This solves an outstanding foundational problem in theoretical computer science, the Sensitivity Conjecture of Nisan and Szegedy.
In the past decade, the catalog of algorithms available to combinatorial optimizers has been substantially extended to settings which allow submodular objective functions. One significant recent result was a tight (1-1/e)-approximation for maximizing a non-negative monotone submodular function subject to a matroid constraint. These algorithmic developments were happening concurrently with research that found a wealth of new applications for submodular optimization in machine learning, recommender systems, and algorithmic game theory.
The related supermodular maximization models also offer an abundance of applications, but they appeared to be highly intractable even under simple cardinality constraints and even when the function has a nice structure. For example, the densest subgraph problem - suspected to be highly intractable - can be expressed as a monotonic supermodular function which has a particularly nice form. Namely, the objective can be expressed by a quadratic form $x^T A x$ where $A$ is a non-negative, symmetric, 0-diagonal matrix. On the other hand, when the entries $A(u,v)$ form a metric, it has been shown that the associated maximization problems have constant factor approximations. Inspired by this, we introduce a parameterized class of non-negative functions called meta-submodular functions that can be approximately maximized within a constant factor. This class includes metric diversity, monotone submodular and other objectives appearing in the machine learning and optimization literature. A general meta-submodular function is neither submodular nor supermodular and so its multi-linear extension does not have the nice convexity/concavity properties which hold for submodular functions. They do, however, have an intrinsic one-sided smoothness property which is essential for our algorithms. This smoothness property might be of independent interest.
We consider the problem of computing unstable manifolds for equilibrium solutions of parabolic PDEs posed on irregular spatial domains. This new approach is based on the parameterization method, a general functional analytic framework for studying invariant manifolds of dynamical systems. The method leads to an infinitesimal invariance equation describing the unstable manifold. A recursive scheme leads to linear homological equations for the jets of the manifold which are solved using the finite element method. One feature of the method is that we recover the dynamics on the manifold in addition to its embedding. We implement the method for some example problems with polynomial and non-polynomial nonlinearities posed on various non-convex two dimensional domains. We provide numerical support for the accuracy of the computed manifolds using the natural notion of a-posteriori error admitted by the parameterization method. This is joint work with J.D. Mireles-James and Necibe Tuncer.
A question going back to Serre asks which groups arise as fundamental groups of smooth, complex projective varieties, or more generally, compact Kaehler manifolds. The most basic examples of these are surface groups, arising as fundamental groups of 1-dimensional projective varieties. We will survey known examples and restrictions on such groups and explain the special role surface groups play in their classification. Finally, we connect this circle of ideas to more general questions about surface bundles and mapping class groups.
It is a fundamental problem in computer vision to describe the geometric relations between two or more cameras that view the same scene -- state of the art methods for 3D reconstruction incorporate these geometric relations in a nontrivial way. At the center of the action is the essential variety: an irreducible subvariety of P^8 of dimension 5 and degree 10 whose homogeneous ideal is minimal generated by 10 cubic equations. Taking a linear slice of complementary dimension corresponds to solving the "minimal problem" of 5 point relative pose estimation. Viewed algebraically, this problem has 20 solutions for generic data: these solutions are elements of the special Euclidean group SE(3) which double cover a generic slice of the essential variety. The structure of these 20 solutions is governed by a somewhat mysterious Galois group (ongoing work with Regan et. al.)
We may ask: what other minimal problems are out there? I'll give an overview of work with Kohn, Pajdla, and Leykin on this question. We have computed the degrees of many minimal problems via computer algebra and numerical methods. I am inviting algebraic geometers at large to attack these problems with "pen and paper" methods: there is still a wide class of problems to be considered, and the more tools we have, the better.
Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that signature need not be multiplicative. These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one. When the dimension of the invariants is at most two, we show that the total space admits a branched covering over a product of curves.
I will describe a few classical problems in capillarity and the associated classical variational framework. These problems include the well-known shape and rise height problems for the meniscus in a tube as well as the problems associated with sessile and pendent drops. I will briefly discuss elements of recent modifications of the variational theory allowing floating objects. Finally, I will describe a few open problems.
We establish an upper bound on the spectral gap for compact quantum graphs which depends only on the diameter and total number of vertices. This bound is asymptotically sharp for pumpkin chains with number of edges tending to infinity. This is a joint work with D. Borthwick and L. Corsi.
In this talk, we discuss about methods for proving existence and uniqueness of a root of a square analytic system in a given region. For a regular root, Krawczyk method and Smale's $\alpha$-theory are used. On the other hand, when a system has a multiple root, there is a separation bound isolating the multiple root from other roots. We define a simple multiple root, a multiple root whose deflation process is terminated by one iteration, and establish its separation bound. We give a general framework to certify a root of a system using these concepts.
Using what we have studied in the Brändén-Huh paper, we will go over the proof of the ultra-log-concavity version of Mason's conjecture.
We study the geometry of minimizers of the interaction energy functional. When the interaction potential is mildly repulsive, it is known to be hard to characterize those minimizers due to the fact that they break the rotational symmetry, suggesting that the problem is unlikely to be resolved by the usual convexity or symmetrization techniques from the calculus of variations. We prove that, if the repulsion is mild and the attraction is sufficiently strong, the minimizer is unique up to rotation and exhibits a remarkable simplex-shape rigid structure. As the first crucial step we consider the maximum variance problem of probability measures under the constraint of bounded diameter, whose answer in one dimension was given by Popoviciu in 1935.
We give a formula relating various notions of heights of abelian varieties. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener, and it extends the Faltings-Silverman formula for elliptic curves. We also discuss the case of Jacobians in some detail, where graphs and electrical networks will play a key role. Based on joint works with Robin de Jong (Leiden).
An expectation-maximization (EM) algorithm is a powerful clustering method that was initially developed to fit Gaussian mixture distributions. In the absence of a particular probability density function, an EM algorithm aims to estimate the "best" function that maximizes the likelihood of data being generated by the model. We present an EM algorithm which addresses the problem of clustering "mutated" substrings of similar parent strings such that each substring is correctly assigned to its parent string. This problem is motivated by the process of simultaneously reading similar RNA sequences during which various substrings of the sequence are produced and could be mutated; that is, a substring may have some letters changed during the reading process. Because the original RNA sequences are similar, a substring is likely to be assigned to the wrong original sequence. We describe our EM algorithm and present a test on a simulated benchmark which shows that our method yields a better assignment of the substrings than what has been achieved by previous methods. We conclude by discussing how this assignment problem applies to RNA structure prediction.
The classical isoperimetric inequality states that in Euclidean space spheres form the least perimeter enclosures for any give volume. We will review the historic development of this result in mathematics, and various approaches to proving it. Then we will discuss how one of these approaches, which is a variational argument, may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, in order to generalize the isoperimetric inequality.
Stephen Smale’s h-cobordism Theorem was a landmark result in the classification of smooth manifolds. It paved the way towards solutions for the topological Poincaré and Schoenflies conjectures in dimensions greater than 5. Later, building on this, Freedman’s work applied these techniques to 4 manifolds. I shall discuss the ideas relating to h-cobordisms and the proof, which is a wonderful application of handlebody theory and the Whitney trick. Time permitting, we shall explore the world of smooth 4 manifolds further, and talk about cork twists.
We will show the sharp estimate on the behavior of the smallest singular value of random matrices under very general assumptions. One of the steps in the proof is a result about the efficient discretization of the unit sphere in an n-dimensional euclidean space. Another step involves the study of the regularity of the behavior of lattice sets. Some elements of the proof will be discussed. Based on the joint work with Tikhomirov and Vershynin.
The classical isoperimetric inequality states that in Euclidean space spheres provide unique enclosures of least perimeter for any given volume. In this talk we discuss how this inequality may be extended to spaces of nonpositive curvature, known as Cartan-Hadamard manifolds, as conjectured by Aubin, Gromov, Burago, and Zalgaller in 1970s and 80s. The proposed proof is based on a comparison formula for total curvature of level sets in Riemannian manifolds, and estimates for the smooth approximation of the signed distance function, via inf-convolution and Reilly type formulas among other techniques. Immediate applications include sharp extensions of Sobolev and Faber-Krahn inequalities to spaces of nonpositive curvature. This is joint work with Joel Spruck.
A graph is $k$-critical if its chromatic number is $k$ but any its proper subgraph has chromatic number less than $k$. Let $k\geq 4$. Gallai asked in 1984 if any $k$-critical graph on $n$ vertices contains at least $n$ distinct $(k-1)$-critical subgraphs. Improving a result of Stiebitz, Abbott and Zhou proved in 1995 that every such graph contains $\Omega(n^{1/(k-1)})$ distinct $(k-1)$-critical subgraphs. Since then no progress had been made until very recently, Hare resolved the case $k=4$ by showing that any $4$-critical graph on $n$ vertices contains at least $(8n-29)/3$ odd cycles. We mainly focus on 4-critical graphs and develop some novel tools for counting cycles of specified parity. Our main result shows that any $4$-critical graph on $n$ vertices contains $\Omega(n^2)$ odd cycles, which is tight up to a constant factor by infinite many graphs. As a crucial step, we prove the same bound for 3-connected non-bipartite graphs, which may be of independent interest. Using the tools, we also give a very short proof to the Gallai's problem for the case $k=4$. Moreover, we improve the longstanding lower bound of Abbott and Zhou to $\Omega(n^{1/(k-2)})$ for the general case $k\geq 5$. Joint work with Tianchi Yang.
Expander decomposition has been a central tool in designing graph algorithms in many fields (including fast centralized algorithms, approximation algorithms and property testing) for decades. Recently, we found that it also gives many impressive applications in dynamic graph algorithms and distributed graph algorithms. In this talk, I will survey these recent results based on expander decomposition, explain the key components for using this technique, and give some toy examples on how to apply these components.
In 1959 Mark Kac introduced a simple model for the evolution
of a gas of hard spheres undergoing elastic collisions. The main
simplification consisted in replacing deterministic collisions with
random Poisson distributed collisions.
It is possible to obtain many interesting results for this simplified
dynamics, like estimates on the rate of convergence to equilibrium and
validity of the Boltzmann equation. The price paid is that this system
has no space structure.
I will review some classical results on the Kac model and report on an
attempt to reintroduce some form of space structure and non-equilibrium
evolution in a way that preserve the mathematical tractability of the
system.
Foundation is a powerful tool to understand the representability of matroids. The foundation of a matroid is a pasture which is an algebraic structure genrealize the field. I will briefly introduce matroids, algebraic structures (especially pastures) and matroid representability. I will also give some examples on how foundation works in representation of matroids.
We present a multiscale modeling and computational scheme for optical-
mechanical responses of nanostructures. The multi-physical nature of
the problem is a result of the interaction between the electromagnetic
(EM) field, the molecular motion, and the electronic excitation. To
balance accuracy and complexity, we adopt the semi-classical approach
that the EM field is described classically by the Maxwell equations,
and the charged particles follow the Schr ̈oidnger equations quantum
mechanically. To overcome the numerical challenge of solving the high
dimensional multi-component many- body Schr ̈odinger equations, we
further simplify the model with the Ehrenfest molecular dynamics to
determine the motion of the nuclei, and use the Time- Dependent
Current Density Functional Theory (TD-CDFT) to calculate the
excitation of the electrons. This leads to a system of coupled
equations that computes the electromagnetic field, the nuclear
positions, and the electronic current and charge densities
simultaneously. In the regime of linear responses, the resonant
frequencies initiating the out-of-equilibrium optical-mechanical
responses can be formulated as an eigenvalue problem. A
self-consistent multiscale method is designed to deal with the well
separated space scales. The isomerization of Azobenzene is presented as a numerical example.
It is a remarkable fact that some compact topological 4-manifolds X admit infinitely many exotic smooth structures, a phenomenon unique to dimension four. Indeed a fundamental open problem in the subject is to give a meaningful description of the set of all such structures on any given X. This talk will describe one approach to this problem when X is simply-connected, via cork twisting. First we'll sketch an argument to show that any finite list of smooth manifolds homeomorphic to X can be obtained by removing a single compact contractible submanifold (or cork) from X, and then regluing it by powers of a boundary diffeomorphism. In fact, allowing the cork to be noncompact, the collection of all smooth manifolds homeomorphic to X can be obtained in this way. If time permits, we will also indicate how to construct a single universal noncompact cork whose twists yield all smooth closed simply-connected 4-manifolds. This is joint work with Hannah Schwartz.
This presentation reviews different concepts of solution of a differential equation, in particular stressing the need to modify the classical theory when we want to deal with discontinuous systems. We will review the concept of classical solution, and then of Caratheodory solution and Filippov solution, motivating with simple examples the need for these extensions.
We will discuss a deterministic, polynomial (in the rank) time approximation algorithm for counting the bases of a given matroid and for counting common bases between two matroids of the same rank. This talk follows the paper (https://arxiv.org/abs/1807.00929) of Nima Anari, Shayan Oveis Gharan, and Cynthia Vinzant.
Inspired by the interval decomposition of persistence modules and the extended Newick format of phylogenetic networks, we show that, inside the larger category of partially ordered Reeb graphs, every Reeb graph with n leaves and first Betti number s, is equal to a coproduct of at most 2s trees with (n + s) leaves. An implication of this result, is that Reeb graphs are fixed parameter tractable when the parameter is the first Betti number. We propose partially ordered Reeb graphs as a natural framework for modeling time consistent phylogenetic networks. We define a notion of interleaving distance on partially ordered Reeb graphs which is analogous to the notion of interleaving distance for ordinary Reeb graphs. This suggests using the interleaving distance as a novel metric for time consistent phylogenetic networks.
NOTE THE UNUSUAL TIME: This seminar takes place from 1:10-1:50 for THIS WEEK ONLY.
Basin of attraction for a stable equilibrium point is an effective concept for stability in deterministic systems. However, it does not contain information on the external perturbations that may affect it. The concept of stochastic basin of attraction (SBA) is introduced by incorporating a suitable probabilistic notion of basin. The criteria for the size of the SBA is based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small intensity but also with whose amplitude is proportional or in general is a function of an order parameter. The efficiency of the concept is presented through two applications.
A sub-Riemannian manifold M is a connected smooth manifold such that the only smooth curves in M which are admissible are those whose tangent vectors at any point are restricted to a particular subset of all possible tangent vectors. Such spaces have several applications in physics and engineering, as well as in the study of hypo-elliptic operators. We will construct a random walk on M which converges to a process whose infinitesimal generator is one of the natural sub-elliptic Laplacian operators. We will also describe these Laplacians geometrically and discuss the difficulty of defining one which is canonical. Examples will be provided. This is a joint work with Tom Laetsch.
It is a classical theorem of Alexander that every closed oriented manifold is a piecewise linear branched covering of the sphere. In this talk, we will discuss some obstructions to realizing a manifold as a branched covering of the sphere if we require additional properties (like being a submanifold) on the branch set.
We will survey different methods of proving functional inequalities for hypoelliptic diffusions and the corresponding heat kernels. Some of these methods rely on geometric methods such as curvature-dimension inequalities (due to Baudoin-Garofalo), and some are probabilistic such as coupling, and finally some use structure theory and a Fourier transform on Lie groups. This is based on joint work with M. Asaad, F. Baudoin, B. Driver, T. Melcher, Ph. Mariano et al.
Consider the random surface given by the interface separating the plus and minus phases in a low-temperature Ising model in dimensions $d\geq 3$. Dobrushin (1972) famously showed that in cubes of side-length $n$ the horizontal interface is rigid, exhibiting order one height fluctuations above a fixed point.
We study the large deviations of this interface and obtain a shape theorem for its pillar, conditionally on it reaching an atypically large height. We use this to analyze the law of the maximum height $M_n$ of the interface: we prove that for every $\beta$ large, $M_n/\log n \to c_\beta$, and $(M_n - \mathbb E[M_n])_n$ forms a tight sequence. Moreover, even though this centered sequence does not converge, all its sub-sequential limits satisfy uniform Gumbel tail bounds. Based on joint work with Eyal Lubetzky.
Tangles capture a notion of high-connectivity in graphs which differs from $k$-connectivity. Instead of requiring that a small vertex set $X$ does not disconnect the graph $G$, a tangle “points” to the connected component of $G-X$ that contains most of the “highly connected part”. Developed initially by Robertson and Seymour in the graph minors project, tangles have proven to be a fundamental tool in studying the general structure of graphs and matroids. Tangles are also useful in proving that certain families of graphs satisfy an approximate packing-covering duality, also known as the Erd\H{o}s-P\'osa property. In this talk I will give a gentle introduction to tangles and describe some basic applications related to the Erd\H{o}s-P\'osa property.
Let X be an elliptic surface with a section defined over a number field. Specialization theorems by Néron and Silverman imply that the rank of the Mordell-Weil group of special fibers is at least equal to the MW rank of the generic fiber. We say that the rank jumps when the former is strictly large than the latter. In this talk, I will discuss rank jumps for elliptic surfaces fibred over the projective line. If the surface admits a conic bundle we show that the subset of the line for which the rank jumps is not thin in the sense of Serre. This is joint work with Dan Loughran.
While most evolutionary studies of host-pathogen dynamics consider pathogen evolution alone or host-pathogen coevolution, for some diseases (e.g., White Nose syndrome in bats), there is evidence that hosts can sometimes evolve more rapidly than their pathogen. In this talk, we will discuss the spatial, temporal, and epidemiological factors may drive the evolutionary dynamics of the host population. We consider a simplified system of two host genotypes that trade off factors of disease robustness and spatial mobility or growth. For diseases that infect hosts for life, we find that migration and disease-driven mortality can have antagonistic effect on host densities when disease selection on hosts is low, but show synergy when selection is high. For diseases that allow hosts to recover with immunity, we explore the conditions under which the disease dies out, becomes endemic, or has periodic outbreaks, and show how these dynamics relate to the relative success of the robust and wild type hosts in the population over time. Overall, we will discuss how combinations of host spatial structure, demography, and epidemiology of infectious disease can significantly influence host evolution and disease prevalence. We will conclude with some profound implications for wildlife conservation and zoonotic disease control.
The Torelli group is the subgroup of the mapping class group acting trivially on homology. We will discuss some basic properties of the Torelli group and explain how to define it for surfaces with boundary. We will also give some Torelli analogues of the Birman exact sequence.
Matroids are combinatorial gadgets that reflect properties of linear algebra in situations where this latter theory is not available. This analogy prescribes that the moduli space of matroids should be a Grassmannian over a suitable base object, which cannot be a field or a ring; in consequence usual algebraic geometry does not provide a suitable framework. In joint work with Matt Baker, we use algebraic geometry over F1, the so-called field with one element, to construct such moduli spaces. As an application, we streamline various results of matroid theory and find simplified proofs of classical theorems, such as the fact that a matroid is regular if and only if it is binary and orientable.
We will dedicate the first half of this talk to an introduction of matroids and their generalizations. Then we will outline how to use F1-geometry to construct the moduli space of matroids. In a last part, we will explain why this theory is so useful to simplify classical results in matroid theory.
A powerful method for analyzing graphs is to first apply regularity lemmas, which roughly state that one can partition the graph into a few parts so that it looks mostly random between the parts, and then apply probabilistic tools from there. The drawback of this approach is that it only works in general when the input graph is very dense: standard regularity lemmas are trivial already for n-node graphs on "only" <= n^{1.99} edges.
In this work we prove extensions of several standard regularity lemmas to sparse graphs, which are nontrivial so long as the graph spectrum is not too far from that of a random graph. We then apply our notion of "spectral pseudorandomness" to port several notable regularity-based results in combinatorics and theoretical computer science down to sparser graphs.
Joint work with Santosh Vempala.
High-dimensional inference problems such as sparse PCA and planted clique often exhibit statistical-vs-computational tradeoffs whereby there is no known polynomial-time algorithm matching the performance of the optimal estimator. I will discuss an emerging framework -- based on the so-called low-degree likelihood ratio -- for precisely predicting these tradeoffs and giving rigorous evidence for computational hardness in the conjectured hard regime. This method was originally proposed in a sequence of works on the sum-of-squares hierarchy, and the key idea is to study whether or not there exists a low-degree polynomial that succeeds at a given statistical task.
In the second part of the talk, I will give an application to the algorithmic problem of finding an approximate ground state of the SK (Sherrington-Kirkpatrick) spin glass model. I will explain two variants of this problem: "optimization" and "certification." While optimization can be solved in polynomial time [Montanari'18], we give rigorous evidence (in the low-degree framework) that certification cannot be. This result reveals a fundamental discrepancy between two classes of algorithms: local search succeeds while convex relaxations fail.
Based on joint work with Afonso Bandeira and Tim Kunisky (https://arxiv.org/abs/1902.07324 and https://arxiv.org/abs/1907.11636).
Breathers are periodic in time spatially localized solutions of evolutionary PDEs. They are known to exist for the sine-Gordon equation but are believed to be rare in other Klein-Gordon equations. Exchanging the roles of time and position, breathers can be interpreted as homoclinic solutions to a steady solution. In this talk, I will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solution when the steady solution has weakly hyperbolic one dimensional stable and unstable manifolds. Their distance is exponentially small with respect to the amplitude of the breather and therefore classical perturbative techniques cannot be applied. This is a joint work with O. Gomide, T. Seara and C. Zeng.
In this talk, I will introduce my old(1.) and current works(2.).
1. Bounds on regularity of quadratic monomial ideals
We can understand invariants of monomial ideals by invariants of clique (or flag) complex of corresponding graphs. In particular, we can bound the Castelnuovo-Mumford regularity (which is a measure of algebraic complexity) of the ideals by bounding homol0gy of corresponding (simplicial) complex. The construction and proof of our main theorem are simple, but it provides (and improves) many new bounds of regularities of quadratic monomial ideals.
2. Pythagoras numbers on projections of Rational Normal Curves
Observe that forms of degree $2d$ are quadratic forms of degree $d$. Therefore, to study the cone of sums of squares of degree $2d$, we may study quadratic forms on Veronese embedding of degree $d$. In particular, the rank of sums of squares (of degree $2d$) can be studied via Pythagoras number (which is a classical notion) on the Veronese embedding of degree $d$. In this part, I will compute the Pythagoras number on rational normal curve (which is a veronese embedding of $\mathbb{P}^1$) and discuss about how Pythagoras numbers are changed when we take some projections away from some points.
We will describe a twisted action of the symmetric group on the polynomial ring in n variables and use it to define a twisted version of Schubert polynomials. These twisted Schubert polynomials are known to be related to the Chern-Schwartz-MacPherson classes of Schubert cells in the flag variety. Using properties of skew divided difference operators, we will show that these twisted Schubert polynomials are monomial positive and give a combinatorial formula for their coefficients.
In this talk we present some recent results which allow to prove
instability in near integrable Hamiltonian systems. We will show how
these mechanisms are suitable to apply to concrete systems but also are
useful to obtain Arnold diffusion in a large set of Hamiltonian systems.
Gromov revolutionized the study of finitely generated groups by showing that an intrinsic metric on a group is intimately connected with the algebra of the group. This point of view has produced deep applications not only in group theory, but also topology, geometry, logic, and dynamical systems. We will start at the beginning of this story with the definitions of these metrics on groups and how notions from classical geometry can be generalized to this context. The focus will be on how the "hyperbolic groups" exhibit geometric and dynamical feature reminiscent of the hyperbolic plane and its isometries.
This talk is based on work in progress with Sara Lamboglia and Faye Simon. We study the tropical convex hull of convex sets and of tropical curves. Basic definitions of tropical convexity and tropical curves will be presented, followed by an overview of our results on the interaction between tropical and classical convexity. Lastly, we study a tropical analogue of an inequality bounding the degree of a projective variety in classical algebraic geometry; we show a tropical proof of this result for a special class of tropical curves.
While producing subgroups of a group by specifying generators is easy, understanding the structure of such a subgroup is notoriously difficult problem. In the case of hyperbolic groups, Gitik utilized a local-to-global property for geodesics to produce an elegant condition that ensures a subgroup generated by two elements (or more generally generated by two subgroups) will split as an amalgamated free product over the intersection of the generators. We show that the mapping class group of a surface and many other important groups have a similar local-to-global property from which an analogy of Gitik's result can be obtained. In the case of the mapping class group, this produces a combination theorem for the dynamically and topologically important convex cocompact subgroups. Joint work with Davide Spriano and Hung C. Tran.
We consider the strong chain recurrent set and the generalized recurrent set for continuous maps of compact metric spaces. Recent work by Fathi and Pageault has shown a connection between the two sets, and has led to new results on them. We discuss a structure theorem for transitive/mixing maps, and classify maps that permit explosions in the size of the recurrent sets.
As a geometric group theorist, my favorite type of manifold is a surface and my favorite way to study surfaces is by considering the mapping class group, which is the collection of symmetries of a surface. In this talk, we will think bigger than your average low-dimensional topologist and consider surfaces of infinite type and their associated “big” mapping class groups.
Mori Dream Spaces are generalizations of toric varieties and, as the name suggests, Mori's minimal model program can be run for every divisor. It is known that for n≥5, the blow-up of Pn at r very general points is a Mori Dream Space iff r≤n+3. In this talk we proceed to blow up points as well as lines, by considering the blow-up X of P3 at 6 points in very general position and all the 15 lines through the 6 points. We find that the unique anticanonical section of X is a Jacobian K3 Kummer surface S of Picard number 17. We prove that there exists an infinite-order pseudo-automorphism of X, whose restriction to S is one of the 192 infinite-order automorphisms constructed by Keum. A consequence is that there are infinitely many extremal effective divisors on X; in particular, X is not a Mori Dream Space. We show an application to the blow-up of Pn (n≥3) at (n+3) points and certain lines. We relate this pseudo-automorphism to the structure of the birational automorphism group of P3. This is a joint work with Lei Yang.
This is quick tutorial on bounding the mixing time of a finite Markov chain in terms of functional inequalities defining the spectral gap and the entropy constant of a Markov chain. The lecture will include some examples, including bounding the mixing time of the random transposition shuffle and (time permitting) that of the basis-exchange walk on a balanced matroid.
This is intended as a review lecture before Nima Anari’s lectures (during Nov. 4-6) on applications of Lorentzian polynomials, including recent breakthrough analyses of the basis-exchange walk on an arbitrary matroid.
I will present a new method of analysis for Einstein’s
constraint equations, referred to as the Seed-to-Solution Method, which
leads to the existence of asymptotically Euclidean manifolds with
prescribed asymptotic behavior. This method generates a (Riemannian)
Einstein manifold from any seed data set consisting of (1): a Riemannian
metric and a symmetric two-tensor prescribed on a topological manifold
with finitely many asymptotically Euclidean ends, and (2): a density
field and a momentum vector field representing the matter content. By
distinguishing between several classes of seed data referred to as tame
or strongly tame, the method encompasses metrics with the weakest
possible decay (infinite ADM mass) or the strongest possible decay
(Schwarzschild behavior). This analysis is based on a linearization of
the Einstein equations (involving several curvature operators from
Riemannian geometry) around a tame seed data set. It is motivated by
Carlotto and Schoen’s pioneering work on the so-called localization
problem for the Einstein equations. Dealing with manifolds with possibly
very low decay and establishing estimates beyond the critical level of
decay requires significantly new ideas to be presented in this talk. As
an application of our method, we introduce and solve a new problem,
referred to as the asymptotic localization problem, at the critical
level of decay. Collaboration with T. Nguyen. Blog: philippelefloch.org
A central pervasive challenge in genomics is that RNA/DNA must be reconstructed from short, often noisy subsequences. In this talk, we describe a new digraph algorithm which enables this "assembly" when analyzing high-throughput transcriptomic sequencing data. Specifically, the Flow Decomposition problem on a directed ayclic graph asks for the smallest set of weighted paths that “cover” a flow (a weight function on the edges where the amount coming into any vertex is equal to the amount leaving). We describe a new linear-time algorithm solving *k*-Flow Decomposition, the variant where exactly *k* paths are used. Further, we discuss how we implemented and engineered a general Flow Decomposition solver based on this algorithm, and describe its performance on RNA-sequence data. Crucially, our solver finds exact solutions while achieving runtimes competitive with a state-of-the-art heuristic, and we discuss the implications of our results on the original model selection for transcript assembly in this setting.
In the first part of this talk, I will give an overview of a theory of harmonic analysis on a class of fractals that includes the Sierpinski gasket. The starting point of the theory is the introduction by J. Kigami of a Laplacian operator on these fractals. After reviewing the construction of this fractal Laplacian, I will survey some of the properties of its spectrum. In the second part of the talk, I will discuss the fractal analogs of the Heisenberg uncertainty principle, and the spectral properties a class of Schr\"odinger operators.
Which manifold can be obtained from surgery on a knot? Many obstructions to this have been studied. We will discuss some of them, and use Heegaard Floer homology to give an infinite family of seifert fibered integer spheres that cannot be obtained by surgery on a knot in S^3. We will also discuss a recipe to compute HF+ of surgery on a knot (Short review on Heegaard Floer homology included).
Sampling is a fundamental algorithmic task. Many modern applications require sampling from complicated probability distributions in high-dimensional spaces. While the setting of logconcave target distribution is well-studied, it is important to understand sampling beyond the logconcavity assumption. We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution on R^n under isoperimetry conditions. We show a convergence guarantee in Kullback-Leibler (KL) divergence assuming the target distribution satisfies log-Sobolev inequality and the log density has bounded Hessian. Notably, we do not assume convexity or bounds on higher derivatives. We also show convergence guarantees in Rényi divergence assuming the limit of ULA satisfies either log-Sobolev or Poincaré inequality. Joint work with Santosh Vempala (arXiv:1903.08568).
Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and Jorgensen's conjecture, Robertson and Seymour asked when does $G$ contain disjoint connected subgraphs $G_1, G_2$, such that $\{a_0, a_1, a_2\}\subseteq V(G_1)$ and $\{b_1, b_2\}\subseteq V(G_2)$. We prove that if $G$ is 6-connected then such $G_1,G_2$ exist. Joint work with Robin Thomas and Xingxing Yu.
Advisor: Dr. Xingxing Yu (School of Mathematics, Georgia Institute of Technology)
Committee: Dr. Robin Thomas (School of Mathematics, Georgia Institute of Technology), Dr. Prasad Tetali (School of Mathematics, Georgia Institute of Technology), Dr. Lutz Warnke (School of Mathematics, Georgia Institute of Technology), Dr. Richard Peng (School of Computer Science, Georgia Institute of Technology)
Reader: Dr. Gexin Yu (Department of Mathematics, College of William and Mary)
Everybody are convinced that everything is known about the simplest random process of coin tossing. I will show that it is not the case. Particularly not everything was known for the simplest chaotic dynamical systems like the tent map (which is equivalent to coin tossing). This new area of finite time predictions emerged from a natural new question in the theory of open dynamical systems.
We propose a Markov chain Monte Carlo (MCMC) algorithm based on third-order Langevin dynamics for sampling from distributions with log-concave and smooth densities. The higher-order dynamics allow for more flexible discretization schemes, and we develop a specific method that combines splitting with more accurate integration. For a broad class of d-dimensional distributions arising from generalized linear models, we prove that the resulting third-order algorithm produces samples from a distribution that is at most \varepsilon in Wasserstein distance from the target distribution in O(d^{1/3}/ \varepsilon^{2/3}) steps. This result requires only Lipschitz conditions on the gradient. For general strongly convex potentials with α-th order smoothness, we prove that the mixing time scales as O (d^{1/3} / \varepsilon^{2/3} + d^{1/2} / \varepsilon^{1 / (\alpha - 1)} ). Our high-order Langevin diffusion reduces the problem of log-concave sampling to numerical integration along a fixed deterministic path, which makes it possible for further improvements in high-dimensional MCMC problems. Joint work with Yi-An Ma, Martin J, Wainwright, Peter L. Bartlett and Michael I. Jordan.
Rank-structured matrix representations, e.g., H2 and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their applications is the expensive computation required to represent a matrix in a rank-structured matrix format which involves compressing specific matrix blocks into low-rank form.
We focus on the study and application of a class of hybrid analytic-algebraic compression methods, called the proxy point method. We address several critical problems concerning this underutilized method which limit its applicability. A general form of the method is proposed, paving the way for its wider application in the construction of different rank-structured matrices with kernel functions that are more general than those usually used. Further, we extend the applicability of the proxy point method to compress matrices defined by electron repulsion integrals, which accelerates one of the main computational steps in quantum chemistry.
Committee members: Prof. Edmond Chow (Advisor, School of CSE, Georgia Tech), Prof. David Sherrill (School of Chemistry and Biochemistry, Georgia Tech), Prof. Jianlin Xia (Department of Mathematics, Purdue University), Prof. Yuanzhe Xi (Department of Mathematics, Emory University), and Prof. Haomin Zhou (School of Mathematics, Georgia Tech).
In this talk, we will focus on the spin dynamics of rigid bodies.
Algorithm part: There are many algorithms designed for N body simulations.
But, to study the climates of a planet, we need to extend the simulation from point mass bodies to rigid bodies.
In the N-rigid-body simulations, we will consider the orientation and angular momentum of the rigid body to understand the spin.
In terms of the algorithm, symplectic integrators are designed by splitting methods.
Physical part: We studied the spin dynamics of an Earth-like planet in circumbinary systems.
Canonical Delaunay variables and Andoyer variables are applied to split the variables to be slow part and fast part.
Applying averaging method, we approximated the spin dynamics.
From the approximated dynamics, we may draw some interesting physical conclusions.
A fundamental question in Dynamical Systems is to identify regions of
phase/parameter space satisfying a given property (stability,
linearization, etc). In this talk, given a family of analytic circle
diffeomorphisms depending on a parameter, we obtain effective (almost
optimal) lower bounds of the Lebesgue measure of the set of parameters
for which that diffeomorphism is conjugate to a rigid rotation.
We estimate this measure using an a-posteriori KAM
scheme that relies on quantitative conditions that
are checkable using computer-assistance. We carefully describe
how the hypotheses in our theorems are reduced to a finite number of
computations, and apply our methodology to the case of the
Arnold family, in the far-from-integrable regime.
This is joint work with Jordi Lluis Figueras and Alejandro Luque.
Heegaard Floer homology gives a powerful invariant of closed 3-manifolds. It is always computable in the purely combinatorial sense, but usually computing it needs a very hard work. However, for certain graph 3-manifolds, its minus-flavored Heegaard Floer homology can be easily computed in terms of lattice homology, due to Nemethi. I plan to give the basic definitions and constructions of Heegaard Floer theory and lattice homology, as well as the isomorphism between those two objects.
By using the representation theory of the symmetric group we try to compare, with respect to two different bases of the vector space of symmetric forms, the cones of symmetric nonnegative forms and symmetric sums of squares of a fixed even degree when the number of variables goes to infinity.
Using the covering involution on the double branched cover of S3 branched along a knot, and adapting ideas of Hendricks-Manolescu and Hendricks-Hom-Lidman, we define new knot (concordance) invariants and apply them to deduce novel linear independence results in the smooth concordance group of knots. This is a joint work with A. Alfieri and A. Stipsicz.
Given a graph X and a group G, a G-cover of X is a morphism of graphs X’ --> X together with an invariant G-action on X’ that acts freely and transitively on the fibers. G-covers are classified by their monodromy representations, and if G is a finite abelian group, then the set of G-covers of X is in natural bijection with the first simplicial cohomology group H1(X,G).
In tropical geometry, we are naturally led to consider more general objects: morphisms of graphs X’ --> X admitting an invariant G-action on X’, such that the induced action on the fibers is transitive, but not necessarily free. A natural question is to classify all such covers of a given graph X. I will show that when G is a finite abelian group, a G-cover of a graph X is naturally determined by two data: a stratification S of X by subgroups of G, and an element of a cohomology group H1(X,S) generalizing the simplicial cohomology group H1(X,G). This classification can be viewed as a tropical version of geometric class field theory, and as an abelianization of Bass--Serre theory.
I will discuss the realizability problem for tropical abelian covers, and the relationship between cyclic covers of a tropical curve C and the corresponding torsion subgroup of Jac(C). The realizability problem for cyclic covers of prime degree turns out to be related to the classical nowhere-zero flow problem in graph theory.
Joint work with Yoav Len and Martin Ulirsch.
Usual statistical inference techniques for the tree of life like maximum likelihood and bayesian inference through Markov chain Monte Carlo (MCMC) have been widely used, but their performance declines as the datasets increase (in number of genes or number of species).
I will present two new approaches suitable for big data: one, importance sampling technique for bayesian inference of phylogenetic trees, and two, a pseudolikelihood method for inference of phylogenetic networks.
The proposed methods will allow scientists to include more species into the tree of life, and thus complete a broader picture of evolution.
One of the simplest and, at the same time, most prominent models for the discrete quasi-periodic Schrodinger operator is the almost Mathieu operator (also called the Harper's model). This simple-looking operator is known to present exotic spectral properties. Three (out of fifteen) of Barry Simon's problems on Schrodinger operators in the 21st century concerns the almost Mathieu operator. In 2014, Artur Avila won a Fields Medal for work including the solutions to these three problems. In this talk, I will concentrate on the one concerning the Lebesgue measure of the spectrum. I will also talk about the difficulties in generalizing this result to the extended Harper's model. Students with background in numerics are especially welcome to attend!
That the ball minimizes surface area among all sets of fixed volume, was known since antiquity; this is equivalent to the fact that the ball is the unique set which yields equality in the isoperimetric inequality. But the isoperimetric inequality is only a very special case of quadratic inequalities about mixed volumes of convex bodies, whose equality cases were unknown since the time of Minkowski. This talk is about these quadratic inequalities and their unusual equality cases which we resolved using degenerate diffusions on the sphere. No background in geometry will be assumed. Joint work with Ramon van Handel.
The talk will discuss the relationship between topology and
geometry of Einstein 4-manifolds such as K3 surfaces.
The Ehrhard-Borell inequality stands at the top of the pyramid of Gaussian inequalities. It is a powerful and delicate statement about the convexity of the Gaussian measure. In this talk I will discuss the inequality and its beautiful proof by Borell. The delicate nature of the inequality however makes the characterization of the equality cases difficult and they were left unknown. I will explain how we solved this problem. Joint work with Ramon van Handel.
This is a talk about 3-manifolds and knots. We will begin by reviewing some basic constructions and motivations in low-dimensional topology, and will then introduce the homology cobordism group, the group of 3-manifolds with the same homology as the 3-dimensional sphere up to a reasonable notion of equivalence. We will discuss what is known about the structure of this group and its connection to higher dimensional topology. We will then discuss some existing invariants of the homology cobordism group coming from gauge theory and symplectic geometry, particularly Floer theory. Finally, we will introduce a new invariant of homology cobordism coming from an equivariant version of the computationally-friendly Floer-theoretic 3-manifold invariant Heegaard Floer homology, and use it to construct a new filtration on the homology cobordism group and derive some structural applications. Parts of this talk are joint work with C. Manolescu and I. Zemke; more recent parts of this talk are joint work with J. Hom and T. Lidman.
I will introduce a minimum l-degree threshold for the existence of a nearly perfect (i.e., covering all but a constant number of vertices) matching in a k-graph where k ≥ 3 and k/2 < l ≤ k − 1. This is joint work with Hongliang Lu and Xingxing Yu.
This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l ≤ k − 1. In some cases, such a matching can in fact be near perfect (i.e., covering all but at most k vertices) and our bound on the minimum l-degree is best possible.
Let X be the number of length 3 arithmetic progressions in a random subset of Z/101Z. Does X take the values 630 and 640 with roughly the same probability?
Let Y denote the number of triangles in a random graph on n vertices. Despite looking similar to X, the local distribution of Y is quite different, as Y obeys a local limit theorem.
We will talk about a method for distinguishing when combinatorial random variables obey local limit theorems and when they do not.
Considering SL(2,R) skew-product maps over circle rotations,
we prove that a renormalization transformation
associated with the golden mean alpha
has a nontrivial periodic orbit of length 3.
We also present some numerical results,
including evidence that this period 3 describes
scaling properties of the Hofstadter butterfly
near the top of the spectrum at alpha,
and scaling properties of the generalized eigenfunction
for this energy.
For an $n\times n$ matrix $A_n$, the $r\to p$ operator norm is defined as $\|A_n\|_{r \to p}= \sup_{\|x\|_r\leq 1 } \|A_n x\|_p$ for $r,p\geq 1$. The $r\to p$ operator norm puts a huge number of important quantities of interest in diverse disciplines under a single unified framework. The application of this norm spans a broad spectrum of areas including data-dimensionality reduction in machine learning, finding oblivious routing schemes in transportation network, and matrix condition number estimation.
In this talk, we will consider the $r\to p$ norm of a class of symmetric random matrices with nonnegative entries, which includes the adjacency matrices of the Erd\H{o}s-R\'enyi random graphs and matrices with sub-Gaussian entries. For $1< p\leq r< \infty$, we establish the asymptotic normality of the appropriately centered and scaled $\|A_n\|_{r \to p}$, as $n\to\infty$. The special case $r=p=2$, which corresponds to the largest singular value of matrices, was proved in a seminal paper by F\"uredi and Koml\'os (1981). Of independent interest, we further obtain a sharp $\ell_\infty$-approximation for the maximizer vector. The results also hold for sparse matrices and further the $\ell_\infty$-approximation for the maximizer vector also holds for a broad class of deterministic sequence of matrices with certain asymptotic `expansion' properties.
This is based on a joint work with Souvik Dhara (MIT) and Kavita Ramanan (Brown U.).
I’ll try and give some background on the definition of knot Floer homology, and perhaps also bordered Heegaard Floer homology if time permits.
The analysis and decomposition of nonstationary and nonlinear signals in the quest for the identification
of hidden quasiperiodicities and trends is of high theoretical and applied interest nowadays.
Linear techniques like Fourier and Wavelet Transform, historically used in signal processing, cannot capture
completely nonlinear and non stationary phenomena.
For this reason in the last few years new nonlinear methods have been developed like the groundbreaking
Empirical Mode Decomposition algorithm, aka Hilbert--Huang Transform, and the Iterative Filtering technique.
In this seminar I will give an overview of this kind of methods and I will introduce two new algorithms,
the Fast Iterative Filtering and the Adaptive Local Iterative Filtering. I will review the main theoretical results
and outline the most intriguing open problems that still need to be tackled in the field.
Some examples of applications of these techniques to both artificial and real life signals
will be shown to give a foretaste of their potential and robustness.
‘Koszul duality’ is a phenomenon which algebraists are fond of, and has previously been studied in the context of '(bordered) Heegaard Floer homology' by Lipshitz, Ozsváth and Thurston. In this talk, I shall discuss an occurrence of Koszul duality which links older constructions in Heegaard Floer homology with the bordered Heegaard Floer homology of three-manifolds with torus boundary. I shan’t assume any existing knowledge of Koszul duality or any form of Heegaard Floer homology.
The Probabilistic Method is a powerful tool for tackling many problems in discrete mathematics and related areas.
Roughly speaking, its basic idea can be described as follows. In order to prove existence of a combinatorial structure with certain properties, we construct an appropriate probability space, and show that a randomly chosen element of this space has the desired property with positive probability.
In this talk we shall give a gentle introduction to the Probabilistic Method, with an emphasis on examples.
We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. Using the tropical Torelli theorem (due to Caporaso and Viviani), this characterization may be phrased in terms of 3-edge connectiviations. We show that being of hyperelliptic type is independent of the edge lengths and is preserved when passing to genus ≥2 connected minors. The main result is an excluded minors characterization of tropical curves of hyperelliptic type.
A fundamental tool used in sampling, counting, and inference problems is the Markov Chain Monte Carlo method, which uses random walks to solve computational problems. The main parameter defining the efficiency of this method is how quickly the random walk mixes (converges to the stationary distribution). The goal of these talks is to introduce a new approach for analyzing the mixing time of random walks on high-dimensional discrete objects. This approach works by directly relating the mixing time to analytic properties of a certain multivariate generating polynomial. As our main application we will analyze basis-exchange random walks on the set of bases of a matroid. We will show that the corresponding multivariate polynomial is log-concave over the positive orthant, and use this property to show three progressively improving mixing time bounds: For a matroid of rank r on a ground set of n elements:
- We will first show a mixing time of O(r^2 log n) by analyzing the spectral gap of the random walk (based on related works on high-dimensional expanders).
- Then we will show a mixing time of O(r log r + r log log n) based on the modified log-sobolev inequality (MLSI), due to Cryan, Guo, Mousa.
- We will then completely remove the dependence on n, and show the tight mixing time of O(r log r), by appealing to variants of well-studied notions in discrete convexity.
Time-permitting, I will discuss further recent developments, including relaxed notions of log-concavity of a polynomial, and applications to further sampling/counting problems.
Based on joint works with Kuikui Liu, Shayan OveisGharan, and Cynthia Vinzant.
We study the mean field limit of large stochastic systems of interacting particles. To treat more general and singular kernels, we propose a modulated free energy combination of the method that we had previously developed and of the modulated energy introduced by S. Serfaty. This modulated free energy may be understood as introducing appropriate weights in the relative entropy to cancel the most singular terms involving the divergence of the flow. Our modulated free energy allows to treat singular potentials which combine large smooth part, small attractive singular part and large repulsive singular part. As an example, a full rigorous derivation (with quantitative estimates) of some chemotaxis models, such as Patlak-Keller-Segel system in the subcritical regimes, is obtained. This is a joint work with D. Bresch and Z. Wang.
A graphical model encodes conditional independence relations among random variables. For an undirected graph these conditional independence relations are represented by a simple polytope known as the graph associahedron, which is a Minkowski sum of standard simplices. We prove that there are analogous polytopes for a much larger class of graphical models. We construct this polytope as a Minkowski sum of matroid polytopes. The motivation came from the problem of learning Bayesian networks from observational data. No background on graphical models will be assumed for the talk. This is a joint work with Fatemeh Mohammadi, Caroline Uhler, and Charles Wang.
(This is Part 2, continuation of Tuesday's lecture.)
A fundamental tool used in sampling, counting, and inference problems is the Markov Chain Monte Carlo method, which uses random walks to solve computational problems. The main parameter defining the efficiency of this method is how quickly the random walk mixes (converges to the stationary distribution). The goal of these talks is to introduce a new approach for analyzing the mixing time of random walks on high-dimensional discrete objects. This approach works by directly relating the mixing time to analytic properties of a certain multivariate generating polynomial. As our main application we will analyze basis-exchange random walks on the set of bases of a matroid. We will show that the corresponding multivariate polynomial is log-concave over the positive orthant, and use this property to show three progressively improving mixing time bounds: For a matroid of rank r on a ground set of n elements:
- We will first show a mixing time of O(r^2 log n) by analyzing the spectral gap of the random walk (based on related works on high-dimensional expanders).
- Then we will show a mixing time of O(r log r + r log log n) based on the modified log-sobolev inequality (MLSI), due to Cryan, Guo, Mousa.
- We will then completely remove the dependence on n, and show the tight mixing time of O(r log r), by appealing to variants of well-studied notions in discrete convexity.
Time-permitting, I will discuss further recent developments, including relaxed notions of log-concavity of a polynomial, and applications to further sampling/counting problems.
Based on joint works with Kuikui Liu, Shayan OveisGharan, and Cynthia Vinzant.
A real matrix is called orthostochastic if it is the entrywise square of an orthogonal matrix. These matrices have been shown to be deeply connected to determinantal representations of polynomials, and also arise naturally in physics. However, the equations defining the real variety are known only up to the 3x3 case. I will show how various techniques of numerical algebraic geometry give a way of finding (set-theoretic) defining equations for the 4x4 orthostochastic variety, which are smaller (both in number and degree) than the naive equations one might initially guess. Based on joint work with Papri Dey.
Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder's inequality or Young's convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts or the multilinear Hilbert transform. We survey some results in the area.
The emergent shape of a knitted fabric is highly sensitive to the underlying stitch pattern. Here, by a stitch pattern we mean a periodic array of symbols encoding a set of rules or instructions performed to produce a swatch or a piece of fabric. So, it is crucial to understand what exactly these instructions mean in terms of mechanical moves performed using a yarn (a smooth piece of string) and a set of knitting needles (oriented sticks). Motivated by the fact that locally every knitting move results in a slip knot, we use tools from topology to model the set of all doubly periodic stitch patterns, knittable & non-knittable, as knots & links in a three manifold. Specifically, we define a map from the set of doubly-periodic stitch patterns to the set of links in S^3 and use link invariants such as the linking number, multivariable Alexander polynomial etc. to characterize them. We focus on such links derived from knitted stitch patterns in an attempt to tackle the question: whether or not a given stitch pattern can be realized through knitting.
We will discuss a novel approach to obtaining non-asymptotic estimates on the lower tail of the least singular value of an $n \times n$ random matrix $M_{n} := M + N_{n}$, where $M$ is a fixed matrix with operator norm at most $O(\exp(n^{c}))$ and $N_n$ is a random matrix, each of whose entries is an independent copy of a random variable with mean 0 and variance 1. This has been previously considered in a series of works by Tao and Vu, and our results improve upon theirs in two ways:
(i) We are able to deal with $\|M\| = O(\exp(n^{c}))$ whereas previous work was applicable for $\|M\| = O(\poly(n))$.
(ii) Even for $\|M\| = O(poly(n))$, we are able to extract more refined information – for instance, our results show that for such $M$, the probability that $M_n$ is singular is $O(exp(-n^{c}))$, whereas even in the case when $N_n$ is an i.i.d. Bernoulli matrix, the results of Tao and Vu only give inverse polynomial singularity probability.
When Alice wants to send a k-bits message v to Bob over a noisy channel, she encodes it as a longer n-bits message Mv, where M is a n times k matrix over F_2. The minimal distance d_M of the linear code M is defined as the minimum Hamming distance between Mw and Mu over all distinct points w,u in F_2^k. In this way, if there are less than d_M/2 corrupted bits in the message, Bob can recover the original message via a nearest neighbor search algorithm.
The classical Gilbert-Varshamov Bound provides a lower bound for d_M if the columns of M are independent copies of X, where X is the random vector uniformly distributed on F_2^n. Under the same assumption on M, we show that the distribution of d_M is essentially the same as the minimum of Hamming weight (Hamming distance to origin) of 2^k-1 i.i.d copies of X.
The result is surprising since M is only generated by k independent copies of X. Furthermore, our results also work for arbitrary finite fields.
This is joint work with Jing Hao, Galyna Livshyts, Konstantin Tikhomirov.
I will talk about algorithms (with unlimited computational power) which adaptively probe pairs of vertices of a graph to learn the presence or absence of edges and whose goal is to output a large clique. I will focus on the case of the random graph G(n,1/2), in which case the size of the largest clique is roughly 2\log(n). Our main result shows that if the number of pairs queried is linear in n and adaptivity is restricted to finitely many rounds, then the largest clique cannot be found; more precisely, no algorithm can find a clique larger than c\log(n) where c < 2 is an explicit constant. I will also discuss this question in the planted clique model. This is based on joint works with Uriel Feige, David Gamarnik, Joe Neeman, Benjamin Schiffer, and Prasad Tetali.
This talk is based on a paper by Grigoriy Blekherman. In most cases, nonnegative polynomials differ from positive polynomials. We will discuss precisely what equations cause these differences, and relate them to the well known Cayley-Bacharach theorem for low degree polynomials.
We discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via an active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations, in a two dimensional open bounded and connected domain. We consider the velocity field steered by a control input that acts tangentially on the boundary of the domain through the Navier slip boundary conditions. This is motivated by mixing within a cavity or vessel by moving the walls or stirring at the boundaries. Our main objective is to design an optimal Navier slip boundary control that optimizes mixing at a given final time. Non-dissipative scalars, both passive and active, governed by the transport equation will be discussed. In the absence of diffusion, transport and mixing occur due to pure advection. This essentially leads to a nonlinear control problem of a semi-dissipative system. We shall provide a rigorous proof of the existence of an optimal controller, derive the first-order necessary conditions for optimality, and present some preliminary results on the numerical implementation.
One strategy for developing a proof of a claimed theorem is to start by understanding what a counter-example should look like. In this talk, we will discuss a few recent results in harmonic analysis that utilize a quantitative version of this approach. A key step is the solution of an inverse problem with the following flavor. Let $T:X \to Y$ be a bounded linear operator and let $0 < a \leq \|T\|$. What can we say about those functions $f \in X$ obeying the reverse inequality $\|Tf\|_Y \geq a\|f\|_X$?
A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points. Stable polynomials have close relations to matroids and hyperbolic programming. We will discuss a generalization of stability to algebraic varieties of codimension larger than one. They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.
I will continue to describe deterministic algorithms for approximately counting common bases of matroids within an exponential factor. This is based on AOVI and previous works of AO.
I will discuss an ongoing project to reconstruct a gene network from time-series data from a mammalian signaling pathway. The data is generated from gene knockouts and the techniques involve computational algebra. Specifically, one creates an pseudomonomial "ideal of non-disposable sets" and applies a analogue of Stanley-Reisner theory and Alexander duality to it. Of course, things never work as well in practice, due to issue such as noise, discretization, and scalability, and so I will discuss some of these challenges and current progress.
Starting from mathematical approaches for image processing, we will discuss different models, analytic aspects of them, and numerical challenges. If time permits we will consider numerical applications to data understanding. A few other applications may be presented.
One of the outstanding open problems of statistical mechanics is about the hard-core model which is a popular topic in mathematical physics and has applications in a number of other disciplines. Namely, do non-overlapping hard disks of the same diameter in the plane admit a unique Gibbs measure at high density? It seems natural to approach this question by requiring the centers to lie in a fine lattice; equivalently, we may fix the lattice, but let the Euclidean diameter D of the hard disks tend to infinity. In two dimensions, it can be a unit triangular lattice A_2 or a unit square lattice Z^2. The randomness is generated by Gibbs/DLR measures with a large value of fugacity which corresponds to a high density. We analyze the structure of high-density hard-core Gibbs measures via the Pirogov-Sinai theory. The first step is to identify periodic ground states, i.e., maximal-density disk configurations which cannot be locally `improved'. A key finding is that only certain `dominant' ground states, which we determine, generate nearby Gibbs measures. Another important ingredient is the Peierls bound separating ground states from other admissible configurations. In particular, number-theoretic properties of the exclusion diameter D turn out to be important. Answers are provided in terms of Eisenstein primes for A_2 and norm equations in the cyclotomic ring Z[ζ] for Z^2, where ζ is the primitive 12th root of unity. Unlike most models in statistical physics, we find non-universality: the number of high-density hard-core Gibbs measures grows indefinitely with D but
non-monotonically. In Z^2 we also analyze the phenomenon of 'sliding' and show it is rare.
This is a joint work with A. Mazel and Y. Suhov.
Let X be a random variable taking values in {0,...,n} and f(z) be its probability generating function. Pemantle conjectured that if the variance of X is large and f has no roots close to 1 in the complex plane, then X must be approximately normal. We will discuss a complete resolution of this conjecture in a strong quantitative form, thereby giving the best possible version of a result of Lebowitz, Pittel, Ruelle and Speer. Additionally, if f has no roots with small argument, then X must be approximately normal, again in a sharp quantitative form. These results also imply a multivariate central limit theorem that answers a conjecture and completes a program of Ghosh, Liggett and Pemantle. This talk is based on joint work with Julian Sahasrabudhe.
In this talk, we provide the details of our faster width-dependent algorithm for mixed packing-covering LPs. Mixed packing-covering LPs are fundamental to combinatorial optimization in computer science and operations research. Our algorithm finds a $1+\eps$ approximate solution in time $O(Nw/ \varepsilon)$, where $N$ is number of nonzero entries in the constraint matrix, and $w$ is the maximum number of nonzeros in any constraint. This algorithm is faster than Nesterov's smoothing algorithm which requires $O(N\sqrt{n}w/ \eps)$ time, where $n$ is the dimension of the problem. Our work utilizes the framework of area convexity introduced in [Sherman-FOCS’17] to obtain the best dependence on $\varepsilon$ while breaking the infamous $\ell_{\infty}$ barrier to eliminate the factor of $\sqrt{n}$. The current best width-independent algorithm for this problem runs in time $O(N/\eps^2)$ [Young-arXiv-14] and hence has worse running time dependence on $\varepsilon$. Many real life instances of mixed packing-covering problems exhibit small width and for such cases, our algorithm can report higher precision results when compared to width-independent algorithms. As a special case of our result, we report a $1+\varepsilon$ approximation algorithm for the densest subgraph problem which runs in time $O(md/ \varepsilon)$, where $m$ is the number of edges in the graph and $d$ is the maximum graph degree.
We report on the discovery of a general principle leading to the unexpected cancellation of oscillating sums. It turns out that sums in the
class we consider are much smaller than would be predicted by certain probabilistic heuristics. After stating the motivation, and our theorem,
we apply it to prove a number of results on integer partitions, the distribution of prime numbers, and the Prouhet-Tarry-Escott Problem. For example, we prove a "Pentagonal Number Theorem for the Primes", which counts the number of primes (with von Mangoldt weight) in a set of intervals very precisely. In fact the result is stronger than one would get using a strong form of the Prime Number Theorem and also the Riemann Hypothesis (where one naively estimates the \Psi function on each of the intervals; however, a less naive argument can give an improvement), since the widths of the intervals are smaller than \sqrt{x}, making the Riemann Hypothesis estimate "trivial".
Based on joint work with Ernie Croot.
The topological entropy of a subshift is the exponential growth rate of the number of words of different lengths in its language. For subshifts of entropy zero, finer growth invariants constrain their dynamical properties. In this talk we will survey how the complexity of a subshift affects properties of the ergodic measures it carries. In particular, we will see some recent results (joint with B. Kra) relating the word complexity of a subshift to its set of ergodic measures as well as some applications.
The Bergman fan is a tropical linear space with trivial valuations describing a matroid combinatorially as it corresponds to a matroid. In this talk, based on a plenty of examples, we study the definition of the Bergman fan and their subdivisions. The talk will be closed with the recent result of the Maclagan-Yu's paper (https://arxiv.org/abs/1908.05988) that the fine subdivision of the Bergman fan of any matroid is r-1 connected where r is the rank of the matroid.
The talk is concerned with low multilinear rank approximations to antisymmetric tensors, that is, multivariate arrays for which the entries change sign when permuting pairs of indices. Such tensors play a major role in quantum chemistry. We show which ranks can be attained by an antisymmetric tensor and discuss the adaption of existing approximation algorithms to preserve antisymmetry, most notably a Jacobi-type algorithm. Particular attention is paid to the special case when choosing the rank equal to the order of the tensor. It is shown that this case can be addressed with an unstructured rank-1 approximation. This allows for the straightforward application of the higher-order power method, for which we discuss effective initialization strategies. This is a joint work with Daniel Kressner (EPFL).
I will outline the construction of some knot concordance invariants based on the Heegaard Floer homology of double branched coverings. The construction builds on some ideas developed by Hendricks, Manolescu, Hom and Lidman. This is joint work with Andras Stipsicz, and Sungkyung Kang.
Surfaces are some of the most basic examples of spaces. Although topologists have studied surfaces for a long time, they continue to fascinate. I'll give an overview of the study of surfaces over the past 150 years by highlighting work of seven mathematicians. We'll discuss the classification of surfaces, and we'll also discuss mapping class groups, which are collections of symmetries of surfaces. I'll also give the flavor of four of my own research projects about surfaces, one for each of four broad mathematical areas: group theory, geometry, topology, and dynamics.
Building on previous work of Rozansky and Willis, we generalise Rasmussen’s s-invariant to connected sums of $S^1 \times S^2$. Such an invariant can be computed by approximating the Khovanov-Lee complex of a link in $\#^r S^1 \times S^2$ with that of appropriate links in $S^3$. We use the approximation result to compute the s-invariant of a family of links in $S^3$ which seems otherwise inaccessible, and use this computation to deduce an adjunction inequality for null-homologous surfaces in a (punctured) connected sum of $\bar{CP^2}$. This inequality has several consequences: first, the s-invariant of a knot in the three-sphere does not increase under the operation of adding a null-homologous full twist. Second, the s-invariant cannot be used to distinguish $S^4$ from homotopy 4-spheres obtained by Gluck twist on $S^4$. We also prove a connected sum formula for the s-invariant, improving a previous result of Beliakova and Wehrli. We define two s-invariants for links in $\#^r S^1 \times S^2$. One of them gives a lower bound to the slice genus in $\natural^r S^1 \times B^3$ and the other one to the slice genus in $\natural^r D^2 \times S^2$ . Lastly, we give a combinatorial proof of the slice Bennequin inequality in $\#^r S^1 \times S^2$.
We show that the dynamics of nonlinear dynamical systems with many degrees of freedom (possibly infinitely many) can be similar to that of ordered system in a surprising fashion. To this aim, in the literature one typically uses techniques from perturbation theory, such as KAM theorem or Nekhoroshev theorem. Unfortunately they are known to be ill-suited for obtaining results in the case of many degrees of freedom. We present here a probabilistic approach, in which we focus on some observables of physical interest (obtained by averaging on the probability distribution on initial data) and for several models we get results of stability on long times similar to Nekhoroshev estimates. We present the example of a nonlinear chain of particles with alternating masses, an hyper-simplified model of diatomic solid. In this case, which is similar to the celebrated Fermi-Pasta-Ulam model and is widely studied in the literature, we show the progress with respect to previous results, and in particular how the present approach permits to obtain theorems valid in the thermodynamic limit, as this is of great relevance for physical implications.
In this talk I will give an introduction to certain aspects of geometric Littlewood-Paley theory, which is an area of harmonic analysis concerned with deriving regularity properties of sets and measures from the analytic behavior of associated operators. The work we shall describe has been carried out in collaboration with Fedor Nazarov, Maria Carmen Reguera, Xavier Tolsa, and Michele Villa.
Function classes are collections of Boolean functions on a finite set. Recently, a method of studying function classes via commutative algebra, by associating a squarefree monomial ideal to a function class, was introduced by Yang. I will describe this connection, as well as some free resolutions and Betti numbers for these ideals for an interesting collection of function classes, corresponding to intersection-closed posets. This is joint work with Chris Eur, Greg Yang, and Mengyuan Zhang.
We consider a physical periodic Ehrenfests' Wind-Tree model where a moving particle is a hard ball rather than (mathematical) point particle. Some dynamics and statistical properties of this model are studied. Moreover, it is shown that it has a new superdiffusive regime where the diffusion coefficient $D(t)\sim(\ln t)^2$ of dynamics seems to be never observed before in any model.
In this talk I'll first give an background overview of Bourgain's approach to prove the invariance of the Gibbs measure for the periodic cubic nonlinear Schrodinger equation in 2D and of the para-controlled calculus of Gubinelli-Imkeller and Perkowski in the context of parabolic stochastic equations. I will then present our resolution of the long-standing problem of proving almost sure global well-posedness (i.e. existence /with uniqueness/) for the periodic nonlinear Schrödinger equation (NLS) in 2D on the support of the Gibbs measure, for any (defocusing and renormalized) odd power nonlinearity. Consequently we get the invariance of the Gibbs measure. This is achieved by a new method we call /random averaging operators /which precisely captures the intrinsic randomness structure of the problematic high-low frequency interactions at the heart of this problem. This is work with Yu Deng (USC) and Haitian Yue (USC).
In both biological brains and artificial neural networks, the representational geometry - the shape and distribution of activity - at different layers in an artificial network or across different populations of neurons in the brain, can reveal important signatures of the underlying computations taking place. In this talk, I will describe how we are developing strategies for comparing and aligning neural representations, using a combination of tools from computational geometry and optimal transport.
Determining when two objects have “the same shape” is difficult; this difficulty depends on the dimension we are working in. While many of the same techniques work to study things in dimensions 5 and higher, we can better understand dimensions 1, 2, and 3 using other methods. We can think of 4-dimensional space as the “bridge” between low-dimensional behavior and high-dimensional behavior.
One way to understand the possibilities in each dimension is to examine objects called cobordisms: if an (n+1)-dimensional space has an ``edge,” which is called a boundary, then that boundary is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the boundary of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form an algebraic structure called a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions.
In this talk, I will discuss different types of cobordism groups and how to study them using tools from a broad range of mathematical areas.
I will discuss the prime decomposition of three-manifolds. First I will define the connect sum operation, irreducible and prime 3-manifolds. Then using the connect sum operation as "multiplication," I will show any closed oriented three-manifold decomposes uniquely into prime factors using spheres. If time permits, I will show another way of decomposing using discs.
The condition number of a matrix A is the quantity κ(A) = smax(A)/smin(A), where smax(A), smin(A) are the largest and smallest singular values of A, respectively. Let A be a random n × n matrix with i.i.d, mean zero, unit variance, subgaussian entries. We will discuss a result by Litvak, Tikhomirov and Tomczak-Jaegermann which states that, in this setting, the condition number satisfies the small ball probability estimate
P{κ(A) ≤ n/t} ≤ 2 exp(−ct^2), t ≥ 1, where c > 0 is a constant depending only on the subgaussian moment.
A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory.
Joint work with David Conlon.
A sequence A of positive integers is r-Ramsey complete if for every r-coloring of A, every sufficiently large integer can be written as a sum of the elements of a monochromatic subsequence. Burr and Erdos proposed several open problems in 1985 on how sparse can an r-Ramsey complete sequence be and which polynomial sequences are r-Ramsey complete. Erdos later offered cash prizes for two of these problems. We prove a result which solves the problems of Burr and Erdos on Ramsey complete sequences. The proof uses tools from probability, combinatorics, and number theory.
Joint work with David Conlon.
Fictitious play is one of the simplest and most natural dynamics for two-player zero-sum games. Originally proposed by Brown in 1949, the fictitious play dynamic has each player simultaneously best-respond to the distribution of historical plays of their opponent. In 1951, Robinson showed that fictitious play converges to the Nash Equilibrium, albeit at an exponentially-slow rate, and in 1959, Karlin conjectured that the true convergence rate of fictitious play after k iterations is O(k^{-1/2}), a rate which is achieved by similar algorithms and is consistent with empirical observations. Somewhat surprisingly, Daskalakis and Pan disproved a version of this conjecture in 2014, showing that an exponentially-slow rate can occur, although their result relied on adversarial tie-breaking. In this talk, we show that Karlin’s conjecture holds if ties are broken lexicographically and the game matrix is diagonal. We also show a matching lower bound under this tie-breaking assumption. This is joint work with Jake Abernethy and Andre Wibisono.
A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all. Erdos and Rado in 1960 proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower. The famous sunflower conjecture is that the bound on the number of sets can be improved to $c^w$ for some constant $c$. Despite much research, the best bounds until recently were all of the order of $w^{cw}$ for some constant c. In this work, we improve the bounds to about $(\log w)^{w}$.
There are two main ideas that underlie our result. The first is a structure vs pseudo-randomness paradigm, a commonly used paradigm in combinatorics. This allows us to either exploit structure in the given family of sets, or otherwise to assume that it is pseudo-random in a certain way. The second is a duality between families of sets and DNFs (Disjunctive Normal Forms). DNFs are widely studied in theoretical computer science. One of the central results about them is the switching lemma, which shows that DNFs simplify under random restriction. We show that when restricted to pseudo-random DNFs, much milder random restrictions are sufficient to simplify their structure.
Joint work with Ryan Alweiss, Kewen Wu and Jiapeng Zhang.
We are studying the asymptotic homotopical complexity of a sequence of billiard flows on the 2D unit torus T^2 with n
circular obstacles. We get asymptotic lower and upper bounds for the radial sizes of the homotopical rotation sets and,
accordingly, asymptotic lower and upper bounds for the sequence of topological entropies. The obtained bounds are rather
close to each other, so this way we are pretty well capturing the asymptotic homotopical complexity of such systems.
Note that the sequence of topological entropies grows at the order of log(n), whereas, in sharp contrast, the order of magnitude of the sequence of metric entropies is only log(n)/n.
Also, we prove the convexity of the admissible rotation set AR, and the fact that the rotation vectors obtained from
periodic admissible trajectories form a dense subset in AR.
One often gains insight into the topology of a manifold by studying its sub-manifolds. Some of the most interesting sub-manifolds of a 3-manifold are the "incompressible surfaces", which, intuitively, are the properly embedded surfaces that can not be further simplified while remaining non-trivial. In this talk, I will present some results on classifying orientable incompressible surfaces in a hyperbolic mapping torus whose fibers are 4-punctured spheres. I will explain how such a surface gives rise to a path satisfying certain combinatorial properties in the arc complex of the 4-punctured sphere, and how we can reconstruct such surfaces from these paths. This extends and generalizes results of Floyd, Hatcher, and Thurston.
Did you know that a wheel or a ball bearing does not need to be round? Convex regions that can roll smoothly come in many remarkable shapes and have practical applications in engineering and science. Moreover, the mathematics used to describe them, known as convex geometry, is a subject that beautifully ties together analysis and geometry. I'll bring some of these objects along and tell the class how to describe them effectively and recount their interesting history.
We will give a brief introduction to Schur polynomials and intersection theory and show how to use Section 10 of the Brändén-Huh paper to obtain log-concavity results for Schur polynomials.
https://arxiv.org/abs/1902.03719
https://arxiv.org/abs/1906.09633
Anosov flows provide beautiful examples of interactions between dynamics, geometry and analysis. In dimension 3 in particular, they are known to have a subtle relation to topology as well. Motivated by a result of Mitsumatsu from 1995, I will discuss their relation to contact and symplectic structures and argue why contact topological methods are natural tools to study the related global phenomena.
I will discuss the orderability of the fundamental groups of knot complements including known results, a useful technique using some ideas of Baumslag, and some interesting questions that have recently arisen from this study.
I will discuss how a graph theoretic construction used by Hirasawa and Murasugi can be used to show that the commutator subgroup of the knot group of a two-bridge knot is a union of an ascending chain of parafree groups. Using a theorem of Baumslag, this implies that the commutator subgroup of a two-bridge knot group is residually torsion-free nilpotent which has applications to the anti-symmetry of ribbon concordance and the bi-orderability of two-bridge knots. In 1973, E. J. Mayland gave a conference talk in which he announced this result. Notes on this talk can be found online. However, this result has never been published, and there is evidence, in later papers, that a proper proof might have eluded Mayland.
Domino tilings of finite grid regions have been studied in many contexts, revealing rich combinatorial structure. They arise in applications spanning physics, computer science and probability theory and recreational mathematics. We will look at questions such as counting and sampling from large combinatorial sets, such as the set of domino tilings, providing a small sample of some of the techniques that are used.
Branched covers are a generalization of covering spaces, and give rise to interesting questions in smooth as well as contact topology. All 3 manifolds arise as branched coverings of the 3-sphere. The talk will involve a discussion of the proof of this fact due to Montesinos, and will explore the work done towards understanding which contact 3 manifolds arise as the branched cover of the standard tight 3 sphere, and how the branch set can be regulated.
The Caffarelli contraction theorem states that the Brenier map sending the
Gaussian measure onto a uniformly log-concave probability measure is
lipschitz. In this talk, I will present a new proof, using entropic
regularization and a variational characterization of lipschitz transport
maps. Based on joint work with Nathael Gozlan and Maxime Prod'homme.
The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. Specifically, neuronal networks at multiple scales utilize their structural complexities to achieve different computational goals. In this talk, I will discuss functional implications that can be inferred from the architecture of brain networks.
The first part of the talk will focus on a generalized problem of linking structure and dynamics of the whole-brain network. By simulating large-scale brain dynamics using a data-driven network of phase oscillators, we show that complexities added to the spatially embedded brain connectome by idiosyncratic long-range connections, enable rapid transitions between local and global synchronizations. In addition to the spatial dependence, I will also discuss hierarchical structure of the brain network. Based on the data-driven layer-specific connectivity patterns, we developed an unsupervised method to find the hierarchical organization of the mouse cortical and thalamic network. The uncovered hierarchy provides insights into the direction of information flow in the mouse brain, which has been less well-defined compared to the primate brain.
Finally, I will discuss computational implications of the hierarchical organization of the brain network. I will focus on a specific type of computation – discrimination of partially occluded objects— carried out by a small cortical circuitry composed of an intermediate visual cortical area V4 and its efferent prefrontal cortex. I will explore how distinct feedforward and feedback signals promote robust encoding of visual stimuli by leveraging predictive coding, a Bayesian inference theory of cortical computation which has been proposed as a method to create efficient neural codes. We implement a predictive coding model of V4 and prefrontal cortex to investigate possible computational roles of feedback signals in the visual system and their potential significance in robust encoding of nosy visual stimuli.
In sum, our results reveal the close link between structural complexity and computational versatility found in brain networks, which may be useful for developing more efficient artificial neural networks and neuromorphic devices.
This talk concerns a naturally occurring family of Calabi-Yau manifolds that degenerates in the sense of metric geometry, algebraic geometry and nonlinear PDE. A primary tool in analyzing their behavior is the recently developed regularity theory. We will give a precise description of arising singularities and explain possible generalizations.
I will introduce an isoperimetric inequality for the Hamming cube and some of its applications. The applications include a “stability” version of Harper’s edge-isoperimetric inequality, which was first proved by Friedgut, Kalai and Naor for half cubes, and later by Ellis for subsets of any size. Our inequality also plays a key role in a recent result on the asymptotic number of maximal independent sets in the cube.
This is joint work with Jeff Kahn.
(This is a joint event of the Combinatorics Seminar Series and the ACO Student Seminar.)
In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdos-Rado “Sunflower Conjecture.”
This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.
(This is a joint event of ACO Student Seminar and the Combinatorics Seminar Series)
In this talk we will prove a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kalai and Kahn. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdős-Rado “Sunflower Conjecture.”
This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.
A group is said to be torsion-free if it has no elements of finite order. An example is the group, under composition, of self-homeomorphisms (continuous maps with continuous inverses) of the interval I = [0, 1] fixed on the boundary {0, 1}. In fact this group has the stronger property of being left-orderable, meaning that the elements of the group can be ordered in a way that is nvariant under left-multiplication. If one restricts to piecewise-linear (PL) homeomorphisms, there exists a two-sided (bi-)ordering, an even stronger property of groups.
I will discuss joint work with Danny Calegari concerning groups of homeomorphisms of the cube [0, 1]^n fixed on the boundary. In the PL category, this group is left-orderable, but not bi-orderable, for all n>1. Also I will report on recent work of James Hyde showing that left-orderability fails for n>1 in the topological category.
Note time and place of seminar
Two of the most basic questions in contact topology are which manifolds admit tight contact structures, and on those that do, can we classify such structures. In dimension 3, these questions have been answered for large classes of manifolds, but with a notable absence of hyperbolic manifolds. In this talk, we will see a new classification of contact structures on an family of hyperbolic 3-manifolds arising from Dehn surgery on the figure-eight knot, and see how it suggests some structural results about tight contact structures. This is joint work with Hyunki Min.
Introduced by Hendricks and Manolescu in 2015, Involutive Heegaard Floer homology is a variation of the 3-manifold invariant Heegaard Floer homology which makes use of the conjugation symmetry of the Heegaard Floer complexes. This theory can be used to obtain two new invariants of homology cobordism. This talk will involve a brief overview of general Heegaard Floer homology, followed by a discussion of the involutive theory and some computations of the homology cobordism invariants.
When a topological object admits a group action, we expect that our invariants reflect this symmetry in their structure. This talk will explore how link symmetries are reflected in three generations of related invariants: the Jones polynomial; its categorification, Khovanov homology; and the youngest invariant in the family, the Khovanov stable homotopy type, introduced by Lipshitz and Sarkar. In joint work with Matthew Stoffregen, we use Lawson-Lipshitz-Sarkar's construction of the Lipshitz-Sarkar Khovanov homotopy type to produce localization theorems and Smith-type inequalities for the Khovanov homology of periodic links.
Lorentzian polynomials link continuous convex analysis and discrete convex analysis via tropical geometry. The tropical connection is used to produce Lorentzian polynomials from discrete convex functions. Although no specific background beyond linear algebra and multivariable calculus will be needed to enjoy the presentation, I advertise the talk to people with interests in at least one of the following topics: graphs, convex bodies, stable polynomials, projective varieties, Potts model partition functions, tropicalizations, Schur polynomials, highest weight representations. Based on joint works with Petter Brändén, Christopher Eur, Jacob Matherne, Karola Mészáros, and Avery St. Dizier.
Elliptic integrands are used to model anisotropic energies in variational problems. These energies are employed in a variety of applications, such as crystal structures, capillarity problems and gravitational fields, to account for preferred inhomogeneous and directionally dependent configurations. After a brief introduction to variational problems involving elliptic integrands, I will present an overview of the techniques I have developed to prove existence, regularity and uniqueness properties of the critical points of anisotropic energies. In particular, I will present the anisotropic extension of Allard's rectifiability theorem and its applications to the Plateau problem. Furthermore, I will describe the anisotropic counterpart of Alexandrov's characterization of volume-constrained critical points. Finally, I will mention some of my ongoing and future research projects.
In various applications involving ranking data, statistical models for mixtures of permutations are frequently employed when the population exhibits heterogeneity. In this talk, I will discuss the widely used Mallows mixture model. I will introduce a generic polynomial-time algorithm that learns a mixture of permutations from groups of pairwise comparisons. This generic algorithm, equipped with a specialized subroutine, demixes the Mallows mixture with a sample complexity that improves upon the previous state of the art.
The basic model of an isolated self-gravitating gaseous star is given by the gravitational Euler-Poisson system. For any value of the adiabatic index strictly between 1 and 4/3 we construct an infinite-dimensional family of collapsing solutions to the Euler-Poisson system whose density is in general space inhomogeneous and undergoes gravitational blowup along a prescribed space-time surface in the Lagrangian coordinates. The leading order singular behaviour is driven by collapsing dust solutions. This is a joint work with Yan Guo (Brown) and Juhi Jang (USC).
We consider the problem of learning optimal reserve price in repeated auctions against non- myopic bidders, who may bid strategically in order to gain in future rounds even if the single- round auctions are truthful. Previous algorithms, e.g., empirical pricing, do not provide non- trivial regret rounds in this setting in general. We introduce algorithms that obtain a small regret against non-myopic bidders either when the market is large, i.e., no single bidder appears in more than a small constant fraction of the rounds, or when the bidders are impatient, i.e., they discount future utility by some factor mildly bounded away from one. Our approach carefully controls what information is revealed to each bidder, and builds on techniques from differentially private online learning as well as the recent line of works on jointly differentially private algorithms.
We shall discuss the recent breakthrough of Annika Heckel on the chromatic number of the binomial random graph G(n,1/2), showing that it is not concentrated on any sequence of intervals of length n^{1/4-o(1)}.
To put this into context, in 1992 Erdos (and also Bollobás in 2004) asked for any non-trivial results asserting a lack of concentration, pointing out that even the weakest such results would be of interest.
Until recently this seemed completely out of reach, in part because there seemed to be no obvious approach/strategy how to get one's foot in the door.
Annika Heckel has now found such an approach, based on a clever coupling idea that compares the chromatic number of G(n,1/2) for different n.
In this informal talk we shall try to say a few words about her insightful proof approach from https://arxiv.org/abs/1906.11808
Please note the unusual room (Skiles 202)
In a recent work with A. Gogolev we found some new form of rigidity for expanding maps through marching of potentials (also named cocycles). In this talk I plan to discuss these rigidity results and explain how this relates to some old results by Shub and Sullivan and de la Llave.
The cosmetic surgery conjecture states that no two different Dehn surgeries on a given knot produce the same oriented 3-manifold (such a pair of surgeries is called purely cosmetic). For knots in S^3, I will describe how knot Floer homology provides a strong obstruction to the existence of purely cosmetic surgeries. For many knots, including all alternating knots with genus not equal to two as well as all but 337 of the first 1.7 million knots, this is enough to confirm the conjecture. For the remaining knots, all but finitely many surgery slopes are obstructed, so checking the conjecture for a given knot reduces to distinguishing finitely many pairs of manifolds. Using a computer search, the conjecture has been verified for all prime knots with up to 16 crossings, as well as for arbitrary connected sums of such knots. These results significantly improve on earlier work of Ni and Wu, who also used Heegaard Floer homology to obstruct purely cosmetic surgeries. The improvement comes from using the full graded Heegaard Floer invariant, which is facilitated by a recent recasting of knot Floer homology as a collection of immersed curves in the punctured torus.
The talk will revolve around combinatorial aspects of Abelian varieties. I will focus on Pryms, a class of Abelian varieties that occurs in the presence of double covers, and have deep connections with torsion points of Jacobians, bi-tangent lines of curves, and spin structures. I will explain how problems concerning Pryms may be reduced, via tropical geometry, to problems on metric graphs. As a consequence, we obtain new results concerning the geometry of special algebraic curves, and bounds on dimensions of certain Brill–Noether loci.
Prime-power-fold cyclic branched covers along smoothly slice knots all bound rational homology balls. This phenomenon, however, does not characterize slice knots: In this talk, we give examples of non-slice knots that have the above property. This is joint work with Aceto, Meier, A. Miller, M. Miller, and Stipsicz.
What is the smallest total width of a collection of strips that cover a disk in the plane? How many lines through the origin pairwise separated by the same angle can be placed in 3-dimensional space? What about higher-dimensions?
These extremal problems in Discrete Geometry look deceitfully simple, yet some of them remain unsolved for an extended period or have been partly solved only recently following great efforts. In this talk, I will discuss two longstanding problems: Fejes Tóth’s zone conjecture and a problem on equiangular lines with a fixed angle.
No specific background will be needed to enjoy the talk.
We consider the Benjamin Ono equation, modeling one-dimensional long interval waves in a stratified fluid, with a slowly-varying potential perturbation. Starting with near soliton initial data, we prove that the solution remains close to a soliton wave form, with parameters of position and scale evolving according to effective ODEs depending on the potential. The result is valid on a time-scale that is dynamically relevant, and highlights the effect of the perturbation. It is proved using a Lyapunov functional built from energy and mass, Taylor expansions, spectral estimates, and estimates for the Hilbert transform.
This talk is centered around the symmetry properties of optimizers for the Caffarelli-Kohn-Nirenberg (CKN) inequalities, a two parameter family of inequalities. After a general overview I will explain some of the ideas on how to obtain the optimal symmetry region in the parameter space and will present an application to non-linear functionals of Aharonov-Bohm type, i.e., to problems that include a magnetic flux concentrated at one point. These functionals are rotationally invariant and, as I will discuss, depending on the magnitude of the flux, the optimizers are radially symmetric or not.
The hyperplane conjecture, raised by Bourgain in 1986, is a major unsolved problem in high-dimensional geometry. It states that every convex set of volume 1 in the Euclidean space has a section that is lower bounded away from 0 uniformly over the dimension. We will present a probabilistic approach to the conjecture.
In 1963, Lieb introduced an effective theory to approximate the ground state energy of a system of Bosons interacting with each other via a repulsive pair potential, in the thermodynamic limit. Lieb showed that in one dimension, this effective theory predicts a ground state energy that differs at most by 20% from its exact value, for any density. The main idea is that instead of considering marginals of the square of the wave function, as in Hartree theory, we consider marginals of the wave function itself, which is positive in the ground state. The effective theory Lieb obtained is a non-linear integro-differential equation, whose non-linearity is an auto-convolution. In this talk, I will discuss some recent work about this effective equation. In particular, we proved the existence of a solution. We also proved that the ground state energy obtained from this simplified equation agrees exactly with that of the full N-body system at asymptotically low and at high densities. In fact, preliminary numerical work has shown that, for some potentials, the ground state energy can be computed in this way with an error of at most 5% over the entire range of densities. This is joint work with E. Carlen and E.H. Lieb.
We show that the Principle of Exchange of Stability holds for convection in a layer of fluids overlaying a porous media with proper interface boundary conditions and suitable assumption on the parameters. The physically relevant small Darcy number regime as well as the dependence of the convection on various parameters will be discussed. A theory on the dependence of the depth ratio of the onset of deep convection will be put forth together with supporting numerical evidence. A decoupled uniquely solvable, unconditionally stable numerical scheme for solving the system will be presented as well.
Note the unusual time
Gordon and Luecke showed that the knot complements determine the isotopy classes of knots in S^3. In this talk, we will study the topology of various knot complements in S^3: torus knots, cable knots, satellite knots, etc. As an application, we will see some knot invariants using knot complements.
We will discuss certain isoperimetric-type problems for convex sets, such as the Log-Brunn-Minkowski conjecture for Lebesgue measure, and will explain the approach to this type of problems via local versions of inequalities and why it arises naturally. We consider a weaker form of the conjecture and prove it in several cases, with elementary geometric methods. We shall also consider several illustrative ``hands on’’ examples. If time permits, we will discuss Bochner’s method approach to the question and formulate some new results in this regard. The second (optional!) part of this talk will be at the High-dimensional seminar right after, and will involve a discussion of more involved methods. Partially based on a joint work with Hosle and Kolesnikov.
The first part of this pair of talks will be given at the Analysis seminar right before; attending it is not necessary, as all the background will be given in this lecture as well, and the talks will be sufficiently independent of each other.
I will discuss the L_p-Brunn-Minkowski inequality for log-concave measures, explain ‘’Bochner’s method’’ approach to this problem and state and prove several new results. This falls into a general framework of isoperimetric type inequalities in high-dimensional euclidean spaces. Joint with Hosle and Kolesnikov.
Abstract: Matroids are combinatorial objects that model various types of independence. They appear several fields mathematics, including graph theory, combinatorial optimization, and algebraic geometry. In this talk, I will introduce the theory of matroids along with the closely related class of polynomials called strongly log-concave polynomials. Strong log-concavity is a functional property of a real multivariate polynomial that translates to useful conditions on its coefficients. Discrete probability distributions defined by these coefficients inherit several of these nice properties. I will discuss the beautiful real and combinatorial geometry underlying these polynomials and describe applications to random walks on the faces of simplicial complexes. Consequences include proofs of Mason's conjecture that the sequence of numbers of independent sets of a matroid is ultra log-concave and the Mihail-Vazirani conjecture that the basis exchange graph of a matroid has expansion at least one. This is based on joint work with Nima Anari, Kuikui Liu, and Shayan Oveis Gharan.
The advances in bioimaging techniques have enabled us to access the 3D shapes of a variety of structures: organs, cells, proteins. Since biological shapes are related to physiological functions, statistical analyses in biomedical research are poised to incorporate more shape data. This leads to the question: how do we define quantitative descriptions of shape variability from images?
Mathematically, landmarks’ shapes, curve shapes, or surface shapes can be seen as the remainder after we have filtered out the corresponding object’s position and orientation. As such, shape data belong to quotient spaces, which are non-Euclidean spaces.
In this talk, I introduce “Geometric statistics”, a statistical theory for data belonging to non-Euclidean spaces. In the context of shape data analysis, I use geometric statistics to prove mathematically and experimentally that the “template shape estimation” algorithm, used for more than 15 years in biomedical imaging and signal processing, has an asymptotic bias. As an alternative, I present variational autoencoders (VAEs) and discuss the accuracy-speed trade-off of these procedures. I show how to use VAEs to estimate biomolecular shapes from cryo-electron microscopy (cryo-EM) images. This study opens the door to unsupervised fast (cryo-EM) biological shape estimation and analysis.
In this talk I will discuss algorithms for a uniform generation of random graphs with a given degree sequence. Let $M$ be the sum of all degrees and $\Delta$ be the maximum degree of a given degree sequence. McKay and Wormald described a switching based algorithm for the generation of graphs with given degrees that had expected runtime $O(M^2\Delta^2)$, under the assumption $\Delta^4=O(M)$. I will present a modification of the McKay-Wormald algorithm that incorporates a new rejection scheme and uses the same switching operation. A new algorithm has expected running time linear in $M$, under the same assumptions.
I will also describe how a new rejection scheme can be integrated into other graph generation algorithms to significantly reduce expected runtime, as well as how it can be used to generate contingency tables with given marginals uniformly at random.
This talk is based on the joint work with Jane Gao and Nick Wormald.
In the first talk I will introduce the main constructions, many of which are classical, from scratch. This part will be introductory and accessible to a general audience with a basic knowledge of topology. This introduction will also serve as preparation for the main talk in which I will outline the proof and discuss some applications.
The goal of this talk is to explain the sense in which the natural algebraic structure of the singular chains on a path-connected space determines its fundamental group functorially. This new basic piece about the algebraic topology of spaces, which tells us that the fundamental group may be determined from homological data, has several interesting and deep implications. An example of a corollary of our statement is the following extension of a classical theorem of Whitehead: a continuous map between path-connected pointed topological spaces is a weak homotopy equivalence if and only if the induced map between the differential graded coalgebras of singular chains is a Koszul weak equivalence (i.e. a quasi-isomorphism after applying the cobar functor). A deeper implication, which is work in progress, is that this allows us to give a complete description of infinity groupoids in terms of homological algebra.
There are three main ingredients that come into play in order to give a precise formulation and proof of our main statement: 1) we extend a classical result of F. Adams from 1956 regarding the “cobar construction” as an algebraic model for the based loop space of a simply connected space, 2) we make use of the homotopical symmetry of the chain approximations to the diagonal map on a space, and 3) we apply a duality theory for algebraic structures known as Koszul duality. This is joint work with Mahmoud Zeinalian and Felix Wierstra.
Although studying numbers seems to have little to do with shapes, geometry has become an indispensable tool in number theory during the last 70 years. Deligne's proof of the Weil Conjectures, Wiles's proof of Fermat's Last Theorem, and Faltings's proof of the Mordell Conjecture all require machinery from Grothendieck's algebraic geometry. It is less frequent to find instances where tools from number theory have been used to deduce theorems in geometry. In this talk, we will introduce one tool from each of these subjects -- Galois representations in number theory and cohomology in geometry -- and explain how arithmetic can be used as a tool to prove some important conjectures in geometry. More precisely, we will discuss ongoing joint work with Laure Flapan in which we prove the Hodge and Tate Conjectures for self-products of 16 K3 surfaces using arithmetic techniques.
We will go over hyperfields and polynomials over hyperfields. We will discuss the current theory as well as new directions.
Pollen grain surface morphologies are famously diverse, and each species displays a unique, replicable pattern. The function of these microstructures, however, has not been elucidated. We show electron microscopy evidence that the templating of these patterns is formed by a phase separation of a polysaccharide mixture on the cell membrane surface. Here we present a Landau theory of phase transitions to ordered states describing all extant pollen morphologies. We show that 10% of all morphologies can be characterized as equilibrium states with a well-defined wavelength of the pattern. The rest of the patterns have a range of wavelengths on the surface that can be recapitulated by exploring the evolution of a conserved dynamics model. We then perform an evolutionary trait reconstruction. Surprisingly, we find that although the equilibrium states have evolved multiple times, evolution has not favored these ordered-polyhedral like shapes and perhaps their patterning is simply a natural consequence of a phase separation process without cross-linkers.
We are interested in arithmetic progressions in positive measure subsets of [0,1]^d. After a counterexample by Bourgain, it seemed as if nothing could be said about the longest interval formed by sizes of their gaps. However, Cook, Magyar, and Pramanik gave a positive result for 3-term progressions if their gaps are measured in the l^p-norm for p other than 1, 2, and infinity, and the dimension d is large enough. We establish an appropriate generalization of their result to longer progressions. The main difficulty lies in handling a class of multilinear singular integrals associated with arithmetic progressions that includes the well-known multilinear Hilbert transforms, bounds for which still constitute an open problem. As a substitute, we use the previous work with Durcik and Thiele on power-type cancellation of those transforms, which was, in turn, motivated by a desire to quantify the results of Tao and Zorin-Kranich. This is joint work with Polona Durcik (Caltech).
In the world of 4 manifolds, finding exotic structures on 4 manifolds is considered one of most interesting and difficult problems. I will give a brief history of this and explain a very interesting tool "knot surgery" defined by Fintushel and Stern. In this talk I will mostly focused on drawing pictures. If time permits, I will talk various interesting applications.
Let $K$ be a n dimensional convex body with of volume $1$. and barycenter of $K$ is the origin. It is known that $|K \cap -K|>2^{-n}$. Via thin shell estimate by Lee-Vempala (earlier versions were done by Guedon-Milman, Fleury, Klartag), we improve the bound by a sub-exponential factor. Furthermore, we can improve the Hadwiger’s Conjecture in the non-symmetric case by a sub-exponential factor. This is a joint work with Boaz A. Slomka, Tomasz Tkocz, and Beatrice-Helen Vritsiou.
The classical isoperimetric problem consists in finding among all sets with the same volume (measure) the one that minimizes the surface area (perimeter measure). In the Euclidean case, balls are known to solve this problem. To formulate the isoperimetric problem, or an isoperimetric inequality, in more general settings, requires in particular a good notion of perimeter measure.
The starting point of this talk will be a characterization of sets of finite perimeter original to Ledoux that involves the heat semigroup associated to a given stochastic process in the space. This approach put in connection isoperimetric problems and functions of bounded variation (BV) via heat semigroups, and we will extend these ideas to develop a natural definition of BV functions and sets of finite perimeter on metric measure spaces. In particular, we will obtain corresponding isoperimetric inequalies in this setting.
The main assumption on the underlying space will be a non-negative curvature type condition that we call weak Bakry-Émery and is satisfied in many examples of interest, also in fractals such as (infinite) Sierpinski gaskets and carpets. The results are part of joint work with F. Baudoin, L. Chen, L. Rogers, N. Shanmugalingam and A. Teplyaev.
Tutte proved that every 4-connected planar graph contains a Hamilton cycle, but
there are 3-connected $n$-vertex graphs whose longest cycles have length
$\Theta(n^{\log_32})$. On the other hand, Jackson and Wormald proved that an
essentially 4-connected $n$-vertex planar graph contains a cycle of
length at least $(2n+4)/5$, which was improved to $5(n+2)/8$ by Fabrici {\it et al}. We improve this bound to $\lceil (2n+6)/3\rceil$ for $n\ge 6$ by proving a quantitative version of a result of Thomassen,
and the bound is best possible.
It is a classical theorem of Roth that every dense subset of $\left\{1,\ldots,N\right\}$ contains a nontrivial three-term arithmetic progression. Quantitatively, results of Sanders, Bloom, and Bloom-Sisask tell us that subsets of relative density at least $1/(\log N)^{1-\epsilon}$ already have this property. In this talk, we will discuss some sets of $N$ integers which unlike $\left\{1,\ldots,N\right\}$ do not contain nontrivial four-term arithmetic progressions, but which still have the property that all of their subsets of density at least $1/(\log N)^{1-\epsilon}$ must contain a three-term arithmetic progression. Perhaps a bit surprisingly, these sets turn out not to have as many three-term progressions as one might be inclined to guess, so we will also address the question of how many three-term progressions can a four-term progression free set may have. Finally, we will also discuss about some related results over $\mathbb{F}_{q}^n$. Based on joint works with Jacob Fox and Oliver Roche-Newton.
While the existence and properties of the SRB measure for the billiard map associated with a periodic Lorentz gas are well understood, there are few results regarding other types of measures for dispersing billiards. We begin by proposing a naive definition of topological entropy for the billiard map, and show that it is equivalent to several classical definitions. We then prove a variational principle for the topological entropy and proceed to construct a unique probability measure which achieves the maximum. This measure is Bernoulli and positive on open sets. An essential ingredient is a proof of the absolute continuity of the unstable foliation with respect to the measure of maximal entropy. This is joint work with Viviane Baladi.
A discussion of Khovanov-Lee homology, how to extract some invariants of braid closures from the homology theory, and motivation for studying both the homology theory and the invariants.
Consider a stochastic process (such as a stochastic differential equation) arising from applications. In practice, we are interested in many things like the invariant probability measure, the sensitivity of the invariant probability measure, and the speed of convergence to the invariant probability measure. Existing rigorous estimates of these problems usually cannot provide enough details. In this talk I will introduce a few data-driven computational methods that solve these problems for a class of stochastic dynamical systems, including but not limited to stochastic differential equations. All these methods are driven by the simulation data, and are less affected by the curse-of-dimensionality than traditional grid-based methods. I will demonstrate a few high (up to 100) dimensional examples in my talk.
Annular Rasmussen invariants are invariants of braid closures which generalize the Rasmussen s invariant and come from an integer bifiltration on Khovanov-Lee homology. In this talk we will explain some connections between the annular Rasmussen invariants and other topological information. Additionally we will state theorems about restrictions on the possible values of annular Rasmussen invariants and a computation of the invariants for all 3-braid closures, or conjugacy classes of 3-braids. Time permitting, we will sketch some proofs.
Originally introduced independently by Hassler Whitney and Takeo Nakasawa, matroids are a combinatorial way of axiomatizing the notion of linear independence in vector spaces. If $K$ is a field and $n$ is a positive integer, any linear subspace of $K^n$ gives rise to a matroid; such matroid are called representable over $K$. Given a matroid $M$, one can ask over which fields $M$ is representable. More generally, one can ask about representability over partial fields in the sense of Semple and Whittle. Pendavingh and van Zwam introduced the universal partial field of a matroid $M$, which governs the representations of $M$ over all partial fields. Unfortunately, most matroids (asymptotically 100%, in fact) are not representable over any partial field, and in this case, the universal partial field gives no information.
Oliver Lorscheid and I have introduced a generalization of the universal partial field which we call the foundation of a matroid. The foundation of $M$ is a type of algebraic object which we call a pasture; pastures include both hyperfields and partial fields. Pastures form a natural class of field-like objects within Lorscheid's theory of ordered blueprints, and they have desirable categorical properties (e.g., existence of products and coproducts) that make them a natural context in which to study algebraic invariants of matroids. The foundation of a matroid $M$ represents the functor taking a pasture $F$ to the set of rescaling equivalence classes of $F$-representations of $M$; in particular, $M$ is representable over a pasture $F$ if and only if there is a homomorphism from the foundation of $M$ to $F$. (In layman's terms, what we're trying to do is recast as much as possible of the theory of matroids and their representations in functorial ``Grothendieck-style'' algebraic geometry, with the goal of gaining new conceptual insights into various phenomena which were previously understood only through lengthy case-by-case analyses and ad hoc computations.)
As a particular application of this point of view, I will explain the classification which Lorscheid and I have recently obtained of all possible foundations for ternary matroids (matroids representable over the field of three elements). The proof of this classification theorem relies crucially on Tutte's celebrated Homotopy Theorem.
Lattice polytopes play an important role in combinatorics due to their intricate geometric structure as well as their enumerative properties. In this talk, we will discuss several instances in which lattice point enumeration of lattice polytopes relates to problems in algebraic combinatorics, particularly the representation theory of GL(n) and related groups. We will also see how certain types of algebraic constructions have polytopal counterparts. This talk is based on joint work with Karola Mészáros and Avery St. Dizier.
The relationship between biodiversity and ecological stability has long interested ecologists. The ongoing biodiversity loss has led to the increasing concern that it may impact ecosystem functioning, including ecosystem stability. Both early conceptual ideas and recent theory suggest a positive relationship between biodiversity and ecosystem stability. While quite a number of empirical studies, particularly experiments that directly manipulated species diversity, support this hypothesis, exceptions are not uncommon. This raises the question of whether there is a general positive diversity-stability relationship.
Literature survey shows that species diversity may not necessarily be an important determinant of ecosystem stability in natural communities. While experiments controlling for other environmental variables often report that ecosystem stability increases with species diversity, these other environmental variables are often more important than species diversity in influencing ecosystem stability. Studies that account for these environmental covariates tend to find a lack of relationship between species diversity and ecosystem stability. An important goal of future studies is to elucidate mechanisms driving the variation in the importance of species diversity in regulating ecosystem stability.
A useful way of studying contact 3 manifolds is by looking at their open book decompositions. A result of Akbulut-Ozbagci, Ghiggini, and Loi-Piergallini showed that the manifold is filled by a Stein manifold if and only if the monodromy of an open book can be factorised as the product of positive Dehn twists. Then, the problem of classifying minimal fillings of contact 3 manifolds, or answering questions about which manifolds can be realised by Legendrian surgery, becomes questions about finding factorisations for a given mapping class. This talk will be expository and expand upon how these mapping classes come up, and also discuss known results, techniques, and future directions for research.
Decoupling is a Fourier analytic tool that has repeatedly proved its extraordinary potential for a broad range of applications to number theory (counting solutions to Diophantine systems, estimates for the growth of the Riemann zeta), PDEs (Strichartz estimates, local smoothing for the wave equation, convergence of solutions to the initial data), geometric measure theory (the Falconer distance conjecture) and harmonic analysis (the Restriction Conjecture). The abstract theorems are formulated and proved in a continuous framework, for arbitrary functions with spectrum supported near curved manifolds. At this level of generality, the proofs involve no number theory, but rely instead on wave packet analysis and incidence geometry related to the Kakeya phenomenon. The special case when the spectrum is localized near lattice points leads to unexpected solutions of conjectures once thought to pertain to the realm of number theory.
(Refreshments will be served at 2:30pm after the lecture.)
In recent years political parties have more and more expertly
crafted political districtings to favor one side or another, while at
the same time, entirely new techniques to detect and measure these
efforts are being developed.
I will discuss a rigorous method which uses Markov chains---random
walks---to statistically assess gerrymandering of political districts
without requiring heuristic validation of the structures of the Markov
chains which arise in the redistricting context. In particular, we will
see two examples where this methodology was applied in successful
lawsuits which overturned district maps in Pennsylvania and North Carolina.
Many functions of interest are in a high-dimensional space but exhibit low-dimensional structures. This work studies regression of a $s$-Hölder function $f$ in $\mathbb{R}^D$ which varies along an active subspace of dimension $d$ while $d\ll D$. A direct approximation of $f$ in $\mathbb{R}^D$ with an $\varepsilon$ accuracy requires the number of samples $n$ in the order of $\varepsilon^{-(2s+D)/s}$. In this work, we modify the Generalized Contour Regression (GCR) algorithm to estimate the active subspace and use piecewise polynomials for function approximation. GCR is among the best estimators for the active subspace, but its sample complexity is an open question. Our modified GCR improves the efficiency over the original GCR and leads to a mean squared estimation error of $O(n^{-1})$ for the active subspace, when $n$ is sufficiently large. The mean squared regression error of $f$ is proved to be in the order of $\left(n/\log n\right)^{-\frac{2s}{2s+d}}$, where the exponent depends on the dimension of the active subspace $d$ instead of the ambient space $D$. This result demonstrates that GCR is effective in learning low-dimensional active subspaces. The convergence rate is validated through several numerical experiments.
This is a joint work with Wenjing Liao.
We show that if $P\neq NP$, then a wide class of TSP heuristics fail to approximate the length of the TSP to asymptotic
optimality, even for random Euclidean instances. Previously, this result was not even known for any heuristics (greedy, etc) used in practice. As an application, we show that when using a heuristic from this class, a natural class of branch-and-bound algorithms takes exponential time to find an optimal tour (again, even on a random point-set), regardless of the particular branching strategy or lower-bound algorithm used.
Given an action of a finite group $G$ on a complex vector space $V$, the Chevalley-Shephard-Todd Theorem gives a beautiful characterization for when the quotient variety $V/G$ is smooth. In his 1986 ICM address, Popov asked whether this criterion could be extended to the case of Lie groups. I will discuss my contributions to this problem and some intriguing questions in combinatorics that this raises. This is based on joint work with Dan Edidin.
We present several new results concerning mixing properties of
hyperbolic systems preserving an infinite measure making a particular
emphasis on mixing for extended systems. This talk is based on a joint
work with Peter Nandori.
Kinetic equations play an important role in multiscale modeling hierarchy. It serves as a basic building block that connects the microscopic particle models and macroscopic continuum models. Numerically approximating kinetic equations presents several difficulties: 1) high dimensionality (the equation is in phase space); 2) nonlinearity and stiffness of the collision/interaction terms; 3) positivity of the solution (the unknown is a probability density function); 4) consistency to the limiting fluid models; etc. I will start with a brief overview of the kinetic equations including the Boltzmann equation and the Fokker-Planck equation, and then discuss in particular our recent effort of constructing efficient and robust numerical methods for these equations, overcoming some of the aforementioned difficulties. This is joint work with Ruiwen Shu (University of Maryland).
Fintushel and Stern showed that the Brieskorn sphere Σ(2, 3, 7) bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to provide the first additional examples of Brieskorn spheres that bound rational homology balls but not integral homology balls, including two infinite families. This is joint work with Selman Akbulut.
We will give a brief introduction to the spectral theory of ergodic operators. Then we discuss several remarkable spectral phenomena present in the class of quasiperiodic operators, as well as the nonperturbative approach to small denominator problems that has been behind much of the related progress. In particular, we will talk about the almost Mathieu (aka Harper's) operator - a model heavily studied in physics literature and linked to several Nobel prizes (in addition to one Fields medal). We will describe several results on this model that resolve some long-standing conjectures.
Given a potential function of three vector arguments, $f(x,y,z)$, which is $O(n)$-invariant, $f(Qx,Qy,Qz)=f(x,y,z)$ for all $Q$ orthogonal, we use semidefinite programming bounds to determine optimizing probability measures for interaction energies of the form $\int\int\int f(x,y,z) d\mu(x)d\mu(y)d\mu(z)$ over the sphere. This approach builds on previous use of such bounds in the discrete setting by Bachoc-Vallentin, Cohn-Woo, and Musin, and is successful for kernels which can be shown to have expansions in a particular basis, for instance certain symmetric polynomials in inner products $u=\langle x,y \rangle$, $v=\langle y,z\rangle$, and $t=\langle z, x \rangle$. For other kernels we pose conjectures on the behavior of optimizers, partially inferred through numerical studies.
This is an ordinary research Geometry/Topology seminar:<br />
https://math.gatech.edu/seminars-colloquia/series/geometry-topology-semi...
Ideas from physics have predicted a number of important properties of random constraint satisfaction problems such as the satisfiability threshold and the free energy (the exponential growth rate of the number of solutions). Another prediction is the condensation regime where most of the solutions are contained in a small number of clusters and the overlap of two random solutions is concentrated on two points. We establish this phenomena for the random regular NAESAT model.
We give an efficient algorithm for robustly clustering of a mixture of two arbitrary Gaussians, a central open problem in the theory of computationally efficient robust estimation, assuming only that the the means of the component Gaussian are well-separated or their covariances are well-separated. Our algorithm and analysis extend naturally to robustly clustering mixtures of well-separated logconcave distributions. The mean separation required is close to the smallest possible to guarantee that most of the measure of the component Gaussians can be separated by some hyperplane (for covariances, it is the same condition in the second degree polynomial kernel). Our main tools are a new identifiability criterion based on isotropic position, and a corresponding Sum-of-Squares convex programming relaxation.
Batchelor's law describes the power law spectrum of the turbulent regime of passive scalars (e.g., temperature or a dilute concentration of some tracer chemical) advected by an incompressible fluid (e.g., the Navier-Stokes equations at fixed Reynolds number), in the limit of vanishingly low molecular diffusivity. Predicted in 1959, it has been confirmed empirically in a variety of experiments, e.g. salinity concentrations among ocean currents. On the other hand, as with many turbulent regimes in physics, a true predictive theory from first principles has been missing (even a non-rigorous one), and there has been some controversy regarding the extent to which Batchelor's law is universal.
In this talk, I will present a program of research, joint with Jacob Bedrossian (UMD) and Sam Punshon-Smith (Brown), which has rigorously proved Batchelor's law for passive scalars advected by the Navier-Stokes equations on the periodic box subjected to Sobolev regular, white-in-time body forces. The proof is a synthesis of techniques from dynamical systems and smooth ergodic theory, stochastics/probability, and fluid mechanics. To our knowledge, this work constitutes the first mathematically rigorous proof of a turbulent power law spectrum. It also establishes a template for predictive theories of passive scalar turbulence in more general settings, providing a strong argument for the universality of Batchelor's law.
String topology studies various algebraic structures given by intersecting loops in a manifold, as well as those on the Hochschild chains or homology of an algebra. In this preparatory talk, we survey a collection of such structures and their relationships with one another.
It's know that when discretizing stochastic ordinary equation with non-globally Lipschitz coefficient, the traditional numerical method, like
Euler method, may be divergent and not converge in strong or weak sense. For stochastic partial different equation with non-globally Lipschitz
coefficient, there exists fewer result on the strong and weak convergence results of numerical methods. In this talk, we will discuss several numerical schemes approximating stochastic Schrodinger Equation. Under certain condition, we show that the exponential integrability preserving schemes are strongly and weakly convergent with positive orders.
Spaces of fatgraphs have long been used to study a variety of topics in math and physics. In this talk, we introduce two spaces of fatgraphs arising in string topology—one which parameterizes operations on chains of the free loop space of a manifold and one which parametrizes operations on Hochschild cochains of a “V-infinity” algebra. We present a conjecture relating these two spaces to one another and to the moduli space of Riemann surfaces. We also introduce polyhedra called “assocoipahedra” which generalize Stasheff’s associahedra to algebras with a compatible co-inner product. Assocoipahedra are used to prove that the dioperad governing V-infinity algebras satisfies certain algebraic properties.
A combinatorial neural code is convex if it arises as the intersection pattern of convex open subsets of Euclidean space. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to feasibility and complexity. By way of this connection, we prove that all convex codes are related to some representable oriented matroid, and we show that deciding whether a neural code is convex is NP-hard.
Noetherian operators are a set of differential operators that encode the scheme structure of a primary ideal. We propose a framework for studying primary ideals numerically by using a combination of witness sets and Noetherian operators. We will also present a method for computing Noetherian operators using numerical data.
The first step in the theory of Noetherian operators are the Macaulay dual spaces. Indeed, for an ideal that is primary over a maximal ideal corresponding to a rational point, the generators of the dual space are a valid set of Noetherian operators. We will start by presenting basic ideas, results and algorithms in the classical dual space theory, and then revisit some of these ideas in the context of Noetherian operators.
Descriptive combinatorics studies the interaction between classical combinatorial concepts, such as graph colorings and matchings, and notions from measure theory and topology. Results in this area enable one to apply combinatorial techniques to problems in other (seemingly unrelated) branches of mathematics, such as the study of dynamical systems. In this talk I will give an introduction to descriptive combinatorics and discuss some recent progress concerning a particular family of combinatorial tools---the probabilistic method---and its applications in the descriptive setting.
An embedding of a manifold into a trivial disc bundle over another manifold is called braided if projection onto the first factor gives a branched cover. This notion generalizes closed braids in the solid torus, and gives an explicit way to construct many embeddings in higher dimensions. In this talk, we will discuss when a covering map of surfaces lift to a braided embedding.
Let $N_n$ be an $n\times n$ matrix whose entries are i.i.d. copies of a random variable $\zeta=\xi+i\xi'$, where $\xi,\xi'$ are i.i.d., mean zero, variance one, subgaussian random variables. We will present a result of Luh, according to which the probability that $N_n$ has a real eigenvalue is exponentially small in $n$. An interesting part of the proof is a small ball probability estimate for the smallest singular value of a complex perturbation $M_n=M+N_n$ of the original matrix.
I will present some recent results, obtained with D. Bresch and Z. Wang, on large stochastic many-particle or multi-agent systems. Because such systems are conceptually simple but exhibit a wide range of emerging macroscopic behaviors, they are now employed in a large variety of applications from Physics (plasmas, galaxy formation...) to the Biosciences, Economy, Social Sciences...
The number of agents or particles is typically quite large, with 10^20-10^25 particles in many Physics settings for example and just as many equations. Analytical or numerical studies of such systems are potentially very complex leading to the key question as to whether it is possible to reduce this complexity, notably thanks to the notion of propagation of chaos (agents remaining almost uncorrelated).
To derive this propagation of chaos, we have introduced a novel analytical method, which led to the resolution of two long-standing conjectures:
_The quantitative derivation of the 2-dimensional incompressible Navier-Stokes system from the point vortices dynamics;
_The derivation of the mean-field limit for attractive singular interactions such as in the Keller-Segel model for chemotaxis and some Coulomb gases.
Given a vertex-weighted graph G= (V, E), the MaximumWeight Internal Spanning Tree (MWIST) problem is to find a spanning tree T of G such that the total weight of internal vertices in T is maximized. The unweighted version of this problem, known as Maxi-mum Internal Spanning Tree (MIST) problem, is a generalization of the Hamiltonian path problem, and hence, it is NP-hard. In the literature lot of research has been done on designing approximation algorithms to achieve an approximation ratio close to 1. The best known approximation algorithm achieves an approximation ratio of 17/13 for the MIST problem for general graphs. For the MWIST problem, the current best approximation algorithm achieves an approximation ratio of 2 for general graphs. Researchers have also tried to design exact/approximation algorithms for some special classes of graphs. The MIST problem parameterized by the number of internal vertices k, and its special cases and variants, have also been extensively studied in the literature. The best known kernel for the general problem has size 2k, which leads to the fastest known exact algorithm with running time O(4^kn^{O(1)}). In this talk, we will talk about some selected recent results on the MWIST problem.
In critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large, it is known that there are a.s. no infinite open clusters. In particular, for n large, every path from the origin to the boundary of $[-n, n]^d$ must contain some closed edges. Let $T_n$ be the (random) minimal number of closed edges in such a path. How does $T_n$ grow with $n$? We present results showing that for d larger than the upper critical dimension for Bernoulli percolation ($d > 6$), $T_n$ is typically of the order $\log \log n$. This is in contrast with the $d = 2$ case, where $T_n$ grows logarithmically. Perhaps surprisingly, the model exhibits another major change in behavior depending on whether $d > 8$.
This is the open forum for Pierre-Emmanuel Jabin (https://home.cscamm.umd.edu/~jabin/)
as a candidate for Elaine M. Hubbard Chair in Mathematics.
In the $k$-cut problem, we are given an edge-weighted graph and want to find the least-weight set of edges whose deletion breaks the graph into $k$ connected components. Algorithms due to Karger-Stein and Thorup showed how to find such a minimum $k$-cut in time approximately $O(n^{2k-2})$. The best lower bounds come from conjectures about the solvability of the $k$-clique problem and a reduction from $k$-clique to $k$-cut, and show that solving $k$-cut is likely to require time $\Omega(n^k)$. Our recent results have given special-purpose algorithms that solve the problem in time $n^{1.98k + O(1)}$, and ones that have better performance for special classes of graphs (e.g., for small integer weights).
In this work, we resolve the problem for general graphs, by showing that for any fixed $k \geq 2$, the Karger-Stein algorithm outputs any fixed minimum $k$-cut with probability at least $\widehat{O}(n^{-k})$, where $\widehat{O}(\cdot)$ hides a $2^{O(\ln \ln n)^2}$ factor. This also gives an extremal bound of $\widehat{O}(n^k)$ on the number of minimum $k$-cuts in an $n$-vertex graph and an algorithm to compute a minimum $k$-cut in similar runtime. Both are tight up to $\widehat{O}(1)$ factors.
The first main ingredient in our result is a fine-grained analysis of how the graph shrinks---and how the average degree evolves---under the Karger-Stein process. The second ingredient is an extremal result bounding the number of cuts of size at most $(2-\delta) OPT/k$, using the Sunflower lemma.
The most popular model for Image Denoising is without any doubt the ROF (for Rudin-OsherFatemi) model. However, since the ROF approach has some drawbacks (the stair-case effect being one of them) practitioners have been looking for alternatives. One of them is the Elastica model, relying on the minimization in an appropriate functional space of the energy functional $J$ defined by
$$ J(v)=\varepsilon \int_{\Omega} \left[ a+b\left| \nabla\cdot \frac{\nabla v}{|\nabla v|}\right|^2 \right]|\nabla v| d\mathbf{x} + \frac{1}{2}\int_{\Omega} |f-v|^2d\mathbf{x} $$
where in $J(v)$: (i) $\Omega$ is typically a rectangular region of $R^2$ and $d\mathbf{x}=dx_1dx_2$. (ii) $\varepsilon, a$ and $b$ are positive parameters. (iii) function $f$ represents the image one intends to denoise.
Minimizing functional $J$ is a non-smooth, non-convex bi-harmonic problem from Calculus of Variations. Its numerical solution is a relatively complicated issue. However, one can achieve this task rather easily by combining operator-splitting and finite element approximations. The main goal of this lecture is to describe such a methodology and to present the results of numerical experiments which validate it.
The past few years have seen the advent of big data, which brings unprecedented convenience to our daily life. Meanwhile, from a computational point of view, a central question arises amid the exploding amount of data: how to tame big data in an economic and efficient way. In the context of matrix computations, the question consists in the ability to handle large dense matrices. In this talk, I will first introduce data-sparse hierarchical representations for dense matrices. Then I will present recent development of a new data-driven algorithm called SMASH to operate dense matrices efficiently in the most general setting. The new method not only outperforms existing algorithms but also works in high dimensions. Various experiments will be provided to justify the advantages of the new method.
A classical theorem of Lie and Tresse states that the algebra of differential invariants of a Lie group or (suitable) Lie pseudo-group action is finitely generated. I will present a fully constructive algorithm, based on the equivariant method of moving frames, that reveals the full structure of such non-commutative differential algebras, and, in particular, pinpoints generating sets of differential invariants as well as their differential syzygies. Some applications and outstanding issues will be discussed.
The dual of the cone of non-negative quadratics (on a variety) is included in the dual of the cone of sums of squares. Moreover, all (points which span) extreme rays of the dual cone of non-negative quadratics is point evaluations on real points of the variety. Therefore, we are interested in extreme rays of the dual cone of sums of squares which do not come from point evaluations. The dual cone of sums of squares on a variety is called the Hankel spectrahetron and the smallest rank of extreme rays which do not come from point evaluations is called Hankel index of the variety. In this talk, I will introduce some algebraic (or geometric) properties which control the Hankel index of varieties and compute the Hankel index of rational curves obtained by projecting rational normal curve away from a point (which has almost minimal degree).
We consider the problem of finding on a given bounded and smooth
Euclidean domain \Omega of dimension n ≥ 3 a complete conformally flat metric whose Schouten
curvature A satisfies some equation of the form f(\lambda(-A)) =1. This generalizes a problem
considered by Loewner and Nirenberg for the scalar curvature. We prove the existence and uniqueness of
locally Lipschitz solutions. We also show that the Lipschitz regularity is in general optimal.
The malaria parasite Plasmodium falciparum requires a vertebrate host, such as a human, and a vector host, the Anopheles mosquito, to complete a full life cycle. The portion of the life cycle in the mosquito harbors both the only time of sexual reproduction, expanding genetic complexity, and the most severe bottlenecks experienced, restricting genetic diversity, across the entire parasite life cycle. In previous work, we developed a two-stage stochastic model of parasite diversity within a mosquito, and demonstrated the importance of heterogeneity amongst parasite dynamics across a population of mosquitoes. Here, we focus on the parasite dynamics component to evaluate the first appearance of sporozoites, which is key for determining the time at which mosquitoes first become infectious. We use Bayesian inference techniques with simple models of within-mosquito parasite dynamics coupled with experimental data to estimate a posterior distribution of parameters. We determine that growth rate and the bursting function are key to the timing of first infectiousness, a key epidemiological parameter.
Two classic questions -- the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem -- both focus on the distance, which is a simple two point configuration. When studying the Falconer distance problem, a geometric averaging operator, namely the spherical averaging operator, arises naturally. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk I will give a brief introduction to the motivating point configuration questions and then report on some novel geometric averaging operators and their mapping properties.
Cosmetic surgeries (purely cosmetic surgeries) are two distinct surgeries on a knot that produce homeomorphic 3-manifolds (as oriented manifolds). It is one of the ways Dehn surgeries on knots could fail to be unique. Gordon conjectured that there are no nontrivial purely cosmetic surgeries on nontrivial knots in S^3. We will recap the progress of the problem over time, and mainly discuss Ni and Wu's results in their paper "Cosmetic surgeries on knots in S^3".
The cornerstone in local theory of Banach spaces is Dvoretzky’s theorem, which asserts that almost euclidean structure is locally present in any high-dimensional normed space. The random version of this remarkable phenomenon was put forth by V. Milman in 70’s, who employed the concentration of measure on the sphere. Purpose of the talk is to present how Gaussian tools from high-dimensional probability (e.g., Gaussian convexity, hypercontractivity, superconcentration) can be exploited for obtaining optimal results in random forms of Dvoretzky’s theorem. Based on joint work(s) with Grigoris Paouris and Konstantin Tikhomirov.
Continuing from Biggs’s Algebraic Graph Theory, we discuss the properties of the Laplacian Matrix of a graph and how it relates to the tree number.
While deviation estimates above the mean is a very well studied subject in high-dimensional probability, for their lower analogues far less are known. However, it has been observed, in several key situations, that lower deviation inequalities exhibit very different and stronger behavior. In this talk I will discuss how convexity can serve as a key feature to (a) explain this distinction, (b) obtain improved lower tail bounds, and (c) characterize the tightness of Gaussian concentration.
No seminar due to the "Special tea" fro the Admitted Student Day, which is 3-4 pm in the Skiles atrium.
A Stein manifold is a complex manifold with particularly nice convexity properties. In real dimensions above 4, the existence of a Stein structure is essentially a homotopical question, but for 4-manifolds the situation is more subtle. In this talk we will consider the existence of such structures in the ambient settings, that is, manifolds/domains with various degree of convexity as open/compact subsets of a complex manifold, e.g. complex 2-space C^2. In particular, I will discuss the following question: Which homology spheres embed in C^2 as the boundary of a Stein domain? This question was first considered and explored in detail by Gompf. At that time, he made a fascinating conjecture that no non-trivial Brieskorn homology sphere, with either orientation, embeds in C^2 as a Stein boundary. In this talk, I will survey what we know about this conjecture, and report on some closely related recent work in progress that ties to an interesting symplectic rigidity phenomena in low dimensions.
Like toric varieties, toric vector bundles are a rich class of algebraic varieties that can be described with combinatorial data. Klyachko gave a classification of toric vector bundles in terms of certain systems of filtrations in a vector space. I'll talk about some recent work with Kiumars Kaveh showing that Klyachko's data has an interesting interpretation in terms of tropical geometry. In particular, we show that toric vector bundles can be classified by points on tropicalized linear spaces over a semifield of piecewise-linear functions. I'll discuss how to use this recipe and a closely related tropicalization map to produce toric vector bundles and more general flat toric families.
We have seen that Hankel index of varieties can be controlled by some invariants such as $$N_{2,p}$$ or p-base point free property. Also, we know that the Hankel index of (a linear join of) variety of minimal degree is infinity (and all invariant above are same as infinity). As next case, I will share some computations of invariants on a variety that projecting rational normal curve away from a point (which is a variety of almost minimal degree).
We consider dihedral branched covers of $S^4$, branched along an embedded surface with one non-locally flat point, modelled on the cone on a knot $K\subset S^3$. Kjuchukova proved that the signature of this cover is an invariant $\Xi_p(K)$ of the $p$-colorable knot $K$. We prove that the values of $\Xi_p(K)$ fall in a bounded range for homotopy-ribbon knots. We also construct a family of (non-slice) knots for which the values of $\Xi_p$ are unbounded. More generally, we introduce the notion of the dihedral 4-genus of a knot, and derive a lower bound on the dihedral 4-genus of $K$ in terms of $\Xi_p(K)$. This work is joint with A. Kjuchukova.
Perspectives from numerical optimization and computational algebra are
brought to bear on a scientific application and a data science
application. In the first part of the talk, I will discuss
cryo-electron microscopy (cryo-EM), an imaging technique to determine
the 3-D shape of macromolecules from many noisy 2-D projections,
recognized by the 2017 Chemistry Nobel Prize. Mathematically, cryo-EM
presents a particularly rich inverse problem, with unknown
orientations, extreme noise, big data and conformational
heterogeneity. In particular, this motivates a general framework for
statistical estimation under compact group actions, connecting
information theory and group invariant theory. In the second part of
the talk, I will discuss tensor rank decomposition, a higher-order
variant of PCA broadly applicable in data science. A fast algorithm
is introduced and analyzed, combining ideas of Sylvester and the power
method.
Phytoplankton are the base of the marine food web. They are also responsible for much of the oxygen we breathe, and they remove carbon dioxide from the atmosphere. The mechanisms that govern the timing of seasonal phytoplankton blooms is one of the most debated topics in oceanography. Here, we present a macroscale plankton ecology model consisting of coupled, nonlinear reaction-diffusion equations with spatially and temporally changing coefficients to offer insight into the causes of phytoplankton blooms. This model simulates biological interactions between nutrients, phytoplankton and zooplankton. It also incorporates seasonally varying solar radiation, diffusion and depth of the ocean’s upper mixed layer because of their impact on phytoplankton growth. The model is analyzed using seasonal oceanic data with the goals of understanding the model’s dependence on its parameters and of understanding seasonal changes in plankton biomass. A study of varying parameter values and the resulting effects on the solutions, the stability of the steady-states, and the timing of phytoplankton blooms is carried out. The model’s simulated blooms result from a temporary attraction to one of the model’s steady-states.
We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.
Since the work of Arnold in 1964, it is known that there are situations where the perturbations can accumulate (Arnold diffusion).
This can be understood by noting that the small perturbations generate some invariant objects in phase space that act as routes which allow accumulation of effects.
We will present some rigorous results about geometric objects lead to Arnold diffusion as well as some computational tools that allow to find them in concrete applications.
Thanks to the work of many people, an area which used to be very speculative, is becoming an applicable tool.
This is a part of IEEE Signal Processing Society Lecture Series, organized by Dr. Alessio Medda (alessiomedda@ieee.org). PLEASE RSVP to https://events.vtools.ieee.org/m/222947
The set of diffeomorphisms of the unit interval, or “warping functions,” plays an important role in many in functional data analysis applications. Most prominently, the problem of registering, or aligning, pairs of functions depends on solving for an element of the diffeomorphism group that, when applied to one function, optimally aligns it to the other.
The registration problem is posed as the unconstrained minimization of a cost function over the infinite dimensional diffeomorphism function space. We make use of its well-known Riemannian geometry to implement an efficient, limited memory Riemannian BFGS optimization scheme. We compare performance and results to the benchmark algorithm, Dynamic Programming, on several functional datasets. Additionally, we apply our methodology to the problem of non-parametric density estimation and compare to the benchmark performance of MATLAB’s built-in kernel density estimator ‘ksdensity’.
The problem of phase retrieval for a set of functions $H$ can be thought of as being able to identify a function $f\in H$ or $-f\in H$ from the absolute value $|f|$. Phase retrieval for a set of functions is called stable if when $|f|$ and $|g|$ are close then $f$ is proportionally close to $g$ or $-g$. That is, we say that a set $H\subseteq L_2({\mathbb R})$ does stable phase retrieval if there exists a constant $C>0$ so that
$$\min\big(\big\|f-g\big\|_{L_2({\mathbb R})},\big\|f+g\big\|_{L_2({\mathbb R})}\big)\leq C \big\| |f|-|g| \big\|_{L_2({\mathbb R})} \qquad\textrm{ for all }f,g\in H.
$$
It is known that phase retrieval for finite dimensional spaces is always stable. On the other hand, phase retrieval for infinite dimensional spaces using a frame or a continuous frame is always unstable. We prove that there exist infinite dimensional subspaces of $L_2({\mathbb R})$ which do stable phase retrieval. This is joint work with Robert Calderbank, Ingrid Daubechies, and Nikki Freeman.
In his book Convex Polyhedra, ch. 7 (end of subsection 2) A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of convex polytopes with given geometric data. As an example of a geometric problem in which variational approach was successfully applied, Aleksandrov quotes the Minkowski problem. He also mentions the Weyl problem of isometric embedding for which a variational approach was proposed (but not fully developed and not completed) by W. Blashke and G. Herglotz. The first goal of this talk is to give a variational formulation and solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gaussian curvature (also posed by Aleksandrov who solved it using topological methods). The second goal of this talk is to show that in variational form the Aleksandrov problem is closely connected to the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations.
<br />
Melvin Leok is a professor in the Department of Mathematics at the University of California, San Diego. His research interests are in computational geometric mechanics, computational geometric control theory, discrete geometry, and structure-preserving numerical schemes, and particularly how these subjects relate to systems with symmetry. He received his Ph.D. in 2004 from the California Institute of Technology in Control and Dynamical Systems under the direction of Jerrold Marsden. He is a three-time NAS Kavli Frontiers of Science Fellow, and has received the NSF Faculty Early Career Development (CAREER) award, the SciCADE New Talent Prize, the SIAM Student Paper Prize, and the Leslie Fox Prize (second prize) in Numerical Analysis. He has given plenary talks at the Society for Natural Philosophy, Foundations of Computational Mathematics, NUMDIFF, and the IFAC Workshop on Lagrangian and Hamiltonian Methods for Nonlinear Control. He serves on the editorial boards of the Journal of Nonlinear Science, the Journal of Geometric Mechanics, and the Journal of Computational Dynamics, and has served on the editorial boards of the SIAM Journal on Control and Optimization, and the LMS Journal of Computation and Mathematics.<br />
Geometric mechanics describes Lagrangian and Hamiltonian mechanics geometrically, and information geometry formulates statistical estimation, inference, and machine learning in terms of geometry. A divergence function is an asymmetric distance between two probability densities that induces differential geometric structures and yields efficient machine learning algorithms that minimize the duality gap. The connection between information geometry and geometric mechanics will yield a unified treatment of machine learning and structure-preserving discretizations. In particular, the divergence function of information geometry can be viewed as a discrete Lagrangian, which is a generating function of a symplectic map, that arise in discrete variational mechanics. This identification allows the methods of backward error analysis to be applied, and the symplectic map generated by a divergence function can be associated with the exact time-$h$ flow map of a Hamiltonian system on the space of probability distributions.
We provide a martingale proof of the fact that the number of descents in random permutations is asymptotically normal with an error bound of order n^{-1/2}. The same techniques are shown to be applicable to other descent and descent-related statistics as they satisfy certain recurrence relation conditions. These statistics include inversions, descents in signed permutations, descents in Stirling permutations, the length of the longest alternating subsequences, descents in matchings and two-sided Eulerian numbers.
Consider a system of rotators subject to a small quasi-periodic forcing which (1) is analytic, (2) satisfies a time-reversibility property, and (3) has a Bryuno frequency vector. Without imposing any non-degeneracy condition, we prove that there exists at least one quasi-periodic solution with the same frequency vector as the forcing. The result can be interpreted as a theorem of persistence of lower-dimensional tori of arbitrary codimension in degenerate cases. This is a joint work with Livia Corsi.
Motivated by the Dikin walk, we develop aspects of an interior-point
theory for sampling in high dimensions. Specifically, we introduce symmetric
and strong self-concordance. These properties imply that the corresponding
Dikin walk mixes in $\tilde{O}(n\bar{\nu})$ steps from a warm start
in a convex body in $\mathbb{R}^{n}$ using a strongly self-concordant barrier
with symmetric self-concordance parameter $\bar{\nu}$. For many natural
barriers, $\bar{\nu}$ is roughly bounded by $\nu$, the standard
self-concordance parameter. We show that this property and strong
self-concordance hold for the Lee-Sidford barrier. As a consequence,
we obtain the first walk to mix in $\tilde{O}(n^{2})$ steps for an
arbitrary polytope in $\mathbb{R}^{n}$. Strong self-concordance for other
barriers leads to an interesting (and unexpected) connection ---
for the universal and entropic barriers, it is implied by the KLS
conjecture.
We study self-similar and fractal networks from the combinatorial perspective. We establish analogues of topological (Lebesgue) and fractal (Hausdorff) dimensions for graphs and demonstrate that they are naturally related to known graph-theoretical characteristics: rank dimension and product (or Prague or Nešetřil-Rödl) dimension. Our approach reveals how self-similarity and fractality of a network are defined by a pattern of overlaps between densely connected network communities. It allows us to identify fractal graphs, explore the relations between graph fractality, graph colorings and graph Kolmogorov complexity, and analyze the fractality of several classes of graphs and network models, as well as of a number of real-life networks. We demonstrate the application of our framework to evolutionary studies by revealing the growth of self-organization of heterogeneous viral populations over the course of their intra-host evolution, thus suggesting mechanisms of their gradual adaptation to the host's environment. As far as the authors know, the proposed approach is the first theoretical framework for study of network fractality within the combinatorial paradigm. The obtained results lay a foundation for studying fractal properties of complex networks using combinatorial methods and algorithms.
Based on joint work with Leonid Bunimovich
A surface of genus $g$ has many symmetries. These form the surface’s mapping class group $Mod(S_g)$, which is finitely generated. The most commonly used generating sets for $Mod(S_g)$ are comprised of infinite order elements called Dehn twists; however, a number of authors have shown that torsion generating sets are also possible. For example, Brendle and Farb showed that $Mod(S_g)$ is generated by six involutions for $g \geq 3$. We will discuss our extension of these results to elements of arbitrary order: for $k > 5$ and $g$ sufficiently large, $Mod(S_g)$ is generated by three elements of order $k$. Generalizing this idea, in joint work with Margalit we showed that for $g \geq 3$ every nontrivial periodic element that is not a hyperelliptic involution normally generates $Mod(S_g)$. This result raises a question: does there exist an $N$, independent of $g$, so that if $f$ is a periodic normal generator of $Mod(S_g)$, then $Mod(S_g)$ is generated by $N$ conjugates of $f$? We show that in general there does not exist such an $N$, but that there does exist such a universal bound for the class of non-involution normal generators.
On June 10, 2000, the Millennium Bridge in London opened to the public. As people crossed the bridge, it wobbled. The sway of the bridge was large enough that prompted many on the bridge to hold on to the rails. Three days later, the bridge closed. It reopened only after modifications to prevent the wobbling were made, eighteen months later. We develop and study a model motivated by this event
The satellite construction, which associates to a pattern knot P in a solid torus and a companion knot K in the 3-sphere the so-called satellite knot P(K), features prominently in knot theory and low-dimensional topology. Besides the intuition that P(K) is “more complicated” than either P or K, one can attempt to quantify how the complexity of a knot changes under the satellite operation. In this talk, I’ll discuss how several notions of complexity based on the minimal genus of an embedded surface change under satelliting. In the case of the classical Seifert genus of a knot, Schubert gives an exact formula. In the 4-dimensional context the situation is more complicated, and depends on whether we work in the smooth or topological category: the smooth category is sometimes asymptotically similar to the classical setting, but our main results show that the topological category is much weirder. This talk is based on joint work with Peter Feller and Juanita Pinzón-Caicedo.
Merris and Watkins interpreted results of Littlewood to give generating functions for symmetric group characters induced from one-dimensional characters of Young subgroups. Beginning with an n by n matrix X of formal variables, one obtains induced sign and trivial characters by expanding sums of products of certain determinants and permanents, respectively. We will look at a new analogous result which holds for hyperoctahedral group characters induced from the four one-dimensional characters of its Young subgroups. This requires a 2n by 2n matrix of formal variables and four combinations of determinants and permanents. This is joint work with Jongwon Kim.
Thesis Defense
Energetic stability of matter in quantum mechanics, which refers to the ques-
tion of whether the ground state energy of a many-body quantum mechanical
system is finite, has long been a deep question of mathematical physics. For a
system of many non-relativistic electrons interacting with many nuclei in the
absence of electromagnetic fields this question traces back to the seminal work
of Tosio Kato in 1951 and Freeman Dyson and Andrew Lenard in 1967/1968.
In particular, Dyson and Lenard showed the ground state energy of the many-
body Schrödinger Hamiltonian is bounded below by a constant times the total
particle number, regardless of the size of the nuclear charges. This says such a
system is energetically stable (of the second kind). This situation changes dra-
matically when electromagnetic fields and spin interactions are present in the
problem. Even for a single electron with spin interacting with a single nucleus
of charge $Z > 0$ in an external magnetic field, Jurg Fröhlich, Elliot Lieb, and
Michael Loss in 1986 showed that there is no ground state energy if $Z$ exceeds
a critical charge $Z_c$ and the ground state energy exists if $Z < Z_c$ . In other
words, if the nuclear charge is too large, the one-electron atom is energetically
unstable.
Another notion of stability in quantum mechanics is that of dynamic stabil-
ity, which refers to the question of global well-posedness for a system of partial
differential equations that models the dynamics of many electrons coupled to
their self-generated electromagnetic field and interacting with many nuclei. The
central motivating question of our PhD thesis is whether energetic stability has
any influence over dynamic stability. Concerning this question, we study the
quantum mechanical many-body problem of $N \geq 1$ non-relativistic electrons with
spin interacting with their self-generated classical electromagnetic field and $K \geq 0$
static nuclei. We model the dynamics of the electrons and their self-generated
electromagnetic field using the so-called many-body Maxwell-Pauli equations.
The main result presented is the construction time global, finite-energy, weak
solutions to the many-body Maxwell-Pauli equations under the assumption that
the fine structure constant $\alpha$ and the nuclear charges are sufficiently small to
ensure energetic stability of this system. This result represents an initial step
towards understanding the relationship between energetic stability and dynamic
stability. If time permits, we will discuss several open problems that remain.
Committee members: Prof. Michael Loss (Advisor, School of Mathematics,
Georgia Tech), Prof. Brian Kennedy (School of Physics, Georgia Tech), Prof.
Evans Harrell (School of Mathematics, Georgia Tech), Prof. Federico Bonetto
(School of Mathematics, Georgia Tech), Prof. Chongchun Zeng (School of Math-
ematics, Georgia Tech).
Energetic stability of matter in quantum mechanics, which refers to the question of whether the ground state energy of a
Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that $\dist(S(t),{\mathcal K}(H))\to 0$ as $t\to\infty$, where ${\mathcal K}(H)$ denotes the space of all compact operators on the underlying Hilbert space.
Abstract: Form methods are most efficient to prove generation theorems for semigroups but also for proving selfadjointness. So far those theorems are based on a coercivity notion which allows the use of the Lax-Milgram Lemma. Here we consider weaker "essential" versions of coerciveness which already suffice to obtain the generator of a semigroup S or a selfadjoint operator. We also show that one of these properties, namely essentially positive coerciveness implies a very special asymptotic behaviour of S, namely asymptotic compactness; i.e. that $\dist(S(t),{\mathcal K}(H))\to 0$ as $t\to\infty$, where ${\mathcal K}(H)$ denotes the space of all compact operators on the underlying Hilbert space.
In this talk we will survey some of the developments of Cheeger and Colding’s conjecture on a sequence of n dimensional manifolds with uniform two sides Ricci Curvature bound, investigated by Anderson, Tian, Cheeger, Colding and Naber among others. The conjecture states that every Gromov-Hausdorff limit of the above-mentioned sequence, which is a metric space with singularities, has the singular set with Hausdorff codimension at least 4. This conjecture was proved by Colding-Naber in 2014, where the ideas and techniques like \epsilon-regularity theory, almost splitting and quantitative stratification were extensively used. I will give an introduction of the background of the conjecture and talk about the idea of the part of the proof that deals with codimension 2 singularities.
We give an efficient algorithm for robustly clustering of a mixture of two arbitrary Gaussians,<br />
a central open problem in the theory of computationally efficient robust estimation, assuming<br />
only that the the means of the component Gaussians are well-separated or their covariances are<br />
well-separated. Our algorithm and analysis extend naturally to robustly clustering mixtures of<br />
well-separated logconcave distributions. The mean separation required is close to the smallest<br />
possible to guarantee that most of the measure of the component Gaussians can be separated<br />
by some hyperplane (for covariances, it is the same condition in the second degree polynomial<br />
kernel). Our main tools are a new identifiability criterion based on isotropic position, and a<br />
corresponding Sum-of-Squares convex programming relaxation. This is joint work with He Jia.
The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator. The main ideas are decoupling of spaces by symmetry and construction of approximate eigenvalues. This is based on two works with Stephen Shipman and Karl-Mikael Perfekt.
The pure spherical p-spin model is a Gaussian random polynomial H of degree p on an N-dimensional sphere, with N large. The sphere is viewed as the state space of a physical system with many degrees of freedom, and the random function H is interpreted as a smooth assignment of energy to each state, i.e. as an energy landscape.
In 2012, Auffinger, Ben Arous and Cerny used the Kac-Rice formula to count the average number of critical points of H having a given index, and with energy below a given value. This number is exponentially large in N for p > 2, and the rate of growth itself is a function of the index chosen and of the energy cutoff. This function, called the complexity, reveals interesting topological information about the landscape H: it was shown that below an energy threshold marking the bottom of the landscape, all critical points are local minima or saddles with an index not diverging with N. It was shown that these finite-index saddles have an interesting nested structure, despite their number being exponentially dominated by minima up to the energy threshold. The total complexity (considering critical points of any index) was shown to be positive at energies close to the lowest. Thus, at least from the perspective of the average number of critical points, these random landscapes are very non-convex. The high-dimensional and rugged aspects of these landscapes make them relevant to the folding of large molecules and the performance of neural nets.
Subag made a remarkable contribution in 2017, when he used a second-moment approach to show that the total number of critical points concentrates around its mean. In light of the above, when considering critical points near the bottom of the landscape, we can view Subag's result as a statement about the concentration of the number of local minima. His result demonstrated that the typical behavior of the minima reflects their average behavior. We complete the picture for the bottom of the landscape by showing that the number of critical points of any finite index concentrates around its mean. This information is important to studying associated dynamics, for instance navigation between local minima. Joint work with Antonio Auffinger and Yi Gu at Northwestern.
Cancelled due to COVID-19
An afternoon of public engagement of mathematics through puzzles, games, and the arts, including: magic (by Matt Baker), juggling and other circus arts, music, dance, an art gallery, and a live construction of a Fibonacci-based sculpture (by Akio Hizume). It is free and open to the public, but our partner the Julia Robinson Mathematics Festival recommends registering at https://jrmf.org/event-details/mathapalooza . If you want to get involved, please contact Evans Harrell directly.
A math-themed variety show including music, improv comedy, a poetry slam, juggling, a fashion show (audience members can join in) and more, right there on the stage of the fabulous Highland Ballroom! Tickets are $10.00.
Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.
This talk was cancelled due to the current status. The following is the original abstract for the talk. The celebrated Brill-Noether theorem says that the space of degree $d$ maps of a general genus $g$ curve to $\mathbb{P}^r$ is irreducible. However, for special curves, this need not be the case. Indeed, for general $k$-gonal curves (degree $k$ covers of $\mathbb{P}^1$), this space of maps can have many components, of different dimensions (Coppens-Martens, Pflueger, Jensen-Ranganathan). In this talk, I will introduce a natural refinement of Brill-Noether loci for curves with a distinguished map $C \rightarrow \mathbb{P}^1$, using the splitting type of push forwards of line bundles to $\mathbb{P}^1$. In particular, studying this refinement determines the dimensions of all irreducible components of Brill-Noether loci of general $k$-gonal curves.
TBD
One of the most challenging aspects of designing human sensitive systems is in designing systems that assist decision makers in applying an effective intervention to a large group of individuals. This design challenge becomes especially difficult when the decision maker must operate under scarce resources and only partial knowledge of how each individual will react to the intervention.
In this talk, I will consider this problem from the perspective of a clinician that is designing a personalized weight loss program. Despite this focus, the precision analytics framework I propose for designing these interventions is quite general and can apply to many settings where a single coordinator must influence agents who make decisions by maximizing utility functions that depend on prior system states, inputs, and other parameters that are initially unknown. This precision analytics framework involves three steps: first, a predictive model that effectively captures the decision-making process of an agent; second, an optimization algorithm to estimate this model’s parameters for each agent and predict their future decisions; and third, an optimization model that uses these predictions to optimize a set of incentives that will be provided to each agent. A key advantage of this framework is that the calculated incentives are adapted as new information is collected. In the case of personalized weight loss interventions, this means that the framework can leverage patient level data from mobile and wearable sensors over the course of the intervention to personalize the recommended treatment for each individual.
I will present theoretical results that show that the incentives computed by this approach are asymptotically optimal with respect to a loss function that describes the coordinator's objective. I will also present an effective decomposition scheme to optimize the agent incentives, where each sub-problem solves the coordinator's problem for a single agent, and the master problem is a pure integer program. To validate this method I will present a numerical study that shows this proposed framework is more cost efficient and clinically effective than simple heuristics in a simulated environment. I will conclude by discussing the results of a randomized control trial (RCT) and pilot study where this precision analytics framework was applied for personalizing exercise goals for UC Berkeley staff and students. The results of these trials show that using personalized step goals calculated by the precision analytics algorithm result in a significant improvement over existing state of the art approaches in a real world setting.
TBA
The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.
We consider rooted subgraphs in random graphs, i.e., extension counts such as (i) the number of triangles containing a given vertex or (ii) the number of paths of length three connecting two given vertices.
In 1989, Joel Spencer gave sufficient conditions for the event that, with high probability, these extension counts are asymptotically equal for all choices of the root vertices.
For the important strictly balanced case, Spencer also raised the fundamental question whether these conditions are necessary.
We answer this question by a careful second moment argument, and discuss some intriguing problems that remain open.
Already for bivariate tropical polynomials, factorization is an NP-Complete problem.In this talk, we will introduce a rich class of tropical polynomials in n variables, which admit factorization and rational factorization into well-behaved factors. We present efficient algorithms of their factorizations with examples. Special families of these polynomials have appeared in economics,discrete convex analysis, and combinatorics. Our theorems rely on an intrinsic characterization of regular mixed subdivisions of integral polytopes, and lead to open problems of interest in discrete geometry.
The talk will be held online via Bluejeans. Use the following link to join the meeting.
TBA
In 1941, Hopf gave a proof of the fact that the rational cohomology of a compact connected Lie group is isomorphic to the cohomology of a product of odd dimensional spheres. The proof is natural in the sense that instead of using the classification of Lie groups, it utilizes the extra algebraic structures, now known as Hopf algebras. In this talk, we will discuss the algebraic background and the proof of the theorem.
TBA
TBA
Right and left eigenvectors of non-Hermitian matrices form a bi-orthogonal system to which one can associate homogeneous quantities known as overlaps. The matrix of overlaps quantifies the stability of the spectrum and characterizes the joint eigenvalues increments under Dyson-type dynamics. Overlaps first appeared in the physics literature: Chalker and Mehlig calculated their conditional expectation for complex Ginibre matrices (1998). For the same model, we extend their results by deriving the distribution of the overlaps and their correlations (joint work with P. Bourgade). Similar results can be obtained for quaternionic Gaussian matrices, as well as matrices from the spherical and truncated-unitary ensembles.
TBA
The main goal of this talk is to discuss my proof of a multiplicity formula for polynomials over a real valued field. I also want to talk about some of the raisons d’être for hyperfields and polynomials over hyperfields. This talk is based on my paper “A Newton Polygon Rule for Formally-Real Valued Fields and Multiplicities over the Signed Tropical Hyperfield” which is in turn based on a paper of Matt Baker and Oliver Lorscheid “Descartes' rule of signs, Newton polygons, and polynomials over hyperfields.”
The talk will be held online via Bluejeans. Use the following link to join the meeting.
Zooplankton is an immensely numerous and diverse group of organisms occupying every corner of the oceans, seas and freshwater bodies on the planet. They form a crucial link between autotrophic phytoplankton and higher trophic levels such as crustaceans, molluscs, fish, and marine mammals. Changing environmental conditions such as rising water temperatures, salinities, and decreasing pH values currently create monumental challenges to their well-being.
A signi cant subgroup of zooplankton are crustaceans of sizes between 1 and 10 mm. Despite their small size, they have extremely acute senses that allow them to navigate their surroundings, escape predators, find food and mate. In a series of joint works with Rudi Strickler (Department of Biological Sciences, University of Wisconsin - Milwaukee) we have investigated various behaviors of crustacean zooplankton. These include the visualization of the feeding current of the copepod Leptodiaptomus sicilis, the introduction of the "ecological temperature" as a descriptor of the swimming behavior of the water flea Daphnia pulicaria and the communication by sex pheromones in the copepod Temora longicornis. The tools required for the studies stem from optics, ecology, dynamical systems, statistical physics, computational fluid dynamics, and computational neuroscience.
Tba
Continuing the theme of Hopf algebras, we will discuss a recipe by Reshetikhin and Turaev for link invariants using representations of quantum groups, which are non-commutative, non-cocommutative Hopf algebras. In the simplest case with the spin 1/2 representation of quantum sl2, we recover the Kauffman bracket and the Jones polynomial when combined with writhe. Time permitting, we will also talk about colored Jones polynomials and connections to 3-manifold invariants.
Tba
An edge-colored graph $G$ is called \textit{rainbow} if every edge of $G$ receives a different color. The \textit{anti-Ramsey} number of $t$ edge-disjoint rainbow spanning trees, denoted by $r(n,t)$, is defined as the maximum number of colors in an edge-coloring of $K_n$ containing no $t$ edge-disjoint rainbow spanning trees. Jahanbekam and West [{\em J. Graph Theory, 2016}] conjectured that for any fixed $t$, $r(n,t)=\binom{n-2}{2}+t$ whenever $n\geq 2t+2 \geq 6$. We show their conjecture is true and also determine $r(n,t)$ when $n = 2t+1$. Together with previous results, this gives the anti-Ramsey number of $t$ edge-disjoint rainbow spanning trees for all values of $n$ and $t$. Joint work with Linyuan Lu.
The seminar is held in BlueJeans: https://bluejeans.com/900271747
A central question in ergodic theory is whether sequences obtained by sampling along the orbits of a given dynamical system behave similarly to sequences of i.i.d. random variables. Here we consider this question from a spectral-theoretic perspective. Specifically, we study large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift on the 2-torus with irrational frequency. We prove that their global eigenvalue distribution converges to the Wigner semicircle law, a hallmark of random matrix statistics, which evidences the quasi-random nature of the skew-shift dynamics. This is joint work with Arka Adhikari and Horng-Tzer Yau.
In this talk we will explore the interplay between tropical convexity and its classical counterpart. In particular, we will focus on the tropical convex hull of convex sets and polyhedral complexes. We give a vertex description of the tropical convex hull of a line segment and of a ray in Rn/R1 and show that tropical convex hull and classical convex hull commute in R3/R1. Finally, we prove results on the dimension of tropical convex fans and give an upper bound on the dimension of the tropical convex hull of tropical curves under certain hypothesis.
The talk will be held online via Bluejeans, use the following link to join the meeting.
TBA
Bordered Floer homology, due to Lipshitz, Ozsváth, and Thurston, is a Heegaard Floer homology theory for 3-manifolds with connected boundary. This theory associates to the boundary surface (with suitable parameterization) a differential graded algebra A(Z). Our invariant comes in two versions: a left differential (type D) module over A(Z), or its dual, a right A-infinity (type A) module over A(Z). In this talk, we will focus on the case of 3-manifolds with torus boundary, and will explicitly describe how to compute type D structures of knot complements.
My thesis studies two topics. In the first part, we study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques such like PCA (Principal Component Analysis). We believe that related algorithms should perform better with more accurate spectrum estimation. There was an approximation formula proposed by Prof. Matzinger. However, they didn't give any proof. In our research, we show why the formula works. And when both number of features and dimension of space go to infinity, we find the order of error for the approximation formula, which is related to a constant C-the ratio of dimension of space and number of features.
In the second part, we focus on some applications of Naive Bayes models in text classification problems. Especially we focus on two special situations: 1) there is insufficient data for model training; 2) partial labeling problem. We choose Naive Bayes as our base model and do some improvement on the model to achieve better performance in those two situations. To improve model performance and to utilize as many information as possible, we introduce a correlation factor, which somehow relaxes the conditional independence assumption of Naive Bayes. The new estimates are biased estimation compared to the traditional Naive Bayes estimate, but have much smaller variance, which give us a better prediction result.
TBA (joint with Stochastics Seminar)
The talk will be held online via Bluejeans, use the following link to join the meeting.
Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which requires expensive computations, numerical techniques often provides faster methods to tackle these problems. We establish numerical techniques to approximate roots of systems of equations and ways to certify its correctness.
As techniques for approximating roots of systems of equations, homotopy continuation method will be introduced. Combining homotopy method with monodromy group action, we introduce techniques for solving parametrized polynomial systems. Since numerical approaches rely on heuristic method, we study how to certify numerical roots of systems of equations. Based on Newton’s method, we study Krawczyk method and Smale’s alpha theory. These two method will be mainly used for certifying regular roots of systems. Furthermore, as an approach for multiple roots, we establish the local separation bound of a multiple root. For multiple roots whose deflation process terminates by only one iteration, we give their local separation bound and study how to certify an approximation of such multiple roots.
TBA
In the setting of manifolds with connected torus boundary, we can reinterpret bordered invariants as immersed curves in the once punctured torus. This machinery, due to Hanselman, Rasmussen, and Watson, is particularly useful in the context of knot complements. We will show how a type D structure can be viewed as a multicurve in the boundary of a manifold, and we will consider how the operation of cabling acts on this new invariant. If time permits, we will discuss how to extract concordance invariants from the curves.
The main work of this thesis is to numerically estimate some conjectured arm exponents when there exist certain number of open paths and closed dual paths that extend to the boundary of a box of sidelength N centering at the origin in bond invasion percolation on a plane square lattice by Monte-Carlo simulations. The result turns out to be supportive for the conjectured value. The numerical estimate for the acceptance profile of invasion percolation at the critical point is also obtained. An efficient algorithm to simulate invasion percolation and to find disjoint paths on most regular 2-dimensional lattices are also discussed.
The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator. The main ideas are decoupling of spaces by symmetry and construction of approximate eigenvalues. This is based on two works with Stephen Shipman and Karl-Mikael Perfekt.
This is the first installment of our CDSNS virtual colloquium, which will be held in a Bluejeans event space on Wednesdays at 9AM (EST).
We will present the "a-posteriori" approach to KAM theory.
We formulate an invariance equation and show that an approximate-enough solution which verifies some non-degeneracy conditions leads to a solution. Note that this does not have any reference to integrable systems and that the non-degeneracy conditions are not global properties of the system, but only properties of the solution. The "automatic reducibility" allows to take advantage of the geometry to develop very efficient Newton methods and show that they converge.
This leads to very efficient numerical algorithms (which moreover can be proved to lead to correct solutions), to validate formal expansions. From a more theoretical point of view, it can be applied to other geometric contexts (conformally symplectic, presymplectic) and other geometric objects such as whiskered tori. One can deal well with degenerate systems, singular perturbation theory and obtain simple proofs of monogenicity and Whitney regularity.
This is joint work with many people.
Every closed 3-manifold admits foliations, where the leaves are surfaces. For a given 3-manifold, which surfaces can appear as leaves? Kerékjártó and Richards gave a classification up to homeomorphism of noncompact surfaces, which includes surfaces with infinite genus and infinitely many punctures. In their 1985 paper "Every surface is a leaf", Cantwell--Conlon prove that for every orientable noncompact surface L and every closed 3-manifold M, M has a foliation where L appears as a leaf. We will discuss their paper and construction and the surrounding context.
Random graphs are the basic mathematical models for large-scale disordered networks in many different fields (e.g., physics, biology, sociology).
Since many real world networks evolve over time, it is natural to study various random graph processes which arise by adding edges (or vertices) step-by-step in some random way.
The analysis of such random processes typically brings together tools and techniques from seemingly different areas (combinatorial enumeration, differential equations, discrete martingales, branching processes, etc), with connections to the analysis of randomized algorithms.
Furthermore, such processes provide a systematic way to construct graphs with "surprising" properties, leading to some of the best known bounds in extremal combinatorics (Ramsey and Turan Theory).
In this talk I shall survey several random graph processes of interest (in the context of the probabilistic method), and give a glimpse of their analysis.
If time permits, we shall also illustrate one of the main proof techniques (the "differential equation method") using a simple toy example.
Online at <br />
<br />
https://bluejeans.com/603353375/4347?src=calendarLink
In this thesis, the Rayleigh-Taylor instability effects in the setting of the Navier-Stokes equations, given some three-dimensional and incompressible fluids, are investigated. The existence and the uniqueness of the temperature variable in the the weak form is established under suitable initial and boundary conditions, and by the contraction mapping principle we investigate further the conditions for the solution to belong to some continuous class; then a positive minimum temperature result can be proved, and with the aid of the RT instability effect in the density and the velocity, the instability for the temperature is established.
Virtual seminar held on BlueJeans
Adaptive control problems arise in many engineering applications in which one needs to design feedback controllers that ensure tracking of desired reference trajectories while at the same time identify unknown parameters such as control gains. This talk will summarize the speaker's work on adaptive tracking and parameter identification, including an application to curve tracking problems in robotics. The talk will be understandable to those familiar with the basic theory of ordinary differential equations. No prerequisite background in systems and control will be needed to understand and appreciate this talk.
The attendee link is https://primetime.bluejeans.com/a2m/live-event/xsgxxwbh
I will discuss random Young towers and prove an quenched Almost Sure Invariant Principle for them, which implies many quenched limits theorems, e.g., Central Limit Theorem, Functional Central Limit Theorem etc.. I will apply my result to some random perturbations of some nonuniformly expanding maps such as unimodal maps, Pomeau-Manneville maps etc..
This thesis has four chapters. The first three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging electrons to minimize their energy. Restricting our potentials to be polynomials in the squared distance between points, we show in the Chapter 1 that there exist discrete minimal energy distributions. In addition we pose a conjecture on discreteness of minimizers for another class of energies while showing these minimizers must have empty interior.
In Chapter 2, we discover that highly symmetric distributions of points minimize energies over probability measures for potentials which are completely monotonic up to some degree, guided by the work of H. Cohn and A. Kumar. We make conjectures about optima for a class of energies calculated by summing absolute values of inner products raised to a positive power. Through reformulation, these observations give rise to new mixed-volume inequalities and conjectures. Our numerical experiments also lead to discovery of a new highly symmetric complex projective design which we detail the construction for. In this chapter we also provide details on a computer assisted argument which shows optimality of the $600$-cell for such energies (via interval arithmetic).
In Chapter 3 we also investigate energies having minimizers with a small number of distinct inner products. We focus here on discrete energies, confirming that for small $p$ the repeated orthonormal basis minimizes the $\ell_p $-norm of the inner products out of all unit norm configurations. These results have analogs for simplices which we also prove.
Finally, in Chapter 4 we show that real tight frames that generate lattices must be rational, and that the same holds for other vector systems with structured matrices of outer products. We describe a construction of lattices from distance transitive graphs which gives rise to strongly eutactic lattices. We discuss properties of this construction and also detail potential applications of lattices generated by incoherent systems of vectors.
(UPDATED Monday 5-18) Note the nonstandard start time of 12PM.
Riemann's non-differentiable function, although introduced as a pathological example in analysis, makes an appearance in a certain limiting regime of the theory of binormal flow for vortex lines. From this physical point of view, it also bears some qualitative similarities to turbulent fluid velocity fields in the infinite Reynolds number limit. In this talk, we'll see how this function arises in the study of the vortex filaments, and how we can adapt the notion of intermittency from the study of turbulent flows to this setting. Then, we'll study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways. One in the physical space, and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0, which highlights the (weak) intermittent nature of Riemann's function.
This is a joint work with Alexandre Boritchev (Université de Lyon) and Daniel Eceizabarrena (BCAM, Bilbao).
We provide an accurate description of the long time dynamics for generalized Korteweg-de Vries and Benjamin-Ono equations on the circle without external parameters and for almost any (in probability and in density) small initial datum. To obtain that result we construct for these two classes of equations and under a very weak hypothesis of non degeneracy of the nonlinearity, rational normal forms on open sets surrounding the origin in high Sobolev regularity. With this new tool we can make precise the long time dynamics of the respective flows. In particular we prove a long-time stability result in Sobolev norm: given a large constant M and a sufficiently small parameter ε, for generic initial datum u(0) of size ε, we control the Sobolev norm of the solution u(t) for time of order ε^{−M}.
Delay differential equations (DDEs) are important in physical applications where there is a time lag in communication between subsystems. From a mathematical point of view DDEs are an interesting source of problems as they provide natural examples of infinite dimensional dynamical systems. I'll discuss some spectral numerical methods for computing invariant manifolds for DDEs and present some applications.
Full-branch uniformly expanding maps and their long-time statistical quantities are commonly used as simple models in the study of chaotic dynamics, as well as being of their own mathematical interest. A wide range of algorithms for computing these quantities exist, but they are typically unspecialised to the high-order differentiability of many maps of interest, and so have a weak tradeoff between computational effort and accuracy.
This talk will cover a rigorous method to calculate statistics of these maps by discretising transfer operators in a Chebyshev polynomial basis. This discretisation is highly efficient: I will show that, for analytic maps, numerical estimates obtained using this discretisation converge exponentially quickly in the order of the discretisation, for a polynomially growing computational cost. In particular, it is possible to produce (non-validated) estimates of most statistical properties accurate to 14 decimal places in a fraction of a second on a personal computer. Applications of the method to the study of intermittent dynamics and the chaotic hypothesis will be presented.
virtual (online) seminar
We investigate dynamical systems obtained by coupling an Anosov diffeomorphism and a N-pole-to-S-pole map of the circle. Both maps are uniformly hyperbolic; however, they have contrasting character, as the first one is chaotic while the second one has “orderly" dynamics. The first thing we show is that even weak coupling can produce interesting phenomena: when the attractor of the uncoupled system is not normally hyperbolic, most small interactions transform it from a smooth surface to a fractal-like set. We then consider stronger couplings in which the action of the Anosov diffeomorphism on the circle map has certain monotonicity properties. These couplings produce genuine obstructions to uniform hyperbolicity; however, the monotonicity conditions make the system amenable to study by leveraging techniques from the geometric and ergodic theories of hyperbolic systems. In particular, we can show existence of invariant cones and SRB measures.
This is joint work with Lai-Sang Young.
When the planar circular restricted 3-body problem (RTBP) is periodically perturbed, most unstable resonant periodic orbits become invariant tori. In this study, we 1) develop a quasi-Newton method which simultaneously solves for the tori and their center, stable, and unstable directions; 2) implement continuation by both perturbation parameter as well as rotation numbers; 3) compute Fourier-Taylor parameterizations of the stable and unstable manifolds; 4) globalize these manifolds; and 5) compute homoclinic and heteroclinic connections. Our methodology improves on efficiency and accuracy compared to prior studies, and applies to a variety of periodic perturbations. We demonstrate the tools on the planar elliptic RTBP. This is based on joint work with R. Anderson and R. de la Llave.
This is an expository talk, to be paired with the CDSNS Colloquium held the next day.
This is a gentle introduction to the classical Oseledets' Multiplicative Ergodic Theorem (MET), which can be viewed as either a dynamical version of the Jordan normal form of a matrix, or a matrix version of the pointwise ergodic theorem (which itself can be viewed as a generalization of the strong law of large numbers). We will also sketch Raghunathan's proof of the MET and discuss how the MET can be applied to smooth ergodic theory.
In 1960, Furstenberg and Kesten introduced the problem of describing the asymptotic behavior of products of random matrices as the number of factors tends to infinity. Oseledets’ proved that such products, after normalization, converge almost surely. This theorem has wide-ranging applications to smooth ergodic theory and rigidity theory. It has been generalized to products of random operators on Banach spaces by Ruelle and others. I will explain a new infinite-dimensional generalization based on von Neumann algebra theory which accommodates continuous Lyapunov distribution. No knowledge of von Neumann algebras will be assumed. This is joint work with Ben Hayes (U. Virginia) and Yuqing Frank Lin (UT Austin, Ben-Gurion U.).
The three-dimensional Poincare conjecture shows that any closed three-manifold other than the three-sphere has non-trivial fundamental group. A natural question is how to measure the non-triviality of such a group, and conjecturally this can be concretely realized by a non-trivial representation to SU(2). We will show that the fundamental groups of three-manifolds with incompressible tori admit non-trivial SU(2) representations. This is joint work with Juanita Pinzon-Caicedo and Raphael Zentner.
The speaker will hold online office hours from 3:15-4:15 pm for interested graduate students and postdocs.
The live talk will be broadcast on Bluejeans: https://gatech.bluejeans.com/759112674
I will present a proof of Gauss's Law of Quadratic Reciprocity based on permutations and the mathematics of dealing cards.
I will compare and contrast a selection of popular equivalence relations on 4 manifolds, and explain some recent progress on classification results.
The speaker will hold online office hours from 3:00-4:00 pm for interested graduate students and postdocs.
Deep learning has achieved great success in recent years. One aspect overlooked by traditional deep-learning methods is uncertainty modeling, which can be very important in certain applications such as medical image classification and reinforcement learning. A standard way for uncertainty modeling is by adopting Bayesian inference. In this talk, I will share some of our recent work on scalable Bayesian inference by sampling, called optimal-transport sampling, motivated from the optimal-transport theory. Our framework formulates Bayesian sampling as optimizing a set of particles, overcoming some intrinsic issues of standard Bayesian sampling algorithms such as sampling efficiency and algorithm scalability. I will also describe how our sampling framework be applied to uncertainty and repulsive attention modeling in state-of-the-art natural-language-processing models.
In the last twenty or so years, a rich theory has emerged concerning combinatorial problems on infinite graphs endowed with extra structure, such as a topology or a measure. It turns out that there is a close relationship between this theory and distributed computing, i.e., the area of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. In this talk I will outline this relationship and present a number of applications.
An infinite-type surface is a surface whose fundamental group is not finitely generated. These surfaces are “big,” having either infinite genus or infinitely many punctures. Recently, it was shown that mapping class groups of these infinite-type surfaces have a wealth of subgroups; for example, there are infinitely many surfaces whose mapping class group contains every countable group as a subgroup. By extending a theorem for finite-type surfaces to the infinite-type case — the Nielsen realization problem — we give topological obstructions to continuous embeddings of topological groups, with a few interesting examples.
How can we recognize a map given certain combinatorial data? The Alexander method gives us the answer for self-homeomorphisms of finite-type surfaces. We can determine a homeomorphism of a surface (up to isotopy) based on how it acts on a finite number of curves. However, is there a way to apply this concept to recognize maps on other spaces? The answer is yes for a special class of maps, post-critically finite quadratic polynomials on the complex plane (Belk-Lanier-Margalit-Winarski).
In this talk, we will discuss Belk-Lanier-Margalit-Winarski’s methods, as well zome difficulties we face when trying to extend their methods to other settings.
Please join using the following link: MS Teams meeting
We shall discuss a recent paper of Wanless and Wood (arXiv:2008.00775), which proves a Lovász Local Lemma type result using inductive counting arguments.
For example, in the context of hypergraph colorings, under LLL-type assumptions their result typically gives exponentially many colorings (usually more than the textbook proof of LLL would give).
We will present a probabilistic proof of the Wanless-Wood result, and discuss some applications to k-SAT, Ramsey number lower bounds, and traversals, as time permits.
We will describe several appearances of Milnor’s invariants in the link Floer complex. This will include a formula that expresses the Milnor triple linking number in terms of the h-function. We will also show that the triple linking number is involved in a structural property of the d-invariants of surgery on certain algebraically split links. We will apply the above properties toward new detection results for the Borromean and Whitehead links. This is joint work with Gorsky, Lidman and Liu.
Join us live via Bluejeans https://bluejeans.com/759112674<br />
for this talk.
Mathematics can help all of us sort through some complicated scenarios, with changing inputs, and changing conclusions. I will illustrate this with some examples. Porker hands and Jury selection bias: Expert testimony that I gave in a death penalty case. Specificity of testing: A random person tests positive for COVID. Do they have the disease? Designing pooled testing for the disease. When is it effective?
Artificial neural networks have gained widespread adoption as a powerful tool for various machine learning tasks in recent years. Training a neural network to approximate a target function involves solving an inherently non-convex problem. In practice, this is done using stochastic gradient descent with random initialization. For the approximation problem with neural networks error rate guarantees are established for different classes of functions however these rates are not always achieved in practice due to many local minima of the resulting optimization problem.
The challenge we address in this work is the following. We want to find small size shallow neural networks that can be trained algorithmically and which achieve guaranteed approximation speed and precision. To maintain the small size we apply penalties on the weights of the network. We show that under minimal requirements, all local minima of the resulting problem are well behaved and possess a desirable small size without sacrificing precision. We adopt the integral neural network framework and use techniques from optimization theory and harmonic analysis to prove our results. In this talk, we will discuss our existing work and possible future promising areas of interest where this approach can potentially be adopted.
A graph G is F-saturated if G is F-free and G+e is not F-free for any edge not in G. The saturation number of F, is the minimum number of edges in an n-vertex F-saturated graph. We consider analogues of this problem in other settings. In particular we prove saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we only require that any extension of the combinatorial structure creates new copies of the forbidden configuration. In this setting, we prove a semisaturation version of the Erdös-Szekeres theorem on convex k-gons, as well as multiple semisaturation theorems for sequences and posets. Joint work with Gábor Damásdi, Balázs Keszegh, David Malec, Casey Tompkins, and Oscar Zamora.
Finding fillings of contact structures is a question that has been studied extensively over the last few decades. In this talk I will discuss some motivations for studying this question, and then visit a few ideas involved in the earliest results, due to Eliashberg and McDuff, that paved the way for a lot of current research in this direction.
This talk's recording is available here.
The harmonic polytope and the bipermutahedron are two related polytopes which arose in our work with Graham Denham and June Huh on the Lagrangian geometry of matroids. This talk will explain their geometric origin and discuss their algebraic and geometric combinatorics.
The bipermutahedron is a (2n−2)-dimensional polytope with (2n!)/2^n vertices and 3^n−3 facets. Its f-polynomial, which counts the faces of each dimension, is given by a simple evaluation of the three variable Rogers-Ramanujan function. Its h-polynomial, which gives the dimensions of the intersection cohomology of the associated topic variety, is real-rooted, so its coefficients are log-concave.
The harmonic polytope is a (2n−2)-dimensional polytope with (n!)^2(1+1/2+...+1/n) vertices and 3^n−3 facets. Its volume is a weighted sum of the degrees of the projective varieties of all the toric ideals of connected bipartite graphs with n edges; or equivalently, a weighted sum of the lattice point counts of all the corresponding trimmed generalized permutahedra.
These two polytopes are related by a surprising fact: in any dimension, the Minkowski quotient of the bipermutahedron and the harmonic polytope equals 2.
The talk will be as self-contained as possible, and will feature joint work with Graham Denham, Laura Escobar, and June Huh.
https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
The definition of hyperbolic polynomials stems from stable polynomials, with many interesting properties related to convex geometry and optimization, including the construction of hyperbolicity cone. We will discuss some of these results and mention the application to locally PSD matrices.
Freiman's theorem characterizes finite subsets of abelian groups that behave "approximately" like subgroups: any such set is (roughly) a sum of arithmetic progressions and a finite subgroup. Quantifying Freiman's theorem is an important area of additive combinatorics; in particular, proving a "polynomial" Freiman theorem would be extremely useful.
The "Helfgott-Lindenstrauss conjecture" describes the structure of finite subsets of non-abelian groups that behave approximately like subgroups: any such set is (roughly) a finite extension of a nilpotent group. Breuillard, Green, and Tao proved a qualitative version of this conjecture. In general, a "polynomial" version of the HL conjecture cannot hold, but we prove that a polynomial version of the HL conjecture is true for linear groups of bounded rank.
In this talk, we will see how the "sum-product phenomenon" and its generalizations play a crucial role in the proof of this result. The amount of group theory needed is minimal.
Thomassen conjectures that every graph of sufficiently large average degree has a subgraph of average degree at least d and girth at least k, for any d and k. What if we want the subgraph to be induced? Large cliques and bicliques are the obvious obstructions; we conjecture there are no others. We survey results in this direction, and we prove that every bipartite graph of sufficiently large average degree has either K_{d,d} or an induced subgraph of average degree at least d and girth at least 6.
Two knots are concordant to each other if they cobound an annulus in the product of S^3. We will discuss a few basic facts about knot concordance and look at J. Levine’s classical result on the knot concordance group.
It has been conjectured that for a sufficiently large $n$, and $p = p_n \in [\log(n)/n, 1/2)$, the probability that a $n\times n$ Bernoulli($p$) matrix $A$ is singular equals to the probability that $A$ contains of a zero row or zero column up to a negligible error.
This conjecture has been recently proved by Litvak-Tikhomirov in the regime $ C\log(n)/ n < p < 1/C$ for some universal constant $C>1$ with their new tool. While for $p = (1+o(1)) \log(n) /n$, it also holds due to a result of Basak-Rudelson. In this talk, we will discuss how to extend their results to fill the gap between these two regions. ( $1\le pn/\log(n) <\infty$ )
Markov chain Monte Carlo (MCMC) methods are state-of-the-art techniques for numerical integration. MCMC methods yield estimators that converge to integrals of interest in the limit of the number of iterations, obtained from Markov chains that converge to stationarity. This iterative asymptotic justification is not ideal. Indeed the literature offers little practical guidance on how many iterations should be performed, despite decades of research on the topic. This talk will describe a computational approach to address some of these issues. The key idea, pioneered by Glynn and Rhee in 2014, is to generate couplings of Markov chains, whereby pairs of chains contract, coalesce or even "meet" after a random number of iterations; we will see that these meeting times, which can be simulated in many practical settings, contain useful information about the finite-time marginal distributions of the chains. This talk will provide an overview of this line of research, joint work with John O'Leary, Yves Atchadé and various collaborators.
The main reference is available here: https://rss.onlinelibrary.wiley.com/doi/abs/10.1111/rssb.12336
Link to meeting: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
Optimization and machine learning algorithms often use real-world data that has been generated through complex socio-economic and behavioral processes. This data, however, is noisy, and naturally encodes difficult-to-quantify systemic biases. In this work, we model and address bias in the secretary problem, which has applications in hiring. We assume that utilities of candidates are scaled by unknown bias factors, perhaps depending on demographic information, and show that bias-agnostic algorithms are suboptimal in terms of utility and fairness. We propose bias-aware algorithms that achieve certain notions of fairness, while achieving order-optimal competitive ratios in several settings.
An n-lift of a graph G is a graph from which there is an n-to-1 covering map onto G. Amit, Linial, and Matousek (2002) raised the question of whether the chromatic number of a random n-lift of K_5 is concentrated on a single value. We consider a more general problem, and show that for fixed d ≥ 3 the chromatic number of a random lift of K_d is (asymptotically almost surely) either k or k+1, where k is the smallest integer satisfying d < 2k log k. Moreover, we show that, for roughly half of the values of d, the chromatic number is concentrated on k. The argument for the upper-bound on the chromatic number uses the small subgraph conditioning method, and it can be extended to random n-lifts of G, for any fixed d-regular graph G. (This is joint work with JD Nir.)
All 3-manifolds can be described as surgery on links in the three-sphere by the celebrated theorem of Lickorish and Wallace. Motivated by the L-space conjecture, it is interesting to understand what surgery manifolds are L-spaces, which have the simplest possible Floer homology such as lens spaces. In this talk, we concentrate on surgeries on a family of links, which are called L-space links, and show possible L-space surgeries on such links. We also give some link detection results in terms of the surgeries.
The weak-type (1,1) estimate for Calderón-Zygmund operators is fundamental in harmonic analysis. We investigate weak-type inequalities for Calderón-Zygmund singular integral operators using the Calderón-Zygmund decomposition and ideas inspired by Nazarov, Treil, and Volberg. We discuss applications of these techniques in the Euclidean setting, in weighted settings, for multilinear operators, for operators with weakened smoothness assumptions, and in studying the dimensional dependence of the Riesz transforms.
Hadwiger's conjecture from 1943 states that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t\ge 1$. In the early 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t\sqrt{\log t})$ and hence is $O(t\sqrt{\log t})$-colorable. In this talk, we show that every graph with no $K_t$ minor is $O(t(\log t)^{\beta})$-colorable for every $\beta > 1/4$, making the first improvement on the order of magnitude of the Kostochka-Thomason bound.
This is joint work with Sergey Norin and Luke Postle.
The preceding talk will be given on Tuesday September 15 at 10:30 am via https://technion.zoom.us/j/99202255210. More info here: http://people.math.gatech.edu/~glivshyts6/AGAonline.html
In this follow-up talk to the talk at the AGA seminar, we will discuss some aspects of a new algorithm for rounding and volume computation, including its proof, an efficient implementation for polytopes and open questions. We will begin the talk with a recap of the algorithm.
Joint work with He Jia, Aditi Laddha and Yin Tat Lee.
Link to attend: https://us02web.zoom.us/j/88203571169
We will discuss geometrical and analytic properties of zero sets of harmonic functions and eigenfunctions of the Laplace operator. For harmonic functions on the plane there is an interestingrelation between local length of the zero set and the growth of harmonic functions. The larger the zero set is, the faster the growth of harmonic function should be and vice versa. Zero sets of Laplace eigenfunctions on surfaces are unions of smooth curves with equiangular intersections. The topology of the zero set could be quite complicated, but Yau conjectured that the total length of the zero set is comparable to the square root of the eigenvalue for all eigenfunctions. We will start with open questions about spherical harmonics and explain some methods to study nodal sets, which are zero sets of solutions of elliptic PDE.
Zoom: https://us02web.zoom.us/j/89107379948
Microsoft Teams Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
Abstract: In this talk we introduce basic definitions in tropical convexity, and give an overview of some of the main results. The focus will then shift to joint work with Faye Pasley Simon and Sara Lamboglia on the interaction between tropical and ordinary convex hull. We will introduce results including the characterization of tropically convex polyhedra and give a lower bound on the degree of a fan tropical curve using only tropical techniques. The talk will conclude with some more recent results and several open questions.
We present a new rounding framework and improve the approximation bounds for min sum vertex cover and generalized min sum set cover, also known as multiple intents re-ranking problem. These classical combinatorial optimization problems find applications in scheduling, speeding up semidefinite-program solvers, and query-results diversification, among others.
Our algorithm is based on transforming the LP solution by a suitable kernel and applying a randomized rounding. It also gives a linear-programming based 4-approximation algorithm for min sum set cover, i.e., best possible due to Feige, Lovász, and Tetali. As part of the analysis of our randomized algorithm we derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which may be of independent interest.
Joint work with Nikhil Bansal, Jatin Batra, and Prasad Tetali. [arXiv:2007.09172]
The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. The dynamics is a global Markov chain that is conjectured to converge quickly to equilibrium even at low temperatures, where the correlations in the system are strong and local chains converge slowly. In this talk, we present new results concerning the speed of convergence of the Swendsen-Wang dynamics under spatial mixing (i.e., decay of correlations) conditions. In particular, we provide tight results for three distinct geometries: the integer d-dimensional integer lattice graph Z^d, regular trees, and random d-regular graphs. Our approaches crucially exploit the underlying geometry in different ways in each case.
The Swendsen-Wang dynamics is a popular algorithm for sampling from the Gibbs distribution for the ferromagnetic Ising and Potts models. The dynamics is a global Markov chain that is conjectured to converge quickly to equilibrium even at low temperatures, where the correlations in the system are strong and local chains converge slowly. In this talk, we present new results concerning the speed of convergence of the Swendsen-Wang dynamics under spatial mixing (i.e., decay of correlations) conditions. In particular, we provide tight results for three distinct geometries: the integer d-dimensional integer lattice graph Z^d, regular trees, and random d-regular graphs. Our approaches crucially exploit the underlying geometry in different ways in each case.
The 2011 PhD thesis of Farris demonstrated that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base, the only known computation of ECH to date which does not rely on toric methods. We extend this result by computing the Z-grading on the chain complex, permitting a finer understanding of this isomorphism. We fill in some technical details, including the Morse-Bott direct limit argument and some writhe bounds. The former requires the isomorphism between filtered Seiberg-Witten Floer cohomology and filtered ECH as established by Hutchings--Taubes. The latter requires the work on higher asymptotics of pseudoholomorphic curves by Cristofaro-Gardiner--Hutchings—Zhang.
A ball and a cube looks so different, but in higher dimension, it turns out a high dimensional ball and a high dimensional cube could be hard to distinguish them. Our intuitions on 3 dimensional geometry often fails in higher dimension! In this talk, we will start from the basic mathematical definition of high dimensional spaces. Then we will explore some phenomenons of high dimensional convex geometry. In the end, we will show how these nice observations could be applied to speed up algorithms in computer science.
An $A$-path is a path whose intersection with a vertex set $A$ is exactly its endpoints. We show that, for all primes $p$, the family of $A$-paths of length $0 \,\mathrm{mod}\, p$ satisfies an approximate packing-covering duality known as the Erdős-Pósa property. This answers a recent question of Bruhn and Ulmer. We also show that, if $m$ is an odd prime power, then for all integers $L$, the family of cycles of length $L \,\mathrm{mod}\, m$ satisfies the Erdős-Pósa property. This partially answers a question of Dejter and Neumann-Lara from 1987 on characterizing all such integer pairs $L$ and $m$. Both results are consequences of a structure theorem which refines the Flat Wall Theorem of Robertson and Seymour to undirected group-labelled graphs analogously to a result of Huynh, Joos, and Wollan in the directed setting. Joint work with Robin Thomas.
In 1925, Heisenberg introduced non-commutativity of coordinates, now known as quantization, to explain the spectral lines of atoms. In topology, finding quantizations of (symplectic or more generally Poisson) spaces can reveal more intricate structures on them. In this talk, we will introduce the main ingredients of quantization. As a concrete example, we will discuss the SL2-character variety, which is closely related to the Teichmüller space, and the skein algebra as its quantization.
Spaces of bounded mean oscillation (BMO) are relatively
large function spaces that are often used in place
of L^\infinity to do basic Fourier analysis.
It is not well-understood how geometric properties
of the underlying point space enters into the functional
analysis of BMO. I will describe recent work with
Galia Dafni and Ryan Gibara, where we take some
steps towards geometric inequalities.
Specifically, we show that the symmetric decreasing
rearrangement in n-dimensions is bounded, but not
continuous in BMO. The question of sharp bounds
remains open.
The talk considers the Popularity Adjusted Block model (PABM) introduced by Sengupta and Chen (2018). We argue that the main appeal of the PABM is the flexibility of the spectral properties of the graph which makes the PABM an attractive choice for modeling networks that appear in, for example, biological sciences. In addition, to the best of our knowledge, the PABM is the only stochastic block model that allows to treat the network sparsity as the structural sparsity that describes community patterns, rather than being an attribute of the network as a whole.
Link to Zoom meeting: https://ucf.zoom.us/j/92646603521?pwd=TnRGSVo1WXo2bjE4Y3JEVGRPSmNWQT09
Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
Noetherian operators are differential operators that encode primary components of a polynomial ideal. We develop a framework, as well as algorithms, for computing Noetherian operators with local dual spaces, both symbolically and numerically. For a primary ideal, such operators provide an alternative representation to one given by a set of generators. This description fits well with numerical algebraic geometry, taking a step toward the goal of numerical primary decomposition. This is joint work with Justin Chen, Robert Krone and Anton Leykin.
I describe my ongoing work using tools from computational and combinatorial algebraic geometry to classify minimal problems and identify which can be solved efficiently. I will not assume any background in algebraic geometry or computer vision.
Structure-from-motion algorithms reconstruct a 3D scene from many images, often by matching features (such as point and lines) between the images. Matchings lead to constraints, resulting in a nonlinear system of polynomial equations that recovers the 3D geometry. Since many matches are outliers, these methods are used in an iterative framework for robust estimation called RANSAC (RAndom SAmpling And Consensus), whose efficiency hinges on using a small number of correspondences in each iteration. As a result, there is a big focus on constructing polynomial solvers for these "minimal problems" that run as fast as possible. Our work classifies these problems in cases of practical interest (calibrated cameras, complete and partial visibility.) Moreover, we identify candidates for practical use, as quantified by "algebraic complexity measures" (degree, Galois group.)
joint w/ Anton Leykin, Kathlen Kohn, Tomas Pajdla arxiv1903.10008 arxiv2003.05015+ Viktor Korotynskiy, TP, and Margaret Regan (ongoing.)
The SL2 skein algebra of a surface is built from diagrams of curves on the surface. To multiply two diagrams, we draw one diagram on top of the other and then resolve the crossings with the Kauffman bracket. If we replace SL2 with another quantum group, we replace curves by embedded graphs on the surface. Recently, Thang Le showed that the SL2 skein algebra has a nice decomposition into simpler algebras whenever the surface has an ideal triangulation. This triangular decomposition is a powerful tool and should help us to study other skein algebras if we are able to show that the necessary ingredients exist. In this talk, I will explain what these ingredients are and how to find them for the SL3 skein algebra of trivalent webs on a surface.
Our mantra throughout the talk will be simple, "Train tracks approximate simple closed curves." Our goal will be to explore some examples of train tracks, draw some meaningful pictures, and develop an analogy between train tracks and another well known method of approximation. No great knowledge of anything is required for this talk as long as one is willing to squint their eyes at their computer's screen a bit at times.
Whitney proved in 1931 that 4-connected planar triangulations are Hamiltonian. Hakimi, Schmeichel, and Thomassen conjectured in 1979 that if $G$ is a 4-connected planar triangulation with $n$ vertices, then $G$ contains at least $2(n-2)(n-4)$ Hamiltonian cycles, with equality if and only if $G$ is a double wheel. On the other hand, a recent result of Alahmadi, Aldred, and Thomassen states that there are exponentially many Hamiltonian cycles in 5-connected planar triangulations. In this paper, we consider 4-connected planar $n$-vertex triangulations $G$ that do not have too many separating 4-cycles or have minimum degree 5. We show that if $G$ has $O(n/\log_2 n)$ separating 4-cycles then $G$ has $\Omega(n^2)$ Hamiltonian cycles, and if $\delta(G) \ge 5$ then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles. Both results improve previous work. Moreover, the proofs involve a "double wheel" structure, providing further evidence to the above conjecture.
Poincare Conjecture, undoubtedly, is the most influential and challenging problem in the world of Geometry and Topology. Over a century, it has left it’s mark on developing the rich theory around it. In this talk I will give a brief history of the development of Topology and then I will focus on the Exotic behavior of manifolds. In the last part of the talk, I will concentrate more on the theory of 4-manifolds.
Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.
Traditionally, the sandpile group is defined on a graph and the Matrix-Tree Theorem says that this group's size is equal to the number of spanning trees. An extension of the Matrix-Tree Theorem gives a relationship between the sandpile group and bases of an arithmetic matroid. I provide a family of combinatorially meaningful maps between these two sets. This generalizes a bijection given by Backman, Baker, and Yuen and extends work by Duval, Klivans, and Martin.
Please note the unusual time/day.
A finite connected acyclic graph is called a tree. Both properties - connectivity and being acyclic - make very good sense in higher dimensions as well. This has led Gil Kalai (1983) to define the notion of a $d$-dimensional hypertree for $d > 1$. The study of hypertrees is an exciting area of research, and I will try to give you a taste of the many open questions and what we know (and do not know) about them. No specific prior background is assumed.
The talk is based on several papers. The list of coauthors on these papers includes Roy Meshulam, Mishael Rosenthal, Yuval Peled, Lior Aronshtam, Tomsz Luczak, Amir Dahari, Ilan Newman and Yuri Rabinovich.
This talk will introduce a precise high-dimensional asymptotic theory for Boosting (AdaBoost) on separable data, taking both statistical and computational perspectives. We will consider the common modern setting where the number of features p and the sample size n are both large and comparable, and in particular, look at scenarios where the data is asymptotically separable. Under a class of statistical models, we will provide an (asymptotically) exact analysis of the generalization error of AdaBoost, when the algorithm interpolates the training data and maximizes an empirical L1 margin. On the computational front, we will provide a sharp analysis of the stopping time when boosting approximately maximizes the empirical L1 margin. Our theory provides several insights into properties of Boosting; for instance, the larger the dimensionality ratio p/n, the faster the optimization reaches interpolation. At the heart of our theory lies an in-depth study of the maximum L1-margin, which can be accurately described by a new system of non-linear equations; we analyze this margin and the properties of this system, using Gaussian comparison techniques and a novel uniform deviation argument. Time permitting, I will present analogous results for a new class of boosting algorithms that correspond to Lq geometry, for q>1. This is based on joint work with Tengyuan Liang.
The goal of this talk is to present a summary of Sam Payne's 2009 paper "Analytification is the limit of all tropicalizations" (Math. Res. Lett. 16, no. 3 543–556). We will introduce Berkovich analytic spaces, tropicalization of projective varieties, and tropicalization of closed subvarieties of toric varieties, as well as the connections between these concepts. We will try to present many examples.
Note: Part I will focus on tropicalization of affine varieties and Berkovich analytic spaces, Part II will focus on tropicalization of toric varieties and discuss Sam Payne's theorem.
Learning latent structures in noisy data has been a central task in statistical and computational sciences. For applications such as ranking, matching and clustering, the structure of interest is non-convex and, furthermore, of combinatorial nature. This talk will be a gentle introduction to selected models and methods for statistical inference of such combinatorial structures. I will particularly discuss some of my recent research interests.
In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In k-clustering, opening k facilities induces an assignment cost vector across the clients. In this paper we consider the following minimum norm optimization problem : given an arbitrary monotone, symmetric norm, find a solution which minimizes the norm of the induced cost-vector. This generalizes many fundamental NP-hard problems. We give a general framework to tackle the minimum norm problem, and illustrate its efficacy in load balancing and, time permitting, in the clustering setting.
In many optimization problems, a feasible solution induces a multi-dimensional cost vector. For example, in load-balancing a schedule induces a load vector across the machines. In k-clustering, opening k facilities induces an assignment cost vector across the clients. In this paper we consider the following minimum norm optimization problem : given an arbitrary monotone, symmetric norm, find a solution which minimizes the norm of the induced cost-vector. This generalizes many fundamental NP-hard problems. We give a general framework to tackle the minimum norm problem, and illustrate its efficacy in load balancing and, time permitting, in the clustering setting.
(The speaker is an ACO alum; after the lecture, the speaker will engage with the ACO students for 30-45 minutes.)
Many data set in image analysis and signal processing is in a high-dimensional space but exhibit low-dimensional structures. For example, data can be modeled as point clouds in a high-dimensional space but are concentrated on a low-dimensional set (or manifold in particular). Our goal is to estimate functions on the low-dimensional manifold from finite samples of data, for statistical inference and prediction. This talk introduces approximation theories of neural networks for functions supported on a low-dimensional manifold. When the function is estimated from finite samples, we give an estimate of the mean squared error for the approximation of these functions. The convergence rate depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that neural networks are adaptive to low-dimensional geometric structures of data.
We improve on some recent results of Sagiv and Steinerberger that quantify the following uncertainty principle: for a function f with mean zero, then either the size of the zero set of the function or the cost of transporting the mass of the positive part of f to its negative part must be big. We also provide a sharp upper estimate of the transport cost of the positive part of an eigenfunction of the Laplacian.
This proves a conjecture of Steinerberger and provides a lower bound of the size of a nodal set of the eigenfunction. Finally, we use a similar technique to provide a measure of how well the points in a design in a manifold are equidistributed. This is a joint work with Tom Carroll and Xavier Massaneda.
A classical question in extremal graph theory asks to maximize the number of induced copies of a given graph or tournament in a large host graph, often expressed as a density. A simple averaging argument shows that the limit of this density exists as the host graph is allowed to grow. Razborov's flag algebra method is well suited to generate bounds on these quantities with the help of semidefinite programming. We will explore this method for a few small examples, and see how to modify it to fit our questions. The extremal graphs show some beautiful structures, sometimes fractal like, sometimes quasi random and sometimes even a combination of both.
Cabling is one of important knot operations. We study various properties of cable knots and how to characterize the cable knots by its complement.
Kernel principal component analysis (KPCA) is a popular non-linear dimensionality reduction technique, which generalizes classical linear PCA by finding functions in a reproducing kernel Hilbert space (RKHS) such that the function evaluation at a random variable $X$ has a maximum variance. Despite its popularity, kernel PCA suffers from poor scalability in big data scenarios as it involves solving a $n \times n$ eigensystem leading to the computational complexity of $O(n^3)$ with $n$ being the number of samples. To address this issue, in this work, we consider a random feature approximation to kernel PCA which requires solving an $m \times m$ eigenvalue problem and therefore has a computational complexity of $O(m^3+nm^2)$, implying that the approximate method is computationally efficient if $m$ < $n$ with $m$ being the number of random features. The goal of this work is to investigate the trade-off between computational and statistical behaviors of approximate KPCA, i.e., whether the computational gain is achieved at the cost of statistical efficiency. We show that the approximate KPCA is both computationally and statistically efficient compared to KPCA in terms of the error associated with reconstructing a kernel function based on its projection onto the corresponding eigenspaces.
Link to Cisco Webex meeting: https://gatech.webex.com/gatech/j.php?MTID=mdd4512d3d11623149a0bd46d9fc086c8
The goal of this talk is to present a summary of Sam Payne's 2009 paper "Analytification is the limit of all tropicalizations" (Math. Res. Lett. 16, no. 3 543–556). We will introduce Berkovich analytic spaces, tropicalization of projective varieties, and tropicalization of closed subvarieties of toric varieties, as well as the connections between these concepts. We will try to present many examples.
Note: Part I will focus on tropicalization of affine varieties and Berkovich analytic spaces, Part II will focus on tropicalization of toric varieties and discuss Sam Payne's theorem.
Semidefinite programming is a powerful optimization tool, which involves optimizing linear functions on a slice of the positive semidefinite matrices. Locally PSD matrices are a natural relaxation of the PSD matrices which can be useful in reducing the space required for semidefinite optimization. We use the theory of hyperbolic polynomials to give precise quantitative bounds on the quality of the approximation resulting from optimizing over the locally-psd cone instead of the PSD cone.
We study discrepancy minimization for vectors in R^n under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Komlós conjecture.
Based on joint work with Alweiss and Sawhney, see https://arxiv.org/abs/2006.14009
Given a contact structure on a bordered 3-manifold, we describe an invariant which takes values in the bordered sutured Floer homology of the manifold. This invariant satisfies a nice gluing formula, and recovers the Oszvath-Szabo contact class in Heegaard Floer homology. This is joint work with Alishahi, Foldvari, Hendricks, Licata, and Vertesi.
Zoom info:
Meeting ID: 980 3103 5804
Passcode: 196398
Many systems of nonlinear PDEs are arising from engineering and biology and have attracted research scientists to study the multiple solution structure such as pattern formation. In this talk, I will present several methods to compute the multiple solutions of nonlinear PDEs. In specific, I will introduce the homotopy continuation technique to compute the multiple steady states of nonlinear differential equations and also to explore the relationship between the number of steady-states and parameters. Then I will also introduce a randomized Newton's method to solve the nonlinear system arising from neural network discretization of the nonlinear PDEs. Several benchmark problems will be used to illustrate these ideas.
In this talk we shall give a brief introduction to the mathematics of soap films (aka minimal surfaces). These are the surfaces that, amongst all possible surfaces with prescribed boundary, have the least area. If one dips a wire mesh into soap solution, then the surface formed is a minimal surface. We shall see how minimal surfaces arise in science and engineering, look at the physical laws that a minimal surface should obey, and see how much mathematicians understand about them.
For positive integers $d < k$ and $n$ divisible by $k$, let $m_d(k,n)$ be the minimum $d$-degree ensuring the existence of a perfect matching in a $k$-uniform hypergraph. In the graph case (where $k=2$), a classical theorem of Dirac says that $m_1(2,n) = \lceil n/2\rceil$. However, in general, our understanding of the values of $m_d(k,n)$ is still very limited, and it is an active topic of research to determine or approximate these values. In the first part of this talk, we discuss a new "transference" theorem for Dirac-type results relative to random hypergraphs. Specifically, we prove that a random $k$-uniform hypergraph $G$ with $n$ vertices and "not too small" edge probability $p$ typically has the property that every spanning subgraph with minimum $d$-degree at least $(1+\varepsilon)m_d(k,n)p$ has a perfect matching. One interesting aspect of our proof is a "non-constructive" application of the absorbing method, which allows us to prove a bound in terms of $m_d(k,n)$ without actually knowing its value.
The ideas in our work are quite powerful and can be applied to other problems: in the second part of this talk we highlight a recent application of these ideas to random designs, proving that a random Steiner triple system typically admits a decomposition of almost all its triples into perfect matchings (that is to say, it is almost resolvable).
Joint work with Asaf Ferber.
https://bluejeans.com/808204151
I will describe results from two recent projects in tropical geometry with relevance in applications. In the first half, I will introduce and give several characterizations for flags of tropical linear spaces, in analogy to Speyer's results for tropical linear spaces. In the second half, I will discuss current work relating tropical fewnomials, vertex bounds of Minkowski sums, and linear regions of maxout neural networks.
(Joint work with Timo Seppäläinen) We establish estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model with i.i.d. exponential weights. There are four estimates: upper and lower bounds on the probabilities of both fast and slow coalescence on the correct spatial scale with exponent 3/2. Our proofs utilize a geodesic duality introduced by Pimentel and properties of the increment-stationary last-passage percolation process. For fast coalescence our bounds are new and they have matching optimal exponential order of magnitude. For slow coalescence, we reproduce bounds proved earlier with integrable probability inputs, except that our upper bound misses the optimal order by a logarithmic factor.
Teams Link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1600608874868?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%223eebc7e2-37e7-4146-9038-a57e56c92d31%22%7d
Locally PSD matrices are a generalization of PSD matrices which appear in sparse semidefinite programming. We will try to explore some connections of extreme rays of this type of matrix with algebraic topology.
Recall that an excedance of a permutation $\pi$ is any position $i$
such that $\pi_i > i$. Inspired by the work of Hopkins, McConville and
Propp (arXiv:1612.06816) on sorting using toppling, we say that
a permutation is toppleable if it gets sorted by a certain sequence of
toppling moves. For the most part of the talk, we will explain the
main ideas in showing that the number of toppleable permutations on n
letters is the same as those for which excedances happen exactly at
$\{1,\dots, \lfloor (n-1)/2 \rfloor\}$. Time permitting, we will give
some ideas showing that this is also the number of acyclic
orientations with unique sink (also known as the Ursell function) of the
complete bipartite graph $K_{\lceil n/2 \rceil, \lfloor n/2 \rfloor + 1}$.
This is joint work with D. Hathcock (CMU) and P. Tetali (Georgia Tech).
There are many ways to study surfaces: topologically, geometrically, dynamically, algebraically, and combinatorially, just to name a few. We will touch on some of the motivation for studying surfaces and their associated mapping class groups, which is the collection of symmetries of a surface. We will also describe a few of the ways that these different perspectives for studying surfaces come together in beautiful ways.
A cobordism between 3-manifolds is ribbon if it is built from handles of index no greater than 2. Such cobordisms arise naturally from several different topological and geometric contexts. In this talk, we discuss these objects and present a few obstructions to their existence, from Thurston geometries, character varieties, and instanton and Heegaard Floer homologies. This is joint work with Aliakbar Daemi, Tye Lidman, and Mike Wong.
Cryo-Electron Microscopy (cryo-EM) is an imaging technology that is revolutionizing structural biology. Cryo-electron microscopes produce a large number of very noisy two-dimensional projection images of individual frozen molecules; unlike related methods, such as computed tomography (CT), the viewing direction of each particle image is unknown. The unknown directions, together with extreme levels of noise and additional technical factors, make the determination of the structure of molecules challenging. While other methods for structure determination, such as x-ray crystallography and nuclear magnetic resonance (NMR), measure ensembles of molecules, cryo-electron microscopes produce images of individual molecules. Therefore, cryo-EM could potentially be used to study mixtures of different conformations of molecules. Indeed, current algorithms have been very successful at analyzing homogeneous samples, and can recover some distinct conformations mixed in solutions, but, the determination of multiple conformations, and in particular, continuums of similar conformations (continuous heterogeneity), remains one of the open problems in cryo-EM. In practice, some of the key components in “molecular machines” are flexible and therefore appear as very blurry regions in 3-D reconstructions of macro-molecular structures that are otherwise stunning in resolution and detail.
We will discuss “hyper-molecules,” the mathematical formulation of heterogenous 3-D objects as higher dimensional objects, and the machinery that goes into recovering these “hyper-objects” from data. We will discuss some of the statistical and computational challenges, and how they are addressed by merging data-driven exploration, models and computational tools originally built for deep-learning.
This is joint work with Joakim Andén and Amit Singer.
Random and irregular growth is all around us: tumor growth, fluid flow through porous media, and the spread of bacterial colonies. Simple models for these processes originated in the '50s with percolation theory and have since given rise to many new models and interesting mathematics. I will introduce a few models (percolation, invasion percolation, first-passage percolation, diffusion-limited aggregation, ...), along with some of their basic properties.
The Yamabe problem asks whether, given a closed Riemannian manifold, one can find a conformal metric of constant scalar curvature (CSC). An affirmative answer was given by Schoen in 1984, following contributions from Yamabe, Trudinger, and Aubin, by establishing the existence of a function that minimizes the so-called Yamabe energy functional; the minimizing function corresponds to the conformal factor of the CSC metric.
We address the quantitative stability of minimizing Yamabe metrics. On any closed Riemannian manifold we show—in a quantitative sense—that if a function nearly minimizes the Yamabe energy, then the corresponding conformal metric is close to a CSC metric. Generically, this closeness is controlled quadratically by the Yamabe energy deficit. However, we construct an example demonstrating that this quadratic estimate is false in the general. This is joint work with Max Engelstein and Luca Spolaor.
In the first part of the talk, I will show that for two bivariate polynomials $P(x,y)$ and $Q(x,y)$ with coefficients in fields with char 0 to simultaneously exhibit small expansion, they must exploit the underlying additive or multiplicative structure of the field in nearly identical fashion. This in particular generalizes the main result of Shen and yields an Elekes-Ronyai type structural result for symmetric nonexpanders, resolving a question mentioned by de Zeeuw (Joint with S. Roy and C-M. Tran). In the second part of the talk, I will show how this sum-product phenomena helps us avoid color-isomorphic even cycles in proper edge colorings of complete graphs (Joint with G. Ge, Z. Xu, and T. Zhang).
Dense graph limit theory is essentially a first-order limit theory analogous to the classical Law of Large Numbers. Is there a corresponding central limit theorem? We believe so. Using the language of Gaussian Hilbert Spaces and the comprehensive theory of generalised U-statistics developed by Svante Janson in the 90s, we identify a collection of Gaussian measures (aka white noise processes) that describes the fluctuations of all orders of magnitude for a broad family of random graphs. We complement the theory with error bounds using a new variant of Stein’s method for multivariate normal approximation, which allows us to also generalise Janson’s theory in some important aspects. This is joint work with Gursharn Kaur.
Please note the unusual time: 5pm
Dense graph limit theory is essentially a first-order limit theory analogous to the classical Law of Large Numbers. Is there a corresponding central limit theorem? We believe so. Using the language of Gaussian Hilbert Spaces and the comprehensive theory of generalised U-statistics developed by Svante Janson in the 90s, we identify a collection of Gaussian measures (aka white noise processes) that describes the fluctuations of all orders of magnitude for a broad family of random graphs. We complement the theory with error bounds using a new variant of Stein’s method for multivariate normal approximation, which allows us to also generalise Janson’s theory in some important aspects. This is joint work with Gursharn Kaur.
Please note the unusual time/day.
This skit recounts one of the foundation stories of mathematics, the puzzle of the Seven Bridges of Königsberg, solved by Euler in 1726. Except that it all takes place in a mad courtroom, and you are the jury!
A sunflower with p petals consists of p sets whose pairwise intersections are all the same set. The goal of the sunflower problem is to find the smallest r = r(p,k) such that every family of at least r^k k-element sets must contain a sunflower with p petals. Major breakthroughs within the last year by Alweiss-Lovett-Wu-Zhang and others show that r = O(p log(pk)) suffices. In this talk, after reviewing the history and significance of the Sunflower Problem, I will present our improvement to r = O(p log k), which we obtained during the 2020 REU at Georgia Tech. As time permits, I will elaborate on key lemmas and techniques used in recent improvements.
Based on joint work with Suchakree Chueluecha (Lehigh University) and Lutz Warnke (Georgia Tech), see https://arxiv.org/abs/2009.09327
We introduce a construction of oriented matroids from any triangulation of a product of two simplices, extending the regular case which follows from signed tropicalization. For this, we use the structure of such a triangulation in terms of polyhedral matching fields. The oriented matroid is composed of compatible chirotopes on the cells in a matroid subdivision of the hypersimplex, which might be of independent interest. We will also describe the extension to matroids over hyperfields and sketch some connections with optimization. This is joint work with Marcel Celaya and Georg Loho; Marcel Celaya will be giving a talk on the topological aspect of the work at the algebra seminar next week.
Please note the unusual time: 4pm
The remarkable success of deep learning in computer science has evinced potentially great applications of deep learning in computational and applied mathematics. Understanding the mathematical principles of deep learning is crucial to validating and advancing deep learning-based scientific computing. We present a few thoughts on the theoretical foundation of this topic and our methodology for designing efficient solutions of high-dimensional and highly nonlinear partial differential equations, mainly focusing on the approximation and optimization of deep neural networks.
The smallest volume cusped hyperbolic 3-manifolds, the figure-eight knot complement and its sister, contain many immersed but no embedded closed totally geodesic surfaces. In this talk we discuss the existence or lack thereof of codimension-1 closed embedded totally geodesic submanifolds in minimal volume cusped hyperbolic 4-manifolds. This talk is based on joint work with Alan Reid.
In 1665, Huygens observed that two pendulum clocks hanging from the same board became synchronized in antiphase after hundreds of swings. On the other hand, modern experiments with metronomes placed on a movable platform show that they tend to synchronize in phase, not antiphase. Here, using a simple model of coupled clocks and metronomes, we calculate the regimes where antiphase and in-phase synchronization are stable. Unusual features of our approach include its treatment of the escapement mechanism, a small-angle approximation up to cubic order, and a three-time scale asymptotic analysis.
For any nonnegative density f and radially decreasing interaction potential W, the celebrated Riesz rearrangement inequality shows the interaction energy E[f] = \int f(x)f(y)W(x-y) dxdy satisfies E[f] <= E[f^*], where f^* is the radially decreasing rearrangement of f. It is a natural question to look for a quantitative version of this inequality: if its two sides almost agree, how close must f be to a translation of f^*? Previously the stability estimate was only known for characteristic functions. I will discuss a recent work with Xukai Yan, where we found a simple proof of stability estimates for general densities.
I will also discuss another work with Matias Delgadino and Xukai Yan, where we constructed an interpolation curve between any two radially decreasing densities with the same mass, and show that the interaction energy is convex along this interpolation. As an application, this leads to uniqueness of steady states in aggregation-diffusion equations with any attractive interaction potential for diffusion power m>=2, where the threshold is sharp.
Erdős, Pach, Pollack and Tuza conjectured that for fixed integers $r$, $\delta \ge 2$, for any connected graph $G$ with minimum degree $\delta$ and order $n$:
(i) If $G$ is $K_{2r}$-free and $\delta$ is a multiple of $(r-1)(3r+2)$, then, as $n$ tends to infinity, the diameter of $G$ is at most $\frac{2(r-1)(3r+2)}{(2r^2-1)} \cdot \frac{n}{\delta} + O(1)$.
(ii) If $G$ is $K_{2r+1}$-free and $\delta$ is a multiple of $3r-1$, then, as $n$ tends to infinity, the diameter of $G$ is at most $\frac{3r-1}{r} \cdot \frac{n}{\delta} + O(1)$.
They created examples that show that the above conjecture, if true, is tight.
No more progress has been reported on this conjecture, except that for $r=2$ in (ii), under a stronger hypothesis ($4$-colorable instead of $K_5$-free), Czabarka, Dankelman and Székely showed that for every connected $4$-colorable graph $G$ of order $n$ and minimum degree $\delta \ge 1$, the diameter of $G$ is at most $\frac{5n}{2\delta} - 1$.
For every $r>1$ and $\delta \ge 2(r-1)$, we create $K_{2r}$-free graphs with minimum degree $\delta$ and diameter $\frac{(6r-5)n}{(2r-1)\delta+2r-3}+O(1)$, which are counterexamples to the conjecture for every $r>1$ and $\delta > 2(r-1)(3r+2)(2r-3)$. We also prove positive results under a stronger hypothesis, $k$-colorability, instead of being $K_{k+1}$-free. We show that the diameter of connected $k$-colorable graphs with minimum degree at least $\delta$ and order $n$ is at most $\left(3-\frac{1}{k-1}\right)\frac{n}{\delta}+O(1)$, while for $k=3$, it is at most $\frac{57n}{23\delta}+O(1)$.
This is joint work with Inne Singgih and László A. Székely.
A classical result on oriented matroids due to Folkman and Lawrence in
1978 states that they are in bijection with pseudosphere arrangements up
to cellular homeomorphism. A more recent result, conjectured by Ardila and
Develin in 2007 and proved by Silke Horn in 2016, states that a similar
result holds for tropical oriented matroids and tropical hyperplane
arrangements. In a joint work with Georg Loho and Chi Ho Yuen, we show how
to unify these two results based on a variant of Viro's patchworking
technique, generalized to complete intersections by Sturmfels, for a
certain class of uniform oriented matroids arising from a product of two
simplices.
In 1946, Dennis Gabor claimed that any Lebesgue square-integrable function can be written as an infinite linear combination of time and frequency shifts of the standard Gaussian. Since then, decomposition methods for larger classes of functions or distributions in terms of various elementary building blocks have lead to an impressive body of work in harmonic analysis. For example, Gabor analysis, which originated from Gabor's claim, is concerned with both the theory and the applications of the approximation properties of sets of time and frequency shifts of a given function. It re-emerged with the advent of wavelets at the end of the last century and is now at the intersection of many fields of mathematics, applied mathematics, engineering, and science. In this talk, I will introduce the fundamentals of the theory highlighting some applications and open problems.
We will discuss a few beautiful questions in high-dimensional convexity, and path their connections to areas such as Analysis, Probability Theory and Differential Geometry. I shall mention some of my recent results too, in particular a new inequality about convex sets in high dimensions. I will describe its relations to one of the difficult problems in the area.
The crossing number of a graph is the minimum number of crossings it can be drawn in a plane. Let $\kappa(n, m)$ be the minimum crossing number of graphs with $n$ vertices and (at least) $m$ edges. Erd\H{o}s and Guy conjectured and Pach, Spencer and T\'oth proved that for any $m = m(n)$ satisfying $n \ll m \ll n^2$, the quatity $\ds\lim_{n \to \infty} \frac{\kappa(n,m) n^2}{m^3}$ exists and is positive. The $k$-planar crossing number of a graph is the minimum crossing number obtained when we partition the edges of the graph into $k$ subgraphs and draw them in $k$ planes. Using designs and a probabilistic algorithm, the guaranteed factor of improvement $\alpha_k$ between the $k$-planar and regular crossing number is $\frac{1}{k^2} (1 + o(1))$, while if we restrict our attention to biplanar graphs, this constant is $\beta_k = \frac{1}{k^2}$ exactly. The lower bound proofs require the existence of a midrange crossing constant. Motivated by this, we show that the midrange crossing constant exists for all graph classes (including bipartite graphs) that satisfy certain mild conditions. The regular midrange crossing constant was shown to be is at most $\frac{8}{9\pi^2}$; we present a probabilistic construction that also shows this bound.
Knots/links associated to overtwisted contact structures have been less explored. There are two types of knots/links in overtwisted contact manifolds, namely loose and non-loose. In this talk, I will start with an overview of these knots and then discuss some of my recent work involving these knots and links. Specifically, I will talk about a coarse classification result of loose, null-homologous Legendrian and transverse links . Next relating them with open book decompositions, I will show that coarse equivalence class of loose null-homologous Legendrian links has support genus zero. I will end with some interesting open questions.
https://gatech.bluejeans.com/759112674
We say that a set is convex if for any two points in the set, the straight line segment connecting them is also contained in the set. For example, a triangle, a square, a cube, a ball are all convex sets. We typically speak of convex sets in Euclidean space with the ordinary addition and multiplication operations. What happens if we replace addition with taking the minimum between two elements, and multiplication with ordinary addition? These are the tropical arithmetic operations and using these we can define tropical convexity. What does it mean for a set to be tropically convex? What does a tropical triangle look like? In this talk we will answer these questions and explore how ordinary and tropical convexity interact.
Neural network-based machine learning methods, inlcuding the most notably deep learning have achieved extraordinary successes in numerious fields. In spite of the rapid development of learning algorithms based on neural networks, their mathematical analysis are far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems.
In this talk, I will demonstrate the power of neural network methods for solving two classes of high dimensional problems: statistical sampling and PDEs. In the first part of the talk, I will present a universal approximation theorem of deep neural networks for representing high dimensional probability distributions. In the second part of the talk, I will discuss a generalization error bound of the Deep Ritz Method for solving high dimensional elliptic problems. For both problems, our theoretical results show that neural networks-based methods can overcome the curse of dimensionality.
Let $F$ be a graph. We say a hypergraph $H$ is a trace of $F$ if there exists a bijection $\phi$ from the edges of $F$ to the hyperedges of $H$ such that for all $xy \in E(F)$, $\phi(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in an $r$-uniform hypergraph that does not contain a trace of $F$. We also obtain better bounds in the case $F = K_{2,t}$. This is joint work with Zoltán Füredi and Sam Spiro.
The motion of a forced vibro-impacting inclined energy harvester is investigated in parameter regimes with asymmetry in the number of impacts on the bottom and top of the device. This motion occurs beyond a grazing bifurcation, at which alternating top and bottom impacts are supplemented by a zero velocity impact with the bottom of the device. For periodic forcing, we obtain semi-analytical expressions for the asymmetric periodic motion with a ratio of 2:1 for the impacts on the device bottom and top, respectively. These expressions are derived via a set of nonlinear maps between different pairs of impacts, combined with impact conditions that provide jump dis continuities in the velocity. Bifurcation diagrams for the analytical solutions are complemented by a linear stability analysis around the 2:1 asymmetric periodic solutions, and are validated numerically. For smaller incline angles, a second grazing bifurcation is numerically detected, leading to a 3:1 asymmetry. For larger incline angles, period doubling bifurcations precede this bifurcation. The converted electrical energy per impact is reduced for the asymmetric motions, and therefore less desirable under this metric.
Bluejeans link: https://bluejeans.com/893955256
We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss some uniqueness results for the underlying (bi-) contact structure for an Anosov flow, and/or a characterization of Anosovity based on Reeb flows.
We consider the ramifications of utilizing biased latent position estimates in subsequent statistical analysis in exchange for sizable variance reductions in finite networks. We establish an explicit bias-variance tradeoff for latent position estimates produced by the omnibus embedding in the presence of heterogeneous network data. We reveal an analytic bias expression, derive a uniform concentration bound on the residual term, and prove a central limit theorem characterizing the distributional properties of these estimates.
Link to the BlueJeans meeting https://bluejeans.com/974631214
Teams meeting link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
If we can write a (homogeneous) polynomial as a sum of squares(SOS), the polynomial is guaranteed to be a non-negative polynomial. However, every non-negative forms does not have to be written as sums of squares in general. This implies that set of sums of square is strictly contained in set of non-negative forms in general. We want to discuss about one way to describe the gaps between the two sets. Since the sets have cone structures, we can consider dual cones of each cones. In particular, the description of dual cone of non-negative polynomials is simple: convex hull of point evaluations. Therefore, we are interested in positive semi-definite quadratic forms that is not point evaluations. We call "Hankel index" the minimal rank of quadratic form (on extreme ray of the dual cone of SOS) which is not a point evaluation. In this talk, we introduce the Hankel index of variety and will discuss about a criterion to obtain the Hankel index of projected rational curves.
The Banach--Tarski paradox is one of the most counterintuitive facts in all of mathematics. It says that it is possible to divide the 3-dimensional unit ball into a finite number of pieces, move the pieces around (without changing their shape), and then put them back together to form two identical copies of the original ball. Many other, equally difficult to believe, equidecomposition statements are also true. For example, a ball of radius 1 can be split into finitely many pieces, which can then be rearranged to form a ball of radius 1000. It turns out that such statements are best understood through the lens of graph theory. I will explain this connection and discuss some recent breakthroughs it has led to.
https://bluejeans.com/819527897/5512
A multi-electrode array-based application for the long-term recording of action potentials from electrogenic cells makes possible exciting cardiac electrophysiology studies in health and disease. With hundreds of simultaneous electrode recordings being acquired over a period of days, the main challenge becomes achieving reliable signal identification and quantification. We set out to develop an algorithm capable of automatically extracting regions of high-quality action potentials from terabyte size experimental results and to map the trains of action potentials into a low-dimensional feature space for analysis. Our automatic segmentation algorithm finds regions of acceptable action potentials in large data sets of electrophysiological readings. We use spectral methods and support vector machines to classify our readings and to extract relevant features. We show that action potentials from the same cell site can be recorded over days without detrimental effects to the cell membrane. The variability between measurements 24 h apart is comparable to the natural variability of the features at a single time point. Our work contributes towards a non-invasive approach for cardiomyocyte functional maturation, as well as developmental, pathological, and pharmacological studies.
This is joint work with Dr. Viviana Zlochiver (Advocate Aurora Research Institute) and John Jurkiewicz (graduate student at UWM).
Meeting room: https://bluejeans.com/819527897/5512
We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.
A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.
Computing, understanding the behavior of concordance invariants obtained from knot Floer homology theories is quite central to the study of the concordance group and low-dimensional topology in general. In this talk, I will describe the method that allows us to compute the concordance invariant epsilon using combinatorial knot Floer homology and talk about some computational results. This is a joint work with S. Dey.
Ranking items from comparisons is a ubiquitous task in many real-world applications. For example, sports teams can be ranked based on outcomes of matches; students' homework solutions can be ranked based on peer grading. In this lecture, I will discuss: (1) how we can design mathematical models for the problem of ranking or rating a set of items from pairwise comparisons between them; (2) how to do statistical inference based on the models. The model we focus on is the Bradley-Terry model proposed in 1952, which is also related to the Elo rating system implemented for the US Chess Federation in 1960.
We will discuss Marstrand's classical theorem concerning the interplay between density of a measure and the Hausdorff dimension of the measure's support in the context of finite-dimensional Banach spaces. This is joint work with David Bate and Tatiana Toro.
Note the unusual time!
The following are two classical questions in the area of universal graphs.
1. What is the minimum number of vertices in a graph that contains all $n$-vertex planar graphs as induced subgraphs?
2. What is the minimum number of edges in a graph that contains all $n$-vertex planar graphs as subgraphs?
We give nearly optimal constructions for each problem, i.e. with $n^{1+o(1)}$ vertices for Question 1 and $n^{1+o(1)}$ edges for Question 2. The proofs combine a recent structure theorem for planar graphs (of independent interest) with techniques from data structures.
Joint work with V. Dujmovic, C. Gavoille, G. Joret, P. Micek, and P. Morin.
TBA
Part 1 of 3-part series
The aim of these two talks is to give an overview of our work on tropical Hodge theory. We show that cohomology groups of smooth projective tropical varieties verify hard Lefschetz property and Hodge-Riemann relations. Providing a description of the Chow groups of matroids in terms of cohomology groups of specific smooth projective tropical varieties, these results can be regarded as a generalization of the work of Adiprasito-Huh-Katz to more general tropical varieties. We also prove that smooth projective tropical varieties verify the analogue in the tropical setting of the weight-monodromy conjecture, affirming a conjecture of Mikhalkin and Zharkov.
BlueJeans link: https://bluejeans.com/476849994
Gradient descent algorithms and their noisy variants, such as the Langevin dynamics or multi-pass SGD, are at the center of attention in machine learning. Yet their behaviour remains perplexing, in particular in the high-dimensional non-convex setting. In this talk, I will present several high-dimensional and non-convex statistical learning problems in which the performance of gradient-based algorithms can be analysed down to a constant. The common point of these settings is that the data come from a probabilistic generative model leading to problems for which, in the high-dimensional limit, statistical physics provides exact closed solutions for the performance of the gradient-based algorithms. The covered settings include the spiked mixed matrix-tensor model and the phase retrieval.
For a certain scaling of the initialization of stochastic gradient descent (SGD), wide neural networks (NN) have been shown to be well approximated by reproducing kernel Hilbert space (RKHS) methods. Recent empirical work showed that, for some classification tasks, RKHS methods can replace NNs without a large loss in performance. On the other hand, two-layers NNs are known to encode richer smoothness classes than RKHS and we know of special examples for which SGD-trained NN provably outperform RKHS. This is true also in the wide network limit, for a different scaling of the initialization.
How can we reconcile the above claims? For which tasks do NNs outperform RKHS? If feature vectors are nearly isotropic, RKHS methods suffer from the curse of dimensionality, while NNs can overcome it by learning the best low-dimensional representation. Here we show that this curse of dimensionality becomes milder if the feature vectors display the same low-dimensional structure as the target function, and we precisely characterize this tradeoff. Building on these results, we present a model that can capture in a unified framework both behaviors observed in earlier work. We hypothesize that such a latent low-dimensional structure is present in image classification. We test numerically this hypothesis by showing that specific perturbations of the training distribution degrade the performances of RKHS methods much more significantly than NNs.
Following the Hilbert Basis theorem and its applications, there has been a vast variety of studies involving the chain conditions over the polynomial or the power series rings. One type of chain condition is the Archimedean condition, which says \cap_n Rt_n = 0for any nonunit element t in the ring R. In this talk, we start with the ascending chain condition on principal ideals (ACCP) over a larger class “skew generalized power series rings”. Then we explain the relation between ACCP rings and Archimedean rings and answer partially to the question “when these properties can be lifted from the ring R to the ring R[[x; α]]? ” In particular we show that if R is an Archimedean reduced ring and satisfy ACC on annihilators, then R[[x]] is also an Archimedean reduced ring.
For computational-intensive mixed integer non-linear optimization problems, a major challenge is to verify/guarantee the quality of any feasible solution under mild assumptions in a tractable fashion. In this talk, we focus on tackling this challenge by constructing tight relaxations and designing approximation algorithms for two different mixed integer non-linear optimization problems.
In the first part, we focus on the (row) sparse principal component analysis (rsPCA) problem. Solving rsPCA is the problem of finding the top-r leading principal components of a covariance matrix such that all these principal components are linear combinations of a subset of k variables. The rsPCA problem is a widely used dimensionality reduction tool with an additional sparsity constraint to enhance its interpretability. We propose: (a) a convex integer programming relaxation of rsPCA that gives upper (dual) bounds for rsPCA, and; (b) a new local search algorithm for finding primal feasible solutions for rsPCA. We also show that, in the worst-case, the dual bounds provided by the convex IP are within an affine function of the global optimal value. We demonstrate our techniques applied to large-scale covariance matrices.
In the second part, we focus on improving the execution speed of compute-intensive numerical code. The compute-intensive numerical code, especially of the variety encountered in deep neural network inference and training, is often written using nested for-loops. One of the main bottlenecks that significantly influence the nested for-loops' execution speed is the so-called memory latency. Iteration space tiling is a common memory management technique used to deal with memory latency. We study the problem of automatically optimizing the implementation of these nested loops by formulating the iteration space tiling problem into an integer geometric programming (IGP) problem. We show how to design an efficient approximation algorithm for this problem and how to use the so-called "non-uniform tiling" technique to improve the execution speed.
The first part of the talk is joint work with Santanu S. Dey, Rahul Mazumder, Macro Molinaro, and the second part of the talk is joint work with Ofer Dekel.
Many real life processes that we would like to model have a self-exciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time. Examples of processes that have this property are earthquakes, crime in a neighborhood, or emails within a company. In 1971, Alan Hawkes first used what is now known as the Hawkes process to model such processes. Since then much work has been done on estimating the parameters of a Hawkes process given a data set and creating variants of the process for different applications.
In this talk, I will be proposing a new variant of a Hawkes process that takes into account the effect of police activity on the underlying crime rate and an algorithm for estimating its parameters given a crime data set.
Self-organization is a common feature in the collective behavior of many animal species, such as flocking birds, herding mammals, and swarming bacteria. As the number of individuals gets large, instead of tracking the movement of each individual, it is more efficient to model the evolution of the whole population density using partial differential equations (PDEs). In this talk, I will introduce some PDE models for collective dynamics, and discuss the challenges in both the modeling part and the mathematical analysis.
We shall discuss the proof of pointwise almost everywhere convergence for the non-conventional (in the sense of Furstenberg) bilinear polynomial ergodic averages. This is my recent work with Ben Krause and Terry Tao.
Call a blowup of a graph $F$ an $n$-blowup if each part has size $n$. For a subgraph $G$ of a blowup of $F$, we define the minimum partial degree of $G$ to be the smallest minimum degree over the bipartite subgraphs of $G$ that correspond to edges of $F$. Johannson proved that if the minimum partial degree of a spanning subgraph of the $n$-blowup of a triangle is $2n/3 + n^{1/2}$, then it contains a collection of $n$ vertex disjoint triangles. Fischer's Conjecture, which was proved by Keevash and Mycroft in 2015, is a generalization of this result to complete graphs larger than the triangle. Another generalization, conjectured independently by Fischer and Häggkvist, is the following: If $G$ is a spanning subgraph of the $n$-blowup of $C_k$ with minimum partial degree $(1 + 1/k)n/2 + 1$, then $G$ contains $n$ vertex disjoint copies of $C_k$ that each intersect each of the $k$ parts. In this talk, we will show that this conjecture holds asymptotically. We will also discuss related conjectures and results.
This is joint work with Beka Ergemlidze.
Grid homology is a purely combinatorial description of knot Floer homology in which the counting of psuedo-holomorphic disks is replaced with a counting of polygons in grid diagrams. This talk will provide an introduction to this theory, and is aimed at an audience with little to no experience with Heegaard Floer homology.
Part 2 of 3-part series
The aim of these two talks is to give an overview of our work on tropical Hodge theory. We show that cohomology groups of smooth projective tropical varieties verify hard Lefschetz property and Hodge-Riemann relations. Providing a description of the Chow groups of matroids in terms of cohomology groups of specific smooth projective tropical varieties, these results can be regarded as a generalization of the work of Adiprasito-Huh-Katz to more general tropical varieties. We also prove that smooth projective tropical varieties verify the analogue in the tropical setting of the weight-monodromy conjecture, affirming a conjecture of Mikhalkin and Zharkov.
BlueJeans link: https://bluejeans.com/476849994
The class which is refereed to as the Cauchy family allows for the simultaneous modeling of the long memory dependence and correlation at short and intermediate lags. We introduce a valid parametric family of cross-covariance functions for multivariate spatial random fields where each component has a covariance function from a Cauchy family. We present the conditions on the parameter space that result in valid models with varying degrees of complexity. Practical implementations, including reparameterizations to reflect the conditions on the parameter space will be discussed. We show results of various Monte Carlo simulation experiments to explore the performances of our approach in terms of estimation and cokriging. The application of the proposed multivariate Cauchy model is illustrated on a dataset from the field of Satellite Oceanography.
Link to Cisco Webex meeting: https://gatech.webex.com/gatech/j.php?MTID=mee147c52d7a4c0a5172f60998fee267a
Various notions of dimension are important throughout mathematics, and for graphs the so-called Prague dimension was introduced by Nesetril, Pultr and Rodl in the 1970s. Proving a conjecture of Furedi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order $n/\log n$ for constant edge-probabilities. The main new proof ingredient is a Pippenger-Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size $O(\log n)$.
Based on joint work with He Guo and Lutz Warnke.
We present a deep learning framework for learning multiscale wave propagation in heterogeneous media. The framework involves the construction of linear feed-forward networks (experts) that specialize in different media groups and a nonlinear "committee" network that gives an improved approximation of wave propagation in more complicated media. The framework is then applied to stabilize the "parareal" schemes of Lions, Maday, and Turinici, which are time-parallelization schemes for evolutionary problems.
Determining when two objects have “the same shape” is difficult; this difficulty depends on the dimension we are working in. While many of the same techniques work to study things in dimensions 5 and higher, we can better understand dimensions 1, 2, and 3 using other methods. We can think of 4-dimensional space as the “bridge” between low-dimensional behavior and high-dimensional behavior. One way to understand the possibilities in each dimension is to examine objects called cobordisms: if an (n+1)-dimensional space has an ``edge,” then that edge is itself an n-dimensional space. We say that two n-dimensional spaces are cobordant if together they form the edge of an (n+1)-dimensional space. Using the idea of spaces related by cobordism, we can form a group. In this way, we can attempt to understand higher dimensions using clues from lower dimensions and organize this information using algebra. In this talk, I will discuss different types of cobordism groups and how to study them using tools from a broad range of mathematical areas.
I will review some results on the question of when the orbits $\{ T^j g : j \in J, g \in G \}$ of a bounded operator $T$ acting on a Hilbert space $\mathcal{H}$ with $G \subset \mathcal{H}$ form a frame of $\mathcal{H}$. I will also comment on recent advances. This is motivated by the Dynamical Sampling problem that consists of recovering a time-evolving signal from its space-time samples.
The notion of weak saturation was introduced by Bollobás in 1968. A graph $G$ on $n$ vertices is weakly $F$-saturated if the edges of $E(K_n) \setminus E(G)$ can be added to $G$, one edge at a time, in such a way that every added edge creates a new copy of $F$. The minimum size of a weakly $F$-saturated graph $G$ of order $n$ is denoted by $\mathrm{wsat}(n, F)$. In this talk, we discuss the weak saturation number of complete bipartite graphs and determine $\mathrm{wsat}(n, K_{t,t})$ whenever $n > 3t-4$. For fixed $1
The L-space conjecture has been in the news a lot lately. It predicts a surprising relationship between the algebraic, geometric, and Floer-homological properties of a 3--manifold Y. In particular, it predicts exactly which 3-manifolds admit a ``taut foliation". In this talk, I'll discuss some of my past and forthcoming work investigating these connections. In particular, I'll discuss a strategy for building taut foliations manifolds obtained by Dehn surgery along knots realized as closures of ``positive braids". As an application, I will show how taut foliations can be used to obstruct positivity for cable knots. All are welcome; no background in foliation or Floer homology theories will be assumed.
https://bccte.zoom.us/j/91883463721
Meeting ID: 918 8346 3721
Over recent years there has been much interest in both Turán and Ramsey properties of vertex ordered graphs (i.e., graphs equipped with an ordering of their vertex set). In a recent paper, József Balogh, Lina Li and I initiated the study of embedding spanning structures into vertex ordered graphs. In particular, we introduced a general framework for approaching the problem of determining the minimum degree threshold for forcing a perfect $H$-tiling in an ordered graph. In this talk I will discuss this work, in particular emphasizing how we adapt the regularity and absorbing methods to be applicable in the ordered setting.
A knot in the 3-sphere is slice if it bounds a smooth disc in the 4-ball. A knot is ribbon if it bounds a self-intersecting disc with only singularities that are closed arcs consisting of intersection points of the disc with itself. Every ribbon knot is a slice knot; the converse is a famous unsolved conjecture of Fox. This talk will show some recent interesting progress around the slice-ribbon conjecture.
In the last years, connections between graphs and Riemann surfaces have been
discovered on several different levels. In particular, graphs are closely related
to singular Riemann surfaces and the boundary in the Deligne–Mumford com-
pactification of moduli spaces. Moreover, in both settings there is a notion of a
canonical measure (the Arakelov–Bergman and Zhang measures) which reflects
crucial geometric information.
In this talk, we focus on the following question: what is the limit of the canon-
ical measures along a family of Riemann surfaces? Combining the canonical
measures on Riemann surfaces and metric graphs, we obtain a full description
and a new compactification of the moduli space of Riemann surfaces in terms
of hybrid curves.
Based on joint work with Omid Amini (École polytechnique).
BlueJeans link: https://bluejeans.com/476849994
In the past several decades, scale invariant stochastic processes have been used in a wide range of applications including internet traffic modeling and hydrology. However, by comparison to univariate scale invariance, far less attention has been paid to characteristically multivariate models that display aspects of scaling behavior the limit theory arguably suggests is most natural.
In this talk, I will introduce a new scale invariance model called operator fractional Lévy motion and discuss some of its interesting features, as well as some aspects of wavelet-based estimation of its scaling exponents. This is related to joint work with Gustavo Didier (Tulane University), Herwig Wendt (CNRS, IRIT Univ. of Toulouse) and Patrice Abry (CNRS, ENS-Lyon).
It is a classical fact that for any c > 0, a random permutation of length n = (1+c)k^2/4 typically contains a monotone subsequence of length k. As a far-reaching generalization, Alon conjectured that for this same n, a typical n-permutation is k-universal, meaning that it simultaneously contains every k-pattern. He also gave a simple proof for the fact that if n is increased to Ck^2 log k, then a typical n-permutation is k-universal. Our main result is that the same statement holds for n = Ck^2 log log k, getting almost all of the way to Alon's conjecture.
In this talk we give an overview of the structure-vs-randomness paradigm which is a key ingredient in the proof, and a sketch of the other main ideas. Based on joint work with Matthew Kwan.
This is the opening talk of the 2020 Tech Topology Conference http://ttc.gatech.edu
For a manifold M, the (generalized) Nielsen realization problem asks if the surjection Diff(M) → π_0 Diff(M) is split, where Diff(M) is the diffeomorphism group. When M is a surface, this question was posed by Thurston in Kirby's problem list and was addressed by Morita. I will discuss some more recent work on Nielsen realization problems with connections to flat fiber bundles, K3 surfaces, and smooth structures on hyperbolic manifolds.
The asymptotic dimension of metric spaces is an important notion in geometric group theory. The metric spaces considered in this talk are the ones whose underlying spaces are the vertex-sets of (edge-)weighted graphs and whose metrics are the distance function in weighted graphs. A standard compactness argument shows that it suffices to consider the asymptotic dimension of classes of finite weighted graphs. We prove that the asymptotic dimension of any minor-closed family of weighted graphs, any class of weighted graphs of bounded tree-width, and any class of graphs of bounded layered tree-width are at most 2, 1,and 2, respectively. The first result solves a question of Fujiwara and Papasoglu; the second and third results solve a number of questions of Bonamy, Bousquet, Esperet, Groenland, Pirot and Scott. These bounds for asymptotic dimension are optimal and generalize and improve some results in the literature, including results for Riemannian surfaces and Cayley graphs of groups with a forbidden minor.
We introduce a class of random graph processes, which we call flip processes. Each such process is given by a rule which is just a function $\mathcal{R}:\mathcal{H}_k\rightarrow \mathcal{H}_k$ from all labelled $k$-vertex graphs into itself ($k$ is fixed). Now, the process starts with a given $n$-vertex graph $G_0$. In each step, the graph $G_i$ is obtained by sampling $k$ random vertices $v_1,\ldots,v_k$ of $G_{i-1}$ and replacing the induced graph $G_{i-1}[v_1,\ldots,v_k]$ by $\mathcal{R}(G_{i-1}[v_1,\ldots,v_k])$. This class contains several previously studied processes including the Erdos-Renyi random graph process and the random triangle removal.
Given a flip processes with a rule $\mathcal{R}$ we construct time-indexed trajectories $\Phi:\mathcal{W}\times [0,\infty)\rightarrow\mathcal{W}$ in the space of graphons. We prove that with high probability, starting with a large finite graph $G_0$ which is close to a graphon $W_0$, the flip process will follow the trajectory $(\Phi(W_0,t))_{t=0}^\infty$ (with appropriate rescaling of the time).
These graphon trajectories are then studied from the perspective of dynamical systems. We prove that two trajectories cannot form a confluence, give an example of a process with an oscilatory trajectory, and study stability and instability of fixed points.
Joint work with Frederik Garbe, Matas Sileikis and Fiona Skerman.
https://zoom.us/j/8833025617?pwd=R1FvQWp1MVlRSTVBdFZNejE3ZURmUT09<br />
<br />
Meeting ID: 883 302 5617
Bestvina--Brady groups are subgroups of right-angled Artin groups, and their Dehn functions are bounded above by quartic functions. There are examples of Bestvina--Brady groups whose Dehn functions are linear, quadratic, cubic, and quartic. In this talk, I will give a class of Bestvina--Brady groups that have polynomial Dehn functions, and we can identify the Dehn functions by the defining graphs of those Bestvina--Brady groups.
We study the greedy independent set algorithm on sparse Erdős-Rényi random graphs G(n,c/n). This range of p is of interest due to the threshold at c=e, beyond which it appears that greedy algorithms are affected by a sudden change in the independent set landscape. A large deviation principle was recently established by Bermolen et al. (2020), however, the proof and rate function are somewhat involved. Upper bounds for the rate function were obtained earlier by Pittel (1982). By discrete calculus, we identify the optimal trajectory realizing a given large deviation and obtain the rate function in a simple closed form. In particular, we show that Pittel's bounds are sharp. The proof is brief and elementary. We think the methods presented here will be useful in analyzing the tail behavior of other random growth and exploration processes.
Based on https://arxiv.org/abs/2011.04613
Frieze showed that the expected weight of the minimum spanning tree (MST) of the uniformly weighted graph converges to ζ(3). Recently, this result was extended to a uniformly weighted simplicial complex, where the role of the MST is played by its higher-dimensional analogue -- the Minimum Spanning Acycle (MSA). In this work, we go beyond and look at the histogram of the weights in this random MSA -- both in the bulk and in the extremes. In particular, we focus on the `incomplete' setting, where one has access only to a fraction of the potential face weights. Our first result is that the empirical distribution of the MSA weights asymptotically converges to a measure based on the shadow -- the complement of graph components in higher dimensions. As far as we know, this result is the first to explore the connection between the MSA weights and the shadow. Our second result is that the extremal weights converge to an inhomogeneous Poisson point process. A interesting consequence of our two results is that we can also state the distribution of the death times in the persistence diagram corresponding to the above weighted complex, a result of interest in applied topology.
Based on joint work with Nicolas Fraiman and Gugan Thoppe, see https://arxiv.org/abs/2012.14122
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
In this talk, we will discuss the global well-posedness issue of the defocusing nonlinear Schrödinger equation (NLS). It is known that for subcritical and critical nonlinearities, the equation is globally well-posed on Euclidean spaces and some bounded domains. The supercritical nonlinearities are by far less understood; few partial or conditional results were established. On the other hand, probabilistic approaches (Gibbs measures, fluctuation-dissipation ...) were developed during the last decades to deal with low regularity settings in the context of dispersive PDEs. However, these approaches fail to apply the supercritical nonlinearities. The aim of this talk is to present a new probabilistic approach recently developed by the author in the context of the energy supercritical NLS. We will review some known results and briefly present earlier probabilistic methods, then discuss the new method and the almost sure global well-posedness consequences for the energy supercritical NLS. The results that will be presented are partly join with Xueying Yu.
BlueJeans Link: https://bluejeans.com/348270750
RiboNucleic Acids (RNAs) are ubiquitous, versatile, and overall fascinating, biomolecules which play central roles in modern molecular biology. They also represent a largely untapped potential for biotechnology and health, substantiated by recent disruptive developments (mRNA vaccines, RNA silencing therapies, guide-RNAs of CRISPR-Cas9 systems...). To address those challenges, one must effectively perform RNA design, generally defined as the determination of an RNA sequence achieving a predefined biological function.
I will focus in this talk on algorithmic results and enumerative properties stemming from the inverse folding, the problem of designing a sequence of nucleotides that fold preferentially and uniquely (with respect to base-pair maximization) into a target secondary structure. Despite the NP-hardness of the problem (+ absence of a Fixed Parameter-Tractable algorithm) we showed that it can be solved in polynomial time for restricted families of structures. Such families are dense in the space of designable 2D structures, so that any structure that admits a solution for the inverse folding can be efficiently designed in an approximated sense.
We show that any 2D structure avoiding two forbidden motifs can be modified into a designable structure by adding at most one extra base-pair per helix. Moreover, both the modification and the design of a sequence for the modified structure can be computed in linear time. Finally, if time allows, I will discuss combinatorial consequences of the existence of undesignable motifs. In particular, it implies an exponentially decreasing density of designable structures amongst secondary structures. Those results extend to virtually any design objectives and energy models.
This is joint work with Cédric Chauve, Jozef Hales, Jan Manuch, Ladislav Stacho (SFU, Canada), Alice Héliou, Mireille Régnier, and Hua-Ting Yao (Ecole Polytechnique, France).
Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. This work is joint with Bill Banks and Terry Tao.
Dan Margalit is inviting you to a scheduled Zoom meeting.<br />
https://zoom.us/j/94410378648?pwd=TVV6UDd0SnU3SnAveHA1NWxYcmlTdz09<br />
<br />
Meeting ID: 944 1037 8648<br />
Passcode: gojackets
In 2010, Bestvina-Bromberg-Fujiwara proved that the mapping class group of a finite type surface has finite asymptotic dimension. In contrast, we will show the mapping class group of an infinite-type surface has infinite asymptotic dimension if it contains an essential shift. This work is joint with Curtis Grant and Kasra Rafi.
In quantum mechanics and the analysis of Markov processes, Monte Carlo methods are needed to identify low-lying eigenfunctions of dynamical generators. The standard Monte Carlo approaches for identifying eigenfunctions, however, can be inaccurate or slow to converge. What limits the efficiency of the currently available spectral estimation methods and what is needed to build more efficient methods for the future? Through numerical analysis and computational examples, we begin to answer these questions. We present the first-ever convergence proof and error bounds for the variational approach to conformational dynamics (VAC), the dominant method for estimating eigenfunctions used in biochemistry. Additionally, we analyze and optimize variational Monte Carlo (VMC), which combines Monte Carlo with neural networks to accurately identify low-lying eigenstates of quantum systems.
The Prague dimension of graphs was introduced by Nešetřil, Pultr and Rödl in the 1970s. Proving a conjecture of Füredi and Kantor, we show that the Prague dimension of the binomial random graph is typically of order $n/\log n$ for constant edge-probabilities. The main new proof ingredient is a Pippenger–Spencer type edge-coloring result for random hypergraphs with large uniformities, i.e., edges of size $O(\log n)$. Based on joint work with Kalen Patton and Lutz Warnke.
An initial result of Bourgain and Chang has lead to a number of striking advances in the understanding of polynomial extensions of Roth's Theorem.
The most striking of these is the result of Peluse and Prendiville which show that sets in [1 ,..., N] with density greater than (\log N)^{-c} contain polynomial progressions of length k (where c=c(k)). There is as of yet no corresponding result for corners, the two dimensional setting for Roth's Theorem, where one would seek progressions of the form(x,y), (x+t^2, y), (x,y+t^3) in [1 ,..., N]^2, for example.
Recently, the corners version of the result of Bourgain and Chang has been established, showing an effective bound for a three term polynomial Roth theorem in the finite field setting. We will survey this area. Joint work with Rui Han and Fan Yang.
The link for the seminar is the following
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
I will introduce Serre spectral sequences, then compute some examples. The talk will be in most part following Allen Hatcher's notes on spectral sequences.
There are two purposes of this talk: 1. to give an example of representation theory in algebraic combinatorics and 2. to explain some of the early work on unimodal/symmetric sequences in combinatorics related to recent work on Hodge theory in combinatorics. We will investigate the structure of graded vector spaces $\bigoplus V_j$ with two "shifting" operators $V_j \to V_{j+1}$ and $V_j → V_{j-1}$. We will see that this leads to a very rich theory of unimodal and symmetric sequences with several interesting connections (e.g. the Edge-Reconstruction Conjecture and Hard Lefschetz). The majority of this talk should be accessible to anyone with a solid knowledge of linear algebra.
The differential equation method is a powerful tool used to study the evolution of random combinatorial processes. By showing that the process is likely to follow the trajectory of an ODE, one can study the deterministic ODE rather than the random process directly. We extend this method to ODEs in infinite-dimensional Banach spaces.
We apply this tool to the classical n-queens problem: Let Q(n) be the number of placements of n non-attacking chess queens on an n x n board. Consider the following random process: Begin with an empty board. For as long as possible choose, uniformly at random, a space with no queens in its row, column, or either diagonal, and place on it a queen. We associate the process with an abstract ODE. By analyzing the ODE we conclude that the process almost succeeds in placing n queens on the board. Furthermore, we can obtain a complete n-queens placement by making only a few changes to the board. By counting the number of choices available at each step we conclude that Q(n) \geq (n/C)^n, for a constant C>0 associated with the ODE. This is optimal up to the value of C.
Based on joint work with Zur Luria.
Given a knot K in the 3-sphere, the 4-genus of K is the minimal genus of an orientable surface embedded in the 4-ball with boundary K. If the knot K has a symmetry (e.g. K is periodic or strongly invertible), one can define the equivariant 4-genus by only minimising the genus over those surfaces in the 4-ball which respect the symmetry of the knot. I'll discuss some work with Keegan Boyle trying to understanding the equivariant 4-genus.
The Teichmüller space is the space of hyperbolic structures on surfaces, and there are different flavors depending on the class of surfaces. In this talk we consider the enhanced Teichmüller space which includes additional data at boundary components. The enhanced version can be parametrized by shear coordinates, and in these coordinates, the Weil-Peterson Poisson structure has a simple form. We will discuss a construction of the quantum Teichmüller space corresponding to this Poisson structure.
Bluejeans: https://bluejeans.com/872588027
This talk will be concerned with nonasymptotic lower bounds for the estimation of principal subspaces. I will start by reviewing some previous methods, including the local asymptotic minimax theorem and the Grassmann approach. Then I will present a new approach based on a van Trees inequality (i.e. a Bayesian version of the Cramér-Rao inequality) tailored for invariant statistical models. As applications, I will provide nonasymptotic lower bounds for principal component analysis and the matrix denoising problem, two examples that are invariant with respect to the orthogonal group. These lower bounds are characterized by doubly substochastic matrices whose entries are bounded by the different Fisher information directions, confirming recent upper bounds in the context of the empirical covariance operator.
Seminar link: https://bluejeans.com/129119189
Homotopy continuation methods are numerical methods for solving polynomial systems of equations in many unknowns. These methods assume a set of start solutions to some start system. The start system is deformed into a system of interest (the target system), and the associated solution paths are approximated by numerical integration (predictor/corrector) schemes.
The most classical homotopy method is the so-called total-degree homotopy. The number of start solutions is given by Bézout's theorem. When the target system has more structure than start system, many paths will diverge, This behavior may be understood by working with solutions in a compact projective space.
In joint work with Telen, Walker, and Yahl, we describe a generalization of the total degree homotopy which aims to track fewer paths by working in a compact toric variety analagous to projective space. This allows for a homotopy that may more closely mirror the structure of the target system. I will explain what this is all about and, time-permitting, touch on a few twists we discovered in this more general setting. The talk will be accessible to a general mathematical audience -- I won't assume any knowledge of algebraic geometry.
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
The theory of nonautonomous dynamical systems has undergone major development during the past 23 years since I talked about attractors of nonautonomous difference equations at ICDEA Poznan in 1998.
Two types of attractors consisting of invariant families of sets have been defined for nonautonomous difference equations, one using pullback convergence with information about the system in the past and the other using forward convergence with information about the system in the future. In both cases, the component sets are constructed using a pullback argument within a positively invariant family of sets. The forward attractor so constructed also uses information about the past, which is very restrictive and not essential for determining future behaviour.
The forward asymptotic behaviour can also be described through the omega-limit set of the system.This set is closely related to what Vishik called the uniform attractor although it need not be invariant. It is shown to be asymptotically positively invariant and also, provided a future uniformity condition holds, also asymptotically positively invariant. Hence this omega-limit set provides useful information about the behaviour in current time during the approach to the future limit.
The prediction of RNA secondary structures from sequence is a well developed task in computational RNA Biology. However, in a cellular environment RNA molecules are not isolated but rather interact with a multitude of proteins. RNA secondary structure affects those interactions with proteins and vice versa proteins binding the RNA affect its secondary structure. We have extended the dynamic programming approaches traditionally used to quantify the ensemble of RNA secondary structures in solution to incorporate protein-RNA interactions and thus quantify these effects of protein-RNA interactions and RNA secondary structure on each other. Using this approach we demonstrate that taking into account RNA secondary structure improves predictions of protein affinities from RNA sequence, that RNA secondary structures mediate cooperativity between different proteins binding the same RNA molecule, and that sequence variations (such as Single Nucleotide Polymorphisms) can affect protein affinity at a distance mediated by RNA secondary structures.
https://gatech.bluejeans.com/348270750
The writhe of a braid (=#pos crossing - #neg crossings) and the fractional Dehn twist coefficient of a braid (a rational number that measures "how much the braid twists") are the two most prominent examples of what is known as a quasimorphism (a map that fails to be a group homomorphism by at most a bounded amount) from Artin's braid group on n-strands to the reals.
We consider characterizing properties for such quasimorphisms and talk about relations to the study of knot concordance. For the latter, we consider inequalities for quasimorphism modelled after the so-called slice-Bennequin inequality:
writhe(B) ≤ 2g_4(K) - 1 + n for all n-stranded braids B with closure a knot K.
Based on work in progress.
Note the unusual time!
Partitioning the edges of a graph into edge disjoint triangles forms a triangle decomposition of the graph. A famous conjecture by Nash-Williams from 1970 asserts that any sufficiently large, triangle divisible graph on $n$ vertices with minimum degree at least $0.75 n$ admits a triangle decomposition. In the light of recent results, the fractional version of this problem is of central importance. A fractional triangle decomposition is an assignment of non-negative weights to each triangle in a graph such that the sum of the weights along each edge is precisely one.
We show that for any graph on n vertices with minimum degree at least $0.827327 n$ admits a fractional triangle decomposition. Combined with results of Barber, Kühn, Lo, and Osthus, this implies that for every sufficiently large triangle divisible graph on $n$ vertices with minimum degree at least $0.82733 n$ admits a triangle decomposition. This is a significant improvement over the previous asymptotic result of Dross showing the existence of fractional triangle decompositions of sufficiently large graphs with minimum degree more than $0.9 n$. This is joint work with Luke Postle.
For (X,d,w) be a space of homogeneous type in the sense of Coifman and Weiss, suppose that u and v are two locally finite positive Borel measures on (X,d,w). Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderon--Zygmund operator T from L^{2}(u) to L^{2}(v) in terms of the A_{2} condition and two testing conditions. The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.
We also give the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.
Zoom link:
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
The group of homeomorphisms on a surface is called the mapping class group. This talk will cover some background and key results to provide a foundation for related talks that will occur later in the semester.
We are motivated by invertible matrix based constructions for expressing the coefficients of ordinary generating functions of special convolution type sums. The sum types we consider typically arise in classical number theoretic applications such as in expressing the Dirichlet convolutions $f \ast 1$ for any arithmetic function $f$. The starting point for this perspective is to consider the so-termed Lambert series generating function (LGF) factorization theorems that have been published over the past few years in work by Merca, Mousavi and Schmidt (collectively). In the LGF case, we are able to connect functions and constructions like divisor sums from multiplicative number theory to standard functions in the more additive theory of partitions. A natural question is to ask how we can replicate this type of unique "best possible", or most expressive expansion relating the generating functions of more general classes of convolution sums? In the talk, we start by summarizing the published results and work on this topic, and then move on to exploring how to define the notion of a "canonically best" factorization theorem to characterize this type of sum in more generality.
BlueJeans link: https://bluejeans.com/936847924
Zoom link: https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
This talk will focus on the multifaceted and mutually perpetuating relationship between ergodic theory, combinatorics and number theory. We will begin by discussing Furstenberg’s ergodic approach to Szemerédi’s Theorem and how it has inspired a recent solution to a long-standing sumset conjecture of Erdős. Thereafter, we will explore a new dynamical framework for treating questions in multiplicative number theory. This leads to a variant of the ergodic theorem that contains the Prime Number Theorem as a special case, and reveals an intriguing new connection between the notion of entropy in dynamical systems and the distribution of the number of prime factors of integers.
URL: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
Even though it is not easy to determine global non-negativity of a polynomial, if the polynomial can be written as a sum of squares(SOS), we certainly see that it must be non-negative(PSD). Representability of polynomials in terms of sums of squares is a good certification for global non-negativity in the sense that any non-negative polynomials is just a sum of squares in some cases. However, there are some non-negative polynomials which cannot be written as sum of squares in general. So, one can ask about when the set of sums of squares is same as the set of non-negative polynomials or describing gap between set of sums of squares and non-negative polynomials if they are different.
In this talk, we will introduce an algebraic invariant (of variety) which can tell us when the two sets are same (or not). Moreover, we will discuss about cases that we can exactly describe structural gaps between the two sets.
URL: Microsoft Teams
Zoom link: https://zoom.us/j/96065531265?pwd=aW5qZW8vUUt3bGRlN29FS0FFVnc1QT09
We obtain Margulis-type asymptotic estimates for the number of free homotopy classes of closed geodesics on certain manifolds without conjugate points. Our results cover all compact surfaces of genus at least 2 without conjugate points. This is based on a join work with Vaughn Climenhaga and Gerhard Knieper.
In this talk, we will discuss the minimum positive value of the stationary distribution of a random walk on a directed random graph with given degrees. While for undirected graphs the stationary distribution is simply determined by the degrees, the graph geometry plays a major role in the directed case. Understanding typical stationary values is key to determining the mixing time of the walk, as shown by Bordenave, Caputo, and Salez. However, typical results provide no information on the minimum value, which is important for many applications. Recently, Caputo and Quattropani showed that the stationary distribution exhibits logarithmic fluctuations provided that the minimum degree is at least 2. In this talk, we show that dropping the minimum degree condition may yield polynomially smaller stationary values of the form n^{-(1+C+o(1))}, for a constant C determined by the degree distribution. In particular, C is the combination of two factors: (1) the contribution of atypically thin in-neighborhoods, controlled by subcritical branching processes; and (2) the contribution of atypically "light" trajectories, controlled by large deviation rate functions. As a by-product of our proof, we also determine the hitting and cover time in random digraphs. This is joint work with Xing Shi Cai.
By the well-known theorem of Brooks, every graph of maximum degree Δ ≥ 3 and clique number at most Δ has chromatic number at most Delta. It is natural to ask (and is the subject of a conjecture of Borodin and Kostochka) whether this bound can be improved for graphs of clique number at most Δ - 1. While there has been little progress on this conjecture, there is a number of interesting results on the analogous question for the fractional chromatic number. We will report on some of them, including a result by myself Bernard Lidický and Luke Postle that except for a finite number of counterexamples, every connected subcubic triangle-free graph has fractional chromatic number at most 11/4.
In this talk we will discuss epidemic modeling in the context of COVID-19. We will review the basics of classical epidemic models, and present joint work with Maria Chikina on the use of age-targeted strategies in the context of a COVID-19-like epidemic. We will also discuss the broader roles epidemic modeling has played over the past year, and the limitations it as presented as a primary lens through which to understand the pandemic.
Link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
We give a purely contact and symplectic geometric characterization of Anosov flows in dimension 3 and set up a framework to use tools from contact and symplectic geometry and topology in the study of questions about Anosov dynamics. If time permits, we will discuss a characterization of Anosovity based on Reeb flows and its consequences.
We prove a new upper bound of $s_r(K_k) = O(k^5 r^{5/2})$ on the Ramsey parameter $s_r(K_k)$ introduced by Burr, Erd\H{o}s and Lov\'{a}sz in 1976, which is defined as the smallest minimum degree of a graph $G$ such that any $r$-colouring of the edges of $G$ contains a monochromatic $K_k$, whereas no proper subgraph of $G$ has this property. This improves the previous upper bound of $s_r(K_k) = O(k^6 r^3)$ proved by Fox et al. The construction used in our proof relies on a group theoretic model of generalised quadrangles introduced by Kantor in 1980.
Talk based on https://arxiv.org/abs/2008.02474
Office hours will be held 3-4pm EST.
Complex dynamics is the study of dynamical systems defined by iterating rational maps on the Riemann sphere. For a rational map f, the Julia set Jf is a beautiful fractal defined as the repeller of the dynamics of f. Fractal invariants of Julia sets, such as Hausdorff dimensions, have information about the complexity of the dynamics of rational maps. Ahlfors-regular conformal dimension, abbreviated by ARconfdim, is the infimum of the Hausdorff dimension in a quasi-symmetric class of Ahlfors-regular metric spaces. The ARconfdim is an important quantity especially in geometric group theory because a natural metric, called a visual metric, on the boundary of any Gromov hyperbolic group is determined up to quasi-symmetry. In the spirit of Sullivan's dictionary, we can use ARconfdim to understand the dynamics of rational maps as well. In this talk, we show that the Julia set of a post-critically finite hyperbolic rational map f has ARconfdim 1 if and only if there is an f-invariant graph G containing the post-critical set such that the dynamics restricted to G has topological entropy zero.
A 3-connected graph is minimally 3-connected if removal of any edge destroys 3-connectivity. We present an algorithm for constructing minimally 3-connected graphs based on the results in (Dawes, JCTB 40, 159-168, 1986) using two operations: adding an edge between non-adjacent vertices and splitting a vertex of degree at least 4. To test sets of vertices and edges for 3-compatibility, which depends on the cycles of the graph, we develop a method for obtaining the cycles of $G'$ from the cycles of $G$, where $G'$ is obtained from $G$ by one of the two operations above. We eliminate isomorphic duplicates using certificates generated by McKay's isomorphism checker nauty. The algorithm consecutively constructs the non-isomorphic minimally 3-connected graphs with $n$ vertices and $m$ edges from the non-isomorphic minimally 3-connected graphs with $n-1$ vertices and $m-2$ edges, $n-1$ vertices and $m-3$ edges, and $n-2$ vertices and $m-3$ edges. In this talk I will focus primarily on the theorems behind the algorithm. This is joint work with Joao Costalonga and Robert Kingan.
The work deals with the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in the presence of the transport term. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without Fredholm property in unbounded domains are used. We discuss how the introduction of the transport term impacts the regularity of solutions.
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.
BlueJeans link: https://bluejeans.com/851535338
The talk considers the equivalence relations of topological conjugacy and measure isomorphism on diffeomorphisms of compact manifolds of small dimension. It is shown that neither is a Borel equivalence relation. As a consequence, there is no inherently countable method that, for general diffeomorphisms $S$ and $T$, determines whether $S\sim T$. It is also shown that the Time Forward/Time Backward problem for diffeomorphisms of the 2-torus encodes most mathematical questions, such as the Riemann Hypothesis.
This work is joint with B Weiss and A Gorodetski.
I will give an account of the recent progress in probability and in number theory to understand the large values of the zeta function in small intervals of the critical line. This problem has interesting connections with the extreme value statistics of IID and log-correlated random variables.
URL: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
We will discuss a new method for proving the existence of diffusion in some systems with Normally Hyperbolic Invariant Manifolds (NHIM). We apply this approach to the generalized standard map to show the existence of drift orbits for an explicit range of actions. The method consists of verifying a finite number of conditions on a computer (keywords: NHIM, shadowing, scattering map, Chirikov Standard model, Parameterization Method, Interval Newton Method).
Live cell imaging and single particle tracking techniques have become increasingly popular amongst the mathematical biology community. We study endocytosis, the cellular internalization and transport of bioparticles. This transport is carried out in membrane-bound vesicles through the use of motor proteins. Lysosomes, known for endocytosis, phagocytic destruction, and autophagy, move about the cell along microtubules. Single particle tracking methods utilize stochastic models to simulate intracellular transport and give rise to rigorous analysis of the resulting properties, specifically related to transitioning between inactive to active states. This confidence in the stochastic modeling of particle tracking is useful not only for particle-containing lysosomes, but also broad questions of cellular transport studied with single particle tracking.
Meeting Link: https://gatech.bluejeans.com/348270750
In 1973, Brown, Erdős and Sós proved that if H is a 3-uniform hypergraph on n vertices which contains no triangulation of the sphere, then H has at most O(n^{5/2}) edges, and this bound is the best possible up to a constant factor. Resolving a conjecture of Linial, also reiterated by Keevash, Long, Narayanan, and Scott, we show that the same result holds for triangulations of the torus. Furthermore, we extend our result to every closed orientable surface S.
Joint work with Alexandr Polyanskii, István Tomon and Dmitriy Zakharov, see https://arxiv.org/abs/2010.07191
We write $F \rightarrow (H,G)$ for graphs $F$, $G$, and $H$, if for any coloring of the edges of $F$ in red and blue, there is either a red induced copy of $H$ or a blue induced copy of $G$. For graphs $G$ and $H$, let the induced Ramsey number $IR(H,G)$ be the smallest number of vertices in a graph $F$ such that $F \rightarrow (H,G)$. Deuber showed in 1975 that $IR(H,G)$ is well-defined for any graphs $H$ and $G$. Still, the determination of $IR(H,G)$ remains a challenge for most graphs. A striking contrast between induced and non-induced Ramsey numbers was demonstrated by Fox and Sudakov in 2008 by showing that $IR(H,G)$ is superlinear in $n$ when $H$ is a matching on $n$ edges and $G$ is a star on $n$ edges.
In this talk, I will address the case when $G= K_{1,n}$, a star on $n$ edges, for large $n$, and $H$ is a fixed graph. We prove that $$ (\chi(H)-1) n \leq IR(H, K_{1,n}) \leq (\chi(H)-1)^2n + \epsilon n,$$ for any $\epsilon>0$, sufficiently large $n$, and $\chi(H)$ denoting the chromatic number of $H$. The lower bound is asymptotically tight for any fixed bipartite $H$. The upper bound is attained up to a constant factor, for example when $H$ is a clique.
This is a joint work with Izolda Gorgol.
The algebraic structure of mapping class groups is deep and beautiful; in this talk, we'll explore some curious conjectures and definite theorems about the structure and quality of different subgroups of the mapping class group.
Matroid theory has seen fruitful developments arising from different algebro-geometric approaches, such as the K-theory of Grassmannians and Chow rings of wonderful compactifications. However, these developments have remained somewhat disjoint. We introduce "tautological bundles of matroids" as a new geometric framework for studying matroids. We show that it unifies, recovers, and extends much of these recent developments including log-concavity statements, as well as answering some open conjectures. This is an on-going work with Andrew Berget, Hunter Spink, and Dennis Tseng.
BlueJeans link: https://bluejeans.com/569437095
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous study of such synchronization and desynchronization scenario. By studying the asymptotics of the Markov perturbed stationary distributions, probabilistic characterizations of the alternating pattern between synchronization and desynchronization behaviors is given. More precisely, it is shown that if a random network without intrinsic noise perturbation is synchronized, then after intrinsic noise perturbation high-probability synchronization and low-probability desynchronization can occur intermittently and alternatively in time, and moreover, both the probability of (de)synchronization and the proportion of time spent in (de)synchrony can be explicitly estimated.
Chip-firing and rotor-routing are two well-studied examples of Abelian networks. We study the complexity of their respective reachability problems. We show that the rotor-routing reachability problem is decidable in polynomial time, and we give a simple characterization of when a chip-and-rotor configuration is reachable from another one. For chip-firing, it has been known that the reachability problem is in P if we have a class of graphs whose period length is polynomial (for example, Eulerian digraphs). Here we show that in the general case, chip-firing reachability is hard in the sense that if the chip-firing reachability problem were in P for general digraphs, then the polynomial hierarchy would collapse to NP.
Talk based on https://arxiv.org/abs/2102.11970
In this talk, we will highlight two different types of movement in viscosity dominated environments: sperm navigation and centrosome clustering in dividing cells. Sperm often interact with chemicals and other proteins in the fluid, changing force generation and emergent swimming trajectories. Recently developed computational methods and asymptotic analysis allow for insight into swimming efficiency and hydrodynamic interactions of swimmers in different fluid environments. We will also show how parameter estimation techniques can be utilized to infer fluid and/or swimmer properties. For the case of centrosome movement, we explore how cancer cells can cluster additional centrosomes and proceed through either a bipolar or multipolar division. The models focus on understanding centrosome movement during cell division, which is the result of complex interactions between stochastic microtubule dynamics and motor proteins in the viscous cytoplasm of the cell.
Meeting Link: https://gatech.bluejeans.com/348270750
The problem of group synchronization asks to recover states of objects associated with group elements given possibly corrupted relative state measurements (or group ratios) between pairs of objects. This problem arises in important data-related tasks, such as structure from motion, simultaneous localization and mapping, Cryo-EM, community detection and sensor network localization. Two common groups in these problems are the rotation and symmetric groups. We propose a general framework for group synchronization with compact groups. The main part of the talk discusses a novel message passing procedure that uses cycle consistency information in order to estimate the corruption levels of group ratios. Under our mathematical model of adversarial corruption, it can be used to infer the corrupted group ratios and thus to solve the synchronization problem. We first explain why the group cycle consistency information is essential for effectively solving group synchronization problems. We then establish exact recovery and linear convergence guarantees for the proposed message passing procedure under a deterministic setting with adversarial corruption. We also establish the stability of the proposed procedure to sub-Gaussian noise. We further establish competitive theoretical results under a uniform corruption model. Finally, we discuss the MPLS (Message Passing Least Squares) or Minneapolis framework for solving real scenarios with high levels of corruption and noise and with nontrivial scenarios of corruption. We demonstrate state-of-the-art results for two different problems that occur in structure from motion and involve the rotation and permutation groups.
Note the unusual time!
A classic theorem of Dirac (1952) states that a graph in which every vertex is connected to half of the other vertices contains a Hamilton cycle. Over the years this result has been generalized in many interesting ways. In this talk, I will give an overview of these efforts and then explore some of the more recent developments.
The Bergman projection is a fundamental operator in complex analysis. It is well-known that in the case of the unit ball, the Bergman projection is bounded on weighted L^p if and only if the weight belongs to the Bekolle-Bonami, or B_p, class. These weights are defined using a Muckenhoupt-type condition. Rahm, Tchoundja, and Wick were able to compute the dependence of the operator norm of the projection in terms of the B_p characteristic of the weight using modern tools of dyadic harmonic analysis. Moreover, their upper bound is essentially sharp. We establish that their results can be extended to a much wider class of domains in several complex variables. A key ingredient in the proof is that favorable estimates on the Bergman kernel have been obtained in these cases. This is joint work with Zhenghui Huo and Brett Wick.
We will explain some basic ways in which a sequence of groups/spaces can be said to be stable and give some natural examples of stability.
Bluejeans link: https://bluejeans.com/872588027
Numerical integration of given dynamic systems can be viewed as a forward problem with the learning of unknown dynamics from available state observations as an inverse problem. The latter has been around in various settings such as the model reduction of multiscale processes. It has received particular attention recently in the data-driven modeling via deep/machine learning. Indeed, solving both forward and inverse problems forms the loop of informative and intelligent scientific computing. A natural question is whether a good numerical integrator for discretizing prescribed dynamics is also good for discovering unknown dynamics. This lecture presents a study in the context of Linear multistep methods (LMMs).
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
We consider a sequence of compositions of orientation preserving diffeomorphisms of the circle chosen randomly with a fixed distribution law. There is naturally associated a Lyapunov exponent, which measures the rate of exponential contractions of the sequence. It is known that under some assumptions, if this Lyapunov exponent is null then all the diffeomorphisms are simultaneously conjugated to rotations. If the Lyapunov exponent is not null but close to 0, we study how well we can approach rotations by a simultaneous conjugation. In particular, our results can apply to random products of matrices 2x2, giving quantitative versions of the classical Furstenberg theorem.
Given a graph G, the clique chromatic number of G is the smallest number of colors needed to color the vertices of G so that no maximal clique containing at least two vertices is monochromatic.
We solve an open question proposed by McDiarmid, the speaker, and Pralat concerning the asymptotic order of the clique chromatic number for binomial random graphs.
More precisely, we find the correct order of the clique chromatic number for most values of the edge-probability p around n^{-1/2}. Furthermore, the gap between upper and lower bounds is at most a logarithmic factor in n in all cases.
Based on joint work in progress with Lyuben Lichev and Lutz Warnke.
(Please note the unusual time from 4-5pm, due to the Virtual Admitted Student Day in the School of Mathematics.)
Explore the light-hearted and artistic sides of math at Mathapalooza on the afternoon of Pi Day! There will be puzzles and games to amuse and challenge everyone from kids to rocket scientists. There will be mathematical music, magic (by Matt Baker), and artwork, and mathematical stories will be recounted on stage through dance, courtroom dramas, and circus acts.
Office hours will be held 3-4 pm.
Surfaces of infinite type, such as the plane minus a Cantor set, occur naturally in dynamics. However, their mapping class groups are much less studied and understood compared to the mapping class groups of surfaces of finite type. Many fundamental questions remain open. We will discuss the mapping class group G of the plane minus a Cantor set, and show that any nontrivial G-action on the circle is semi-conjugate to its action on the so-called simple circle. Along the way, we will discuss some structural results of G to address the following questions: What are some interesting subgroups of G? Is G generated by torsion elements? This is joint work with Danny Calegari.
Knot Floer homology is an invariant for knots introduced by Ozsváth-Szabó and, independently, Rasmussen. We will give a general introduction to the theory, sketching the definition and highlight its major properties and applications.
This talk will be given via BlueJeans: https://bluejeans.com/531363037
In the talk, I will present an equidistribution result for families of (non-degenerate) subvarieties in a (general) family of abelian varieties. This extends a result of DeMarco and Mavraki for curves in fibered products of elliptic surfaces. Using this result, one can deduce a uniform version of the classical Bogomolov conjecture for curves embedded in their Jacobians, namely that the number of torsion points lying on them is uniformly bounded in the genus of the curve. This has been previously only known in few cases by work of David--Philippon and DeMarco--Krieger--Ye. Finally, one can obtain a rather uniform version of the Mordell-Lang conjecture as well by complementing a result of Dimitrov--Gao--Habegger: The number of rational points on a smooth algebraic curve defined over a number field can be bounded solely in terms of its genus and the Mordell-Weil rank of its Jacobian. Again, this was previously known only under additional assumptions (Stoll, Katz--Rabinoff--Zureick-Brown).
Piecewise polynomials with certain global smoothness can be given by traditional finite element methods and also by neural networks with some power of ReLU as activation function. In this talk, I will present some recent results on the connections between finite element and neural network functions and comparative studies of their approximation properties and applications to numerical solution of partial differential equations of high order and/or in high dimensions.
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
One of the most fundamental issues in fluid dynamics is whether or not an initially smooth fluid flow can evolve over time to arrive at a singularity -- a state for which the classical equations of fluid mechanics break down and the flow field no longer makes physical sense. While proof remains an open question, numerical evidence strongly suggests that a singularity arises at the boundary of a flow like that found in a stirred cup of tea. The analysis here focuses on the interplay between inertia and pressure, rather than on vorticity. A model is presented based on a primitive-variables formulation of the Euler equations on the cylinder wall, with closure coming from how pressure is determined from velocity. The model generalizes Burger's equation and captures key features in the mechanics of the blowup scenario.
The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. Specifically, neuronal networks at multiple scales utilize their structural complexities to achieve different computational goals. I will first introduce functional implications that can be inferred from a weighted and directed “single” network representation of the brain. Then, I will consider a more detailed and realistic network representation of the brain featuring multiple types of connection between a pair of brain regions, which enables us to uncover the hierarchical structure of the brain network using an unsupervised method. Finally, if time permits, I will discuss computational implications of the hierarchical organization of the brain network, focusing on a specific type of visual computation- predictive coding.
Meeting Link: https://gatech.bluejeans.com/348270750
Drift analysis is the name for a collection of tools that allow to translate information about the one-step progress of a randomized process into information about first hitting times. Drift analysis is successfully used in the mathematical analysis of randomized search heuristics, most notably, evolutionary algorithms, but (for unclear reasons) much less in discrete mathematics or other areas of algorithms.
In this talk, I will give a brief introduction to drift analysis, show some classic and recent applications, and describe some open problems, both concerning drift methods and the mathematical runtime analysis of randomized search heuristics.
This pre-talk will be an introduction to infinite-type surfaces and big mapping class groups. I will have a prepared talk, but it will be extremely informal, and I am more than happy to take scenic diversions if the audience so desires!
A pre-talk will be given at 1 and office hours will be held at 3 (following the seminar talk).
In the world of finite-type surfaces, one of the key tools to studying the mapping class group is to study its action on the curve graph. The curve graph is a combinatorial object intrinsic to the surface, and its appeal lies in the fact that it is infinite-diameter and $\delta$-hyperbolic. For infinite-type surfaces, the curve graph disappointingly has diameter 2. However, all hope is not lost! In this talk I will introduce the omnipresent arc graph and we will see that for a large collection of infinite-type surfaces, the graph is infinite-diameter and $\delta$-hyperbolic. The talk will feature a new characterization of infinite-type surfaces, which provided the impetus for this project.
This is joint work with Federica Fanoni and Alan McLeay
A hereditary graph property is a class of finite graphs closed under isomorphism and induced subgraphs. Given a hereditary graph property H, the speed of H is the function which sends an integer n to the number of distinct elements in H with underlying set {1,...,n}. Not just any function can occur as the speed of hereditary graph property. Specifically, there are discrete "jumps" in the possible speeds. Study of these jumps began with work of Scheinerman and Zito in the 90's, and culminated in a series of papers from the 2000's by Balogh, Bollobás, and Weinreich, in which essentially all possible speeds of a hereditary graph property were characterized. In contrast to this, many aspects of this problem in the hypergraph setting remained unknown. In this talk we present new hypergraph analogues of many of the jumps from the graph setting, specifically those involving the polynomial, exponential, and factorial speeds. The jumps in the factorial range turned out to have surprising connections to the model theoretic notion of mutual algebricity, which we also discuss. This is joint work with Chris Laskowski.
Dynamical sampling is a framework for studying the sampling and reconstruction problems for vectors that evolve under the action of a linear operator. In the first part of the talk I will review a few specific problems that have been part of the framework or motivated by it. In the second part of the talk I will concentrate on the problem of recovering a burst-like forcing term in an initial value problem for an abstract first order differential equation on a Hilbert space. We will see how the ideas of dynamical sampling lead to algorithms that allow one to stably and accurately approximate the burst-like portion of a forcing term as long as the background portion is sufficiently smooth.
After reviewing classical results about existence of completions of varieties, I will talk about a class of degenerations of toric varieties which have a combinatorial classification - normal toric varieties over rank one valuation rings. I will then discuss recent results about the existence of equivariant completions of such degenerations. In particular, I will show a result from my thesis about the existence of normal equivariant completions of these degenerations.
BlueJeans link: https://bluejeans.com/909590858?src=join_info
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09<br />
<br />
Meeting ID: 977 3221 5148<br />
Passcode: 801074
In 1980, Lennart Carleson introduced the following problem for the free Schrödinger equation: when does the solution converge to the initial datum pointwise almost everywhere? Of course, the answer is immediate for regular functions like Schwartz functions. However, the question of what Sobolev regularity is necessary and sufficient for convergence turned out to be highly non-trivial. After the work of many people, it has been solved in 2019, following important advances in harmonic analysis. But interesting variations of the problem are still open. For instance, what happens with periodic solutions in the torus? And what if we refine the almost everywhere convergence to convergence with respect to fractal Hausdorff measures? Together with Renato Lucà (BCAM, Spain), we tackled these two questions. In the talk, I will present our results after explaining the basics of the problem.
Aldous-Broder algorithm is a famous algorithm used to sample a uniform spanning tree of any finite connected graph G, but it is more general: it states that given a reversible M Markov chain on G started at r, the tree rooted at r formed by the steps of successive first entrance in each node (different from the root) has a probability proportional to $\prod_{e=(e1,e2)∈Edges(t,r)}M_{e1,e2}$ , where the edges are directed toward r. As stated, it allows to sample many distributions on the set of spanning trees. In this paper we extend Aldous-Broder theorem by dropping the reversibility condition on M. We highlight that the general statement we prove is not the same as the original one (but it coincides in the reversible case with that of Aldous and Broder). We prove this extension in two ways: an adaptation of the classical argument, which is purely probabilistic, and a new proof based on combinatorial arguments. On the way we introduce a new combinatorial object that we call the golf sequences.
Based on joint work with Luis Fredes, see https://arxiv.org/abs/2102.08639
We study braided embeddings, which is a natural generalization of closed braids in three dimensions. Braided embeddings give us an explicit way to construct lots of higher dimensional embeddings; and may turn out to be as instrumental in understanding higher dimensional embeddings as closed braids have been in understanding three and four dimensional topology. We will discuss two natural questions related to braided embeddings, the isotopy and lifting problem.
Evolutionary dynamics permeates life and life-like systems. Mathematical methods can be used to study evolutionary processes, such as selection, mutation, and drift, and to make sense of many phenomena in life sciences. I will present two very general types of evolutionary patterns, loss-of-function and gain-of-function mutations, and discuss scenarios of population dynamics -- including stochastic tunneling and calculating the rate of evolution. I will also talk about evolution in random environments. The presence of temporal or spatial randomness significantly affects the competition dynamics in populations and gives rise to some counterintuitive observations. Applications include origins of cancer, passenger and driver mutations, and how aspirin might help prevent cancer.
Bluejeans Link: https://gatech.bluejeans.com/348270750
Many scientific problems involve invariant structures, and learning functions that rely on a much lower dimensional set of features than the data itself. Incorporating these invariances into a parametric model can significantly reduce the model complexity, and lead to a vast reduction in the number of labeled examples required to estimate the parameters. We display this benefit in two settings. The first setting concerns ReLU networks, and the size of networks and number of points required to learn certain functions and classification regions. Here, we assume that the target function has built in invariances, namely that it only depends on the projection onto a very low dimensional, function defined manifold (with dimension possibly significantly smaller than even the intrinsic dimension of the data). We use this manifold variant of a single or multi index model to establish network complexity and ERM rates that beat even the intrinsic dimension of the data. We should note that a corollary of this result is developing intrinsic rates for a manifold plus noise data model without needing to assume the distribution of the noise decays exponentially, and we also discuss implications in two-sample testing and statistical distances. The second setting for building invariances concerns linearized optimal transport (LOT), and using it to build supervised classifiers on distributions. Here, we construct invariances to families of group actions (e.g., shifts and scalings of a fixed distribution), and show that LOT can learn a classifier on group orbits using a simple linear separator. We demonstrate the benefit of this on MNIST by constructing robust classifiers with only a small number of labeled examples. This talk covers joint work with Timo Klock, Xiuyuan Cheng, and Caroline Moosmueller.
The shift locus of (monic and centered) complex polynomials of degree d > 1 is the set of polynomials whose filled-in Julia set contains no critical points. Traversing a loop in the shift locus gives rise to a holomorphic motion of Cantor Julia sets, which can be extended to a homeomorphism of the plane minus a Cantor set up to isotopy. Therefore there is a well-defined monodromy representation from the fundamental group of the shift locus to the mapping class group of the plane minus a Cantor set. In this talk, I will discuss the image and the kernel of this map as well as a combinatorial model for the shift locus. This is joint work with J. Bavard, D. Calegari, S. Koch and A. Walker.
A family of subsets of $[n]$ is intersecting if every pair of its members has a non-trivial intersection. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran independently showed that for $n \geq 2k + c\sqrt{k\ln k}$, almost all $k$-uniform intersecting families are stars. Significantly improving their results, we show that the same conclusion holds for $n \geq 2k + 100 \ln k$. Our proof uses the Sapozhenko’s graph container method and the Das-Tran removal lemma.
This is joint work with József Balogh, Ramon I. Garcia and Adam Zsolt Wagner.
Embedding problems, of an n-manifold into an m-manifold, can be heuristically thought to belong to a spectrum, from rigid, to flexible. Euclidean embeddings define the rigid end of the spectrum, meaning you can only translate or rotate an object into the target. Symplectic embeddings, depending on the object, and target, can show up anywhere on the spectrum, and it is this flexible vs rigid philosophy, and techniques developed to study them, that has lead to a lot of interesting mathematics. In this talk I will make this heuristic clearer, and show some examples and applications of these embedding problems.
For some nonlocal PDEs, its steady states can be seen as critical points of an associated energy functional. Therefore, if one can construct perturbations around a function such that the energy decreases to first order along the perturbation, this function cannot be a steady state. In this talk, I will discuss how this simple variational approach has led to some recent progresses in the following equations, where the key is to carefully construct a suitable perturbation.
I will start with the aggregation-diffusion equation, which is a nonlocal PDE driven by two competing effects: nonlinear diffusion and long-range attraction. We show that all steady states are radially symmetric up to a translation (joint with Carrillo, Hittmeir and Volzone), and give some criteria on the uniqueness/non-uniqueness of steady states within the radial class (joint with Delgadino and Yan).
I will also discuss the 2D Euler equation, where we aim to understand under what condition must a stationary/uniformly-rotating solution be radially symmetric. Using a variational approach, we settle some open questions on the radial symmetry of rotating patches, and also show that any smooth stationary solution with compactly supported and nonnegative vorticity must be radial (joint with Gómez-Serrano, Park and Shi).
Ranking from pairwise comparisons is a central problem in a wide range of learning and social contexts. Researchers in various disciplines have made significant methodological and theoretical contributions to it. However, many fundamental statistical properties remain unclear especially for the recovery of ranking structure. This talk presents two recent projects towards optimal ranking recovery, under the Bradley-Terry-Luce (BTL) model.
In the first project, we study the problem of top-k ranking. That is, to optimally identify the set of top-k players. We derive the minimax rate and show that it can be achieved by MLE. On the other hand, we show another popular algorithm, the spectral method, is in general suboptimal.
In the second project, we study the problem of full ranking among all players. The minimax rate exhibits a transition between an exponential rate and a polynomial rate depending on the magnitude of the signal-to-noise ratio of the problem. To the best of our knowledge, this phenomenon is unique to full ranking and has not been seen in any other statistical estimation problem. A divide-and-conquer ranking algorithm is proposed to achieve the minimax rate.
Identifiability is a crucial property for a statistical model since distributions in the model uniquely determine the parameters that produce them. In phylogenetics, the identifiability of the tree parameter is of particular interest since it means that phylogenetic models can be used to infer evolutionary histories from data. In this paper we introduce a new computational strategy for proving the identifiability of discrete parameters in algebraic statistical models that uses algebraic matroids naturally associated to the models. We then use this algorithm to prove that the tree parameters are generically identifiable for 2-tree CFN and K3P mixtures. We also show that the k-cycle phylogenetic network parameter is identifiable under the K2P and K3P models. This is joint work with Benjamin Hollering.
Meeting Link: https://gatech.bluejeans.com/348270750
In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and the location of each hole for B(t). We discuss the theoretical background of this work, the algorithm we used to conduct simulations, and include an extensive discussion of our simulation results. Our results support some of the prior conjectures and further introduce several interesting open problems.
This defense will be conducted on bluejeans, at https://bluejeans.com/611615950.
I will present MCMC algorithms as optimization over the KL-divergence in the space of probabilities. By incorporating a momentum variable, I will discuss an algorithm which performs accelerated gradient descent over the KL-divergence. Using optimization-like ideas, a suitable Lyapunov function is constructed to prove that an accelerated convergence rate is obtained. I will then discuss how MCMC algorithms compare against variational inference methods in parameterizing the gradient flows in the space of probabilities and how it applies to sequential decision making problems.
Fibered knots in a three-manifold Y can be thought of as the binding of an open book decomposition for Y. A basic question to ask is how properties of the open book decomposition relate to properties of the corresponding knot. In this talk I will describe joint work with Dongtai He and Linh Truong that explores this: specifically, we can give a sufficient condition for the monodromy of an open book decomposition of a fibered knot to be right-veering from the concordance invariant Upsilon. I will discuss some applications of this work, including an application to the Slice-Ribbon conjecture.
We study qualitative and quantitative properties of stationary/uniformly-rotating solutions of the 2D incompressible Euler equation.
For qualitative properties, we aim to establish sufficient conditions for such solutions to be radially symmetric. The proof is based on variational argument, using the fact that a uniformly-rotating solution can be formally thought of as a critical point of an energy functional. It turns out that if positive vorticity is rotating with angular velocity, not in (0,1/2), then the corresponding energy functional has a unique critical point, while radial ones are always critical points. We apply similar ideas to more general active scalar equations (gSQG) and vortex sheet equation. We also prove that for rotating vortex sheets, there exist non-radial rotating vortex sheets, bifurcating from radial ones. This work is based on the joint work with Javier Gomez-Serrano, Jia Shi and Yao Yao.
It is well-known that there are non-radial rotating patches with angular velocity in (0,1/2). Using the variational argument, we derive some quantitative estimates for their angular velocities and the difference from the radial ones.
Link: https://bluejeans.com/974226566
A conjecture by Alon, Krivelevich, and Sudakov in 1999 states that for any graph $H$, there is a constant $c_H > 0$ such that if $G$ is $H$-free of maximum degree $\Delta$, then $\chi(G) \leq c_H \Delta / \log\Delta$. It follows from work by Davies et al. in 2020 that this conjecture holds for $H$ bipartite (specifically $H = K_{t, t}$), with the constant $c_H = (t+o(1))$. We improve this constant to $1 + o(1)$ so it does not depend on $H$, and extend our result to the DP-coloring (also known as correspondence coloring) case introduced by Dvořák and Postle. That is, we show for every bipartite graph $B$, if $G$ is $B$-free with maximum degree $\Delta$, then $\chi_{DP}(G) \leq (1+o(1))\Delta/\log(\Delta)$.
This is joint work with Anton Bernshteyn and Abhishek Dhawan.
One of interesting topic in low-dimensional topology is to study exotic smooth structures on closed 4-manifolds. In this talk, we will see an example to distinguish exotic smooth structure using H-slice knots.
A covering system of the integers is a finite collection of arithmetic progressions whose union is the integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked a number of beautiful questions about them. Most famously, his so-called "minimum modulus problem" was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$.
In this talk I will present a variant of Hough's method, which turns out to be both simpler and more powerful. In particular, I will sketch a short proof of Hough's theorem, and discuss several further applications. I will also discuss a related result, proved using a different method, about the number of minimal covering systems.
Joint work with Paul Balister, Béla Bollobás, Julian Sahasrabudhe and Marius Tiba.
We will discuss a conjectured sharp version of an Ehrhard-type inequality for symmetric convex sets, its connections to other questions, and partial progress towards it. We also discuss some new estimates for non-gaussian measures.
Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, patterns can be assigned to a spectrum. On one extreme, patterns are completely described by algebraic systems where each individual pattern is obtained by repeatedly applying simple operations on primitive elements. On the other extreme, patterns are perceived as visual or frequency regularities without any prior knowledge of the underlying mechanisms. In this presentation, we aim at demonstrating some mathematical techniques for representing patterns traversing the aforementioned spectrum, which leads to qualitative analysis of the patterns’ properties and quantitative prediction of the modeled behaviors from various perspectives. We investigate lattice patterns from material science, shape patterns from computer graphics, submanifold patterns encountered in point cloud processing, color perception patterns applied in underwater image processing, dynamic patterns from spatial-temporal data, and low-rank patterns exploited in medical image reconstruction. For different patterns and based on their dependence on structured or unstructured data, we introduce suitable mathematical representations using techniques ranging from group theory to deep neural networks.
Join Zoom Meeting
https://zoom.us/j/97642529845?pwd=aS9aTGloQnBGVVNQMHd6d0I4eGFNQT09
Meeting ID: 976 4252 9845
Passcode: 42PzXb
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
The classical Livshits theorem characterizes coboundaries over a transitive Anosov flow as precisely those functions which integrate to zero over all periodic orbits of the flow. I will present a variant of this theorem which uses a weaker assumption and gives a weaker conclusion that the function is an ``abelian coboundary.” Such weaker version corresponds to studying the cohomological equation on infinite abelian covers e.g., for geodesic flows on abelian covers of hyperbolic surfaces. I will also discuss a connection to the marked length spectrum rigidity of Riemannian metrics. Joint work with Federico Rodriguez Hertz.
Note special day and time.
In this talk I will discuss some new properties of an invariant for 4-manifold with boundary which was originally defined by Nobuo Iida. As one of the applications of this new invariant I will demonstrate how one can obstruct a knot from being h-slice (i.e bound a homologically trivial disk) in 4-manifolds. Also, this invariant can be useful to detect exotic smooth structures of 4-manifolds. This a joint work with Nobuo Iida and Masaki Taniguchi.
We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning. By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved. Connections to geometric analysis, stochastic processes, and deep learning are emphasized. The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points. Applications to image processing and bioinformatics will be shown, demonstrating state-of-the-art empirical performance. Extensions to active learning, generative modeling, and computational geometry will be discussed.
I'm interested in the smooth mapping class group of S^4, i.e. pi_0(Diff^+(S^4)); we know very little about this group beyond the fact that it is abelian (proving that is a fun warm up exercise). We also know that every orientation preserving diffeomorphism of S^4 is pseudoisotopic to the identity (another fun exercise, starting with the fact that there are no exotic 5-spheres). Cerf theory studies the problem of turning pseudoisotopies into isotopies using parametrized Morse theory. Most of what works in Cerf theory works in dimension 5 and higher, but with a little digging one discovers statements that work in dimension 4 as well. Putting all this stuff together we can show that there is a surjective homomorphism from (a certain limit of) fundamental groups of spaces of embeddings of 2-spheres in connected sums of S^2XS^2 onto this smooth mapping class group of S^4. Furthermore, we can identify two natural, and in some sense complementary, subgroups of this fundamental group, one in the kernel of this homomorphism and one whose image we can understand explicitly in terms of Dehn twist-like diffeomorphisms supported near pairs of embedded S^2's in S^4 (Montesinos twins).
Let $G$ be a simple graph with maximum degree $\Delta(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>\Delta(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $\Delta(G)>n/3$ has chromatic index $\Delta(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. We show that the conjecture of Glock, Kühn and Osthus is affirmative.
We introduce the averages $K_N f (x) = \frac{1}{D(N)} \sum _{n \leq N} d(n) f(x+n)$, where $d(n)$ denotes the divisor function and $D(N) = \sum _{n=1} ^N d(n) $. We shall see that these averages satisfy a uniform, scale free, $\ell^p$-improving estimate for $p \in (1,2)$, that is
$$ \Bigl( \frac{1}{N} \sum |K_Nf|^{p'} \Bigl)^{1/p'} \leq C \Bigl(\frac{1}{N} \sum |f|^p \Bigl)^{1/p} $$
as long as $f$ is supported on the interval $[0,N]$.
We will also see that the associated maximal function $K^*f = \sup_N |K_N f|$ satisfies $(p,p)$ sparse bounds for $p \in (1,2)$, which implies that $K^*$ is bounded on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.
The seminar will be held on Zoom, and can be accessed by the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, together with M. Loss and others, we worked to extend the analysis to more "realistic" versions of the original Kac model. I will give a brief overview of kinetic theory, introduce the Kac model and explain the standard results on it. Finally I will present to new papers with M. Loss and R. Han and with J. Beck.
At each site of Z^d, initially there is a car with probability p or a vacant parking spot with probability (1-p), and the choice is independent for all sites. Cars perform independent simple, symmetric random walks, which do not interact directly with one another, and parking spots do not move. When a car enters a site that contains a vacant spot, then the car parks at the spot and the spot is filled – both the car and the spot are removed from the system, and other cars can move freely through the site. This model exhibits a phase transition at p=1/2: all cars park almost surely if and only if p\le 1/2, and all vacant spots are filled almost surely if and only if p \ge 1/2. We study the rates of decay of cars and vacant spots at, below and above p=1/2. In many cases these rates agree with earlier findings of Bramson—Lebowitz for two-type annihilating systems wherein both particle types perform random walks at equal speeds, though we identify significantly different behavior when p<1/2. Based on joint works with Damron, Gravner, Johnson, Junge and Lyu.
Online at https://bluejeans.com/129119189
A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree.
In 1981, Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles.
We prove an approximate version of this conjecture: for every $\epsilon>0$ there exists $n_0$ such that every diregular bipartite tournament on $2n>n_0$ vertices contains a collection of $(1/2-\epsilon)n$ cycles of length at least $(2-\epsilon)n$.
Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every $c>1/2$ and $\epsilon>0$ there exists $n_0$ such that every $cn$-regular bipartite digraph on $2n>n_0$ vertices contains $(1-\epsilon)cn$ edge-disjoint Hamilton cycles.
Base on joint work with Yanitsa Pehova, see https://arxiv.org/abs/1907.08479
Please note the special time/day: Thursday 6pm
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
Consider the three body problem with masses $m_0,m_1,m_2>0$. Take units such that $m_0+m_1+m_2 = 1$. In 1922 Chazy classified the possible final motions of the three bodies, that is the behaviors the bodies may have when time tends to infinity. One of them are what is known as oscillatory motions, that is, solutions of the three body problem such that the positions of the bodies $q_0, q_1, q_2$ satisfy
\[
\liminf_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|<+\infty \quad \text{ and }\quad
\limsup_{t\to\pm\infty}\sup_{i,j=0,1,2, i\neq j}\|q_i-q_j\|=+\infty.
\] At the time of Chazy, all types of final motions were known, except the oscillatory ones. We prove that, if all three masses $m_0,m_1,m_2>0$ are not equal to $1/3$, such motions exist. In fact, we prove more, since our result is based on the construction of a hyperbolic invariant set whose dynamics is conjugated to the Bernouilli shift of infinite symbols, we prove (if all masses are not all three equals to $1/3$) 1) the existence of chaotic motions and positive topological entropy for the three body problem, 2) the existence of periodic orbits of arbitrarily large period in the 3BP. Reversing time, Chazy's classification describes ``starting'' motions and then, the question if starting and final motions need to coincide or may be different arises. We also prove that one can construct solutions of the three body problem whose starting and final motions are of different type.
Suppose that n sample genomes are collected from the same population. The expected sample frequency spectrum (SFS) is the vector of probabilities that a mutation chosen at random will appear in exactly k out of the n individuals. This vector is known to be highly dependent on the population size history (demography); for this reason, geneticists have used it for demographic inference. What does the set of all possible vectors generated by demographies look like? What if we specify that the demography has to be piecewise-constant with a fixed number of pieces? We will draw on tools from convex and algebraic geometry to answer these and related questions.
Meeting Link: https://gatech.bluejeans.com/348270750
The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space $\ell_1$. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real sequence space $\ell_1^n$. We prove that for any $N \geq n$, FUNTFs of length $N$ exist for real $\ell_1^n$. To show the existence of FUNTFs, specific examples are constructed for various lengths. These constructions involve repetitions of frame elements. However, a different method of frame constructions allows us to prove the existence of FUNTFs for real $\ell_1^n$ of lengths $2n-1$ and $2n-2$ that do not have repeated elements.
The second main topic of this thesis pertains to normalized unconditional Schauder frames for the sequence space $\ell_1$. A Schauder frame provides a reconstruction formula for elements in the space, but need not be associated with a frame inequality. Our main theorem on this topic establishes a set of conditions under which an $\ell_1$-type of frame inequality is applicable towards unconditional Schauder frames. A primary motivation for choosing this set of hypotheses involves appropriate modifications of the Rademacher system, a version of which we prove to be an unconditional Schauder frame that does not satisfy an $\ell_1$-type of frame inequality.
Bluejeans link to meeting: https://bluejeans.com/544995272
We construct integrators to be used in Hamiltonian (or Hybrid) Monte Carlo sampling. The new integrators are easily implementable and, for a given computational budget, may deliver five times as many accepted proposals as standard leapfrog/Verlet without impairing in any way the quality of the samples. They are based on a suitable modification of the processing technique first introduced by J.C. Butcher. The idea of modified processing may also be useful for other purposes, like the construction of high-order splitting integrators with positive coefficients.
Joint work with Mari Paz Calvo, Fernando Casas, and Jesús M. Sanz-Serna
All 3-manifolds bound 4-manifolds, and many construction of 3-manifolds automatically come with a 4-manifold bounding it. Often times these 4-manifolds have definite intersection form. Using Heegaard Floer correction terms and an analysis of short characteristic covectors in bimodular lattices, we give an obstruction for a 3-manifold to bound a definite 4-manifold, and produce some concrete examples. This is joint work with Kyle Larson.
Note the unusual time: 5:45pm.
This talk will survey recent results that describe graphs as subgraphs of products of simpler graphs. The starting point is the following theorem: every planar graph is a subgraph of the strong product of some treewidth 7 graph and a path. This result has been the key to solving several open problems, for example, regarding the queue-number and nonrepetitive chromatic number of planar graphs. The result generalises to provide a universal graph for planar graphs. In particular, if $T^7$ is the universal treewidth 7 graph (which is explicitly defined), then every countable planar graph is a subgraph of the strong product of $T^7$ and the infinite 1-way path. This result generalises in various ways for many sparse graph classes: graphs embeddable on a fixed surface, graphs excluding a fixed minor, map graphs, etc. For example, we prove that for every fixed graph $X$, there is an explicitly defined countable graph $G$ that contains every countable $X$-minor-free graph as a subgraph, and $G$ has several desirable properties such as every $n$-vertex subgraph of $G$ has a $O(\sqrt{n})$-separator. On the other hand, as a lower bound we strengthen a theorem of Pach (1981) by showing that if a countable graph $G$ contains every countable planar graph, then $G$ must contain an infinite complete graph as a minor.
Intermittency is a property observed in the study of turbulence. Two of the most popular ways to measure it are based on the concept of flatness, one with structure functions in the physical space and the other one with high-pass filters in the frequency space. Experimental and numerical simulations suggest that the two approaches do not always give the same results. In this talk, we prove they are not analytically equivalent. For that, we first adapt them to a rigorous mathematical language, and we test them with generalizations of Riemann’s non-differentiable function. This work is motivated by the discovery of Riemann’s non-differentiable function as a trajectory of polygonal vortex filaments.
The seminar will be held on Zoom. Here is the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Given a surface S, the Alexander method is a combinatorial tool used to determine whether two homeomorphisms are isotopic. This statement was formalized in A Primer on Mapping Class Groups in the case that S is of finite type. We extend the Alexander method to include infinite-type surfaces, which are surfaces with infinitely generated fundamental groups.
In this talk, we will introduce a technique useful in proofs dealing with infinite-type surfaces. Then, we provide a "proof by example" of an infinite-type analogue of the Alexander method.
Note the unusual time: 12:00pm.
In the first part of the talk, we discuss the use of a penalized Huber M-estimator for high-dimensional linear regression. We explain how a fairly straightforward analysis yields high-probability error bounds that hold even when the additive errors are heavy-tailed. However, the parameter governing the shape of the Huber loss must be chosen in relation to the scale of the error distribution. We discuss how to use an adaptive technique, based on Lepski's method, to overcome the difficulties traditionally faced by applying Huber M-estimation in a context where both location and scale are unknown.
In the second part of the talk, we turn to a more complicated setting where both the covariates and responses may be heavy-tailed and/or adversarially contaminated. We show how to modify the Huber regression estimator by first applying an appropriate "filtering" procedure to the data based on the covariates. We prove that in low-dimensional settings, this filtered Huber regression estimator achieves near-optimal error rates. We further show that the commonly used least trimmed squares and least absolute deviation estimators may similarly be made robust to contaminated covariates via the same covariate filtering step. This is based on joint work with Ankit Pensia and Varun Jog.
We will consider a model of mixtures of Gaussian distributions, called Multi-Reference Alignment, which has been motivated by imaging techniques in chemistry. In that model, the centers are all related with each other by the action of a (known) group of isometries. In other words, each observation is a noisy version of an isometric transformation of some fixed vector, where the isometric transformation is taken at random from some group of isometries and is not observed. Our goal is to learn that fixed vector, whose orbit by the action of the group determines the set of centers of the mixture. First, we will discuss the asymptotic performances of the maximum-likelihood estimator, exhibiting two scenarios that yield different rates. We will then move on to a non-asymptotic, minimax approach of the problem.
Ever since Eliashberg distinguished overtwisted from tight contact structures in dimension 3, there has been an ongoing project to determine which closed, oriented 3–manifolds support a tight contact structure, and on those that do, whether we can classify them. This thesis studies tight contact structures on an infinite family of hyperbolic L-spaces, which come from surgeries on the Whitehead link. We also present partial results on symplectic fillability on those manifolds.
Bluejeans link to meeting: https://bluejeans.com/855793422
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
In this talk I will present some recent results on the Kirchhoff equation with periodic boundary conditions, in collaboration with Pietro Baldi.
Computing the first step of quasilinear normal form, we erase from the equation all the cubic terms giving nonzero contribution to the energy estimates; thus we deduce that, for small initial data of size $\varepsilon$ in Sobolev class, the time of existence of the solution is at least of order $\varepsilon^{-4}$ (which improves the lower bound $\varepsilon^{-2}$ coming from the linear theory).
In the second step of normal form, there remain some resonant terms (which cannot be erased) that give a non-trivial contribution to the energy estimates; this could be interpreted as a sign of non-integrability of the equation. Nonetheless, we show that small initial data satisfying a suitable nonresonance condition produce solutions that exist over a time of order at least $\varepsilon^{-6}$.
Meeting link: https://bluejeans.com/7708995345
Langevin dynamics-based sampling algorithms are arguably among the most widely-used Markov Chain Monte Carlo (MCMC) methods. Two main directions of the modern study of MCMC methods are (i) How to scale MCMC methods to big data applications, and (ii) Tight convergence analysis of MCMC algorithms, with explicit dependence on various characteristics of the target distribution, in a non-asymptotic manner.
This thesis continues the previous efforts in these two lines and consists of three parts. In the first part, we study stochastic gradient MCMC methods for large-scale applications. We propose a non-uniform subsampling of gradients scheme to approximately match the transition kernel of a base MCMC base with full gradient, aiming for better sample quality. The demonstration is based on underdamped Langevin dynamics.
In the second part, we consider an analog of Nesterov's accelerated algorithm in optimization for sampling. We derive a dynamics termed Hessian-Free-High-Resolution (HFHR) dynamics, from a high-resolution ordinary differential equation description of Nesterov's accelerated algorithm. We then quantify the acceleration of HFHR over underdamped Langevin dynamics at both continuous dynamics level and discrete algorithm level.
In the third part, we study a broad family of bounded, contractive-SDE-based sampling algorithms via mean-square analysis. We show how to extend the applicability of classical mean-square analysis from finite time to infinite time. Iteration complexity in the 2-Wasserstein distance is also characterized and when applied to the Langevin Monte Carlo algorithm, we obtain an improved iteration complexity bound.
Following the approach of Nabutovsky and Rotman for the curve-shortening flow on geodesic nets, we'll show that the shortest closed geodesic on a 2-sphere with non-negative curvature has length bounded above by three times the diameter. On the pinched curvature setting, we prove a bound on the first eigenvalue of the Laplacian and use it to prove a new isoperimetric inequality for pinched 2-spheres sufficiently close to being round. This allows us to improve our bound on the length of the shortest closed geodesic in the pinched curvature setting. This is joint work with Ian Adelstein.
In this talk, we discuss variants of the rigid registration problem, i.e aligning objects via rigid transformation. In the simplest scenario of point-set registration where the correspondence between points are known, we investigate the robustness of registration to outliers. We also study a convex programming formulation of point-set registration with exact recovery, in the situation where both the correspondence and alignment are unknown. This talk is based on joint works with Ankur Kapoor, Cindy Orozco, and Lexing Ying.
A triangle $T'$ is $\varepsilon$-similar to another triangle $T$ if their angles pairwise differ by at most $\varepsilon$. Given a triangle $T$, $\varepsilon >0$ and $n \in \mathbb{N}$, Bárány and Füredi asked to determine the maximum number of triangles $h(n,T,\varepsilon)$ being $\varepsilon$-similar to $T$ in a planar point set of size $n$. We show that for almost all triangles $T$ there exists $\varepsilon=\varepsilon(T)>0$ such that $h(n,T,\varepsilon)=n^3/24 (1+o(1))$. Exploring connections to hypergraph Turán problems, we use flag algebras and stability techniques for the proof. This is joint work with József Balogh and Felix Christian Clemen.
Two of the most famous families of polynomials in combinatorics are Macdonald polynomials and Schubert polynomials. Macdonald polynomials are a family of orthogonal symmetric polynomials which generalize Schur and Hall-Littlewood polynomials and are connected to the Hilbert scheme. Schubert polynomials also generalize Schur polynomials, and represent cohomology classes of Schubert varieties in the flag variety. Meanwhile, the asymmetric exclusion process (ASEP) is a model of particles hopping on a one-dimensional lattice, which was initially introduced by Macdonald-Gibbs-Pipkin to provide a model for translation in protein synthesis. In my talk I will explain how two different variants of the ASEP have stationary distributions which are closely connected to Macdonald polynomials and Schubert polynomials, respectively. This leads to new formulas and new conjectures.
This talk is based on joint work with Corteel-Mandelshtam, and joint work with Donghyun Kim.
Zoom link: https://zoom.us/j/97732215148?pwd=Z0FBNXNFSy9mRUx3UVk4alE4MlRHdz09
In this talk we present the formation of steady waves in two-dimensional fluids under a current with mean velocity $c$ flowing over a periodic bottom. Using a formulation based on the Dirichlet-Neumann operator, we establish the unique continuation of a steady solution from the trivial solution for a flat bottom, with the exception of a sequence of velocities $c_{k}$. Furthermore, we prove that at least two steady solutions for a near-flat bottom persist close to a non-degenerate $S^1$-orbit of steady waves for a flat bottom. As a consequence, we obtain the persistence of at least two steady waves close to a non-degenerate $S^1$-orbit of Stokes waves bifurcating from the velocities $c_{k}$ for a flat bottom. This is a joint work with W. Craig.
Organismal development is a complex process, involving a vast number of molecular constituents interacting on multiple spatio-temporal scales in the formation of intricate body structures. Despite this complexity, development is remarkably reproducible and displays tolerance to both genetic and environmental perturbations. This robustness implies the existence of hidden simplicities in developmental programs. Here, using the Drosophila wing as a model system, we develop a new quantitative strategy that enables a robust description of biologically salient phenotypic variation. Analyzing natural phenotypic variation across a highly outbred population, and variation generated by weak perturbations in genetic and environmental conditions, we observe a highly constrained set of wing phenotypes. Remarkably, the phenotypic variants can be described by a single integrated mode that corresponds to a non-intuitive combination of structural variations across the wing. This work demonstrates the presence of constraints that funnel environmental inputs and genetic variation into phenotypes stretched along a single axis in morphological space. Our results provide quantitative insights into the nature of robustness in complex forms while yet accommodating the potential for evolutionary variations. Methodologically, we introduce a general strategy for finding such invariances in other developmental contexts. -- https://www.biorxiv.org/content/10.1101/2020.10.13.333740v3
Meeting Link: https://gatech.bluejeans.com/348270750
The colored Jones polynomial is a generalization of the Jones polynomial from the finite-dimensional representations of Uq(sl2). One motivating question in quantum topology is to understand how the polynomial relates to other knot invariants. An interesting example is the strong slope conjecture, which relates the asymptotics of the degree of the polynomial to the slopes of essential surfaces of a knot. Motivated by the recent progress on the conjecture, we discuss a connection from the colored Jones polynomial of a knot to the normal surface theory of its complement. We give a map relating generators of a state-sum expansion of the polynomial to normal subsets of a triangulation of the knot complement. Besides direct applications to the slope conjecture, we will also discuss applications to colored Khovanov homology.
A graph $H$ is $k$-common if the number of monochromatic copies of $H$ in a $k$-edge-coloring of $K_n$ is asymptotically minimized by a random coloring. For every $k$, we construct a connected non-bipartite $k$-common graph. This resolves a problem raised by Jagger, Stovicek and Thomason. We also show that a graph $H$ is $k$-common for every $k$ if and only if $H$ is Sidorenko and that $H$ is locally $k$-common for every $k$ if and only if H is locally Sidorenko.
The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$. In this talk, I will present the history of this conjecture and sketch our proof in a special case.
We consider functional differential equations which come from adding delay-related perturbations to ODEs or evolutionary PDEs, which is a singular perturbation problem. We prove that for small enough perturbations, some invariant objects (e.g. periodic orbits, slow stable manifolds) of the unperturbed equations persist and depend on the parameters with high regularity. The results apply to state-dependent delay equations and equations which arise in electrodynamics. We formulate results in a posteriori format. The proof is constructive and leads to algorithms.
This is based on joint works with Joan Gimeno and Rafael de la Llave.
Link: https://bluejeans.com/137621769
Costas arrays are useful in radar and sonar engineering and many other settings in which optimal 2-D autocorrelation is needed: they are permutation matrices in which the vectors joining different pairs of ones are all distinct.
In this talk we discuss some of these applications, and prove that the density of Costas arrays among permutation matrices decays exponentially, solving a core problem in the theory of Costas arrays.
The proof is probabilistic, and combines ideas from random graph theory with tools from probabilistic combinatorics.
Based on joint work in progress with Bill Correll, Jr. and Lutz Warnke.
Description:This talk is part of the Round the World Relay in Combinatorics
The talk starts with Rödl's Theorem that graphs with huge chromatic number contain triangle-free subgraphs with large chromatic number. We will look at various related results and conjectures, with a notable matroid bias; the new results are joint work with Peter Nelson and Raphael Steiner.
The probabilistic method is one of the most powerful tools in combinatorics: it has been used to show the existence of many hard-to-construct objects with exciting properties. It also attracts broad interests in designing and analyzing algorithms to find and construct these objects in an efficient way. In this dissertation we obtain four results using algorithmic approaches in probabilistic method:
1. We study the structural properties of the triangle-free graphs generated by a semirandom variant of triangle-free process and obtain a packing extension of Kim’s famous R(3, t) results. This allows us to resolve a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabo, and answer a problem in extremal graph theory by Esperet, Kang, and Thomasse.
2. We determine the order of magnitude of Prague dimension, which concerns efficient encoding and decomposition of graphs, of binomial random graph with high probability. We resolve conjectures by Furedi and Kantor. Along the way, we prove a Pippenger-Spencer type edge coloring result for random hypergraphs with edges of size O(log n).
3. We analyze the number set generated by r-AP free process, which answers a problem raised by Li and has connection with van der Waerden number in additive combinatorics and Ramsey theory.
4. We study a refined alteration approach to construct H-free graphs in binomial random graphs, which has applications in Ramsey games.
The Bluejeans link of the defense is https://gatech.bluejeans.com/233874892
Final doctoral examination and defense of dissertation of Timothy Duff, June 16, 2021
Date: June 16, 2021, 12:00pm EST
Bluejeans Link is https://bluejeans.com/151393219/
Title: Applications of monodromy in solving polynomial systems
Advisor: Dr. Anton Leykin, School of Mathematics, Georgia Institute of Technology
Committee:
Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
Dr. Gregory Blekherman, School of Mathematics, Georgia Institute of Technology
Dr. Richard Peng, School of Computer Science, Georgia Institute of Technology
Dr. Rekha Thomas, Department of Mathematics, University of Washington
Dr. Josephine Yu, School of Mathematics, Georgia Institute of Technology
Reader: Dr. Rekha Thomas, Department of Mathematics, University of Washington
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The thesis is available here:
fhttps://timduff35.github.io/timduff35/thesis.pdf
Summary:
Polynomial systems of equations that occur in applications frequently have a special structure. Part of that structure can be captured by an associated Galois/monodromy group. This makes numerical homotopy continuation methods that exploit this monodromy action an attractive choice for solving these systems; by contrast, other symbolic-numeric techniques do not generally see this structure. Naturally, there are trade-offs when monodromy is chosen over other methods. Nevertheless, there is a growing literature demonstrating that the trade can be worthwhile in practice.
In this thesis, we consider a framework for efficient monodromy computation which rivals the state-of-the-art in homotopy continuation methods. We show how its implementation in the package MonodromySolver can be used to efficiently solve challenging systems of polynomial equations. Among many applications, we apply monodromy to computer vision---specifically, the study and classification of minimal problems used in RANSAC-based 3D reconstruction pipelines. As a byproduct of numerically computing their Galois/monodromy groups, we observe that several of these problems have a decomposition into algebraic subproblems. Although precise knowledge of such a decomposition is hard to obtain in general, we determine it in some novel cases.
BlueJeans link: https://bluejeans.com/298474885/8484
This dissertation consists of various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. We give a broad introduction and discuss connections to applied algebraic geometry, polyhedral, and tropical geometry. The individual topics discussed are as follows:
Thesis may be viewed here.
BlueJeans link
This thesis focuses on analyzing the physics and designing multiscale methods for nonlinear dynamics in mechanical systems, such as those in astronomy. The planetary systems (e.g. the Solar System) are of great interest as rich dynamics of different scales contribute to many interesting physics. Outside the Solar System, a bursting number of exoplanets have been discovered in recent years, raising interest in understanding the effects of the spin dynamics to the habitability. In part I of this thesis, we investigate the spin dynamics of circumbinary exoplanets, which are planets that orbit around stellar binaries. We found that habitable zone planets around the stellar binaries in near coplanar orbits may hold higher potential for stable climate compared to their single star analogues. And in terms of methodology, secular theory of the slow dominating dynamics is calculated via averaging. Beyond analyzing the dynamics mathematically, to simulate the spin-orbit dynamics for long term accurately, symplectic Lie-group (multiscale) integrators are designed to simulate systems consisting of gravitationally interacting rigid bodies in part II of the thesis. Schematically, slow and fast scales are tailored to compose efficient algorithms. And the integrators are tested via our package GRIT. For the systems that are almost impossible to simulate (e.g. the Solar System with the asteroid belt), how can we understand the dynamics from the observations? In part III, we consider the learning and prediction of nonlinear time series purely from observations of symplectic maps. We represent the symplectic map by a generating function, which we approximate by a neural network (hence the name GFNN). And we will prove, under reasonable assumptions, the global prediction error grows at most linearly with long prediction time as the prediction map is symplectic.
A 3-manifold is called an L-space if its Heegaard Floer homology is "simple." No characterization of all such "simple" 3-manifolds is known. Manifolds obtained as the double-branched cover of alternating knots in the 3-sphere give examples of L-spaces. In this talk, I'll discuss the search for L-spaces among higher index branched cyclic covers of knots. In particular, I'll give new examples of knots whose branched cyclic covers are L-spaces for every index n. I will also discuss an application to "visibility" of certain periodic symmetries of a knot. Some of this work is joint with Ahmad Issa.
This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a ``hole" of positive measure in the phase space which some portion of points in phase space escapes through that hole at each iterate of the dynamical system map. Here, we study the escape rate (a quantity that presents at what rate points in phase space escape through the hole) and various estimations of the escape rate of an open dynamical system. We uncover a reason why the escape rate is faster than expected, which is the convexity of the function defining escape rate. Moreover, exact computations of escape rate and its estimations are present for the skewed tent map and Arnold’s cat map.
In the second part of the thesis, we study physical billiards where the moving particle has a finite nonzero size. In contrast to mathematical billiards where a trajectory is excluded when it hits a corner point of the boundary, in physical billiards reflection of the physical particle (a ball) off a visible corner point is well-defined. Initially, we study properties of such reflections in a physical billiards. Our results confirm that the reflection considered in the literature about physical billiards are indeed no-slip friction-free (elastic) collisions.
In the third part of the thesis, we study physical Ehrenfests' wind-tree models, where we show that physical wind-tree models are dynamically richer than the well-known Lorentz gas model. More precisely, when we replace the point particle by a physical one (a ball), the wind-tree models show a new superdiffusive regimes that never been observed in any other model such as Lorentz gas.
Finally, we prove that typical physical polygonal billiard is hyperbolic at least on a subset of positive measure and therefore has a positive Kolmogorov-Sinai entropy for any positive radius of the moving particle.
In this dissertation, we present, analyze, and implement a quadratically convergent algorithm to compute the invariant circle and the foliation by stable manifolds for 2-dimensional maps. The 2-dimensional maps we are considering are mainly motivated by oscillators subject to periodic perturbation.
The algorithm is based on solving an invariance equation using a quasi-Newton method, and the algorithm works irrespective of whether the dynamics on the invariant circle conjugates to a rotation or is phase-locked, and thus we expect only finite regularity on the invariant circle but analytic on the stable manifolds.
More specifically, the dissertation is divided into the following two parts.
In the theoretical part, we derive our quasi-Newton algorithm and prove that starting from an initial guess that satisfies the invariance equation very approximately, the algorithm converges quadratically to a true solution which is close to the initial guess. The proof of the convergence is based on an abstract Nash-Moser Implicit Function Theorem specially tailored for this problem.
In the numerical part, we discuss some implementation details regarding our algorithm and implemented it on the dissipative standard map. We follow different continuation paths along the perturbation and drift parameter and explore the "bundle merging" scenario when the hyperbolicity of the map losses due to the increase of the perturbation. For non-resonant eigenvalues, we also generalize the algorithm to 3-dimension and implemented it on the 3-D Fattened Arnold Family.
Let, for every positive integer d, a tuple of events A_1,...,A_d be given. Let X_d be the number of events that occur. We state new sufficient conditions for the following extremal independence property: |P(X_d=0)-\prod_{i=1}^d(1-P(A_i))|\to 0. These conditions imply a series of results on asymptotic distributions of certain maximum statistics. In particular, for the maximum number X_n of cliques sharing one vertex in G(n,p), we find sequences a_n and b_n such that (X_n-a_n)/b_n converges in distribution to a standard Gumbel random variable.
We consider the problem of following quasi-periodic tori in perturbations of Hamiltonian systems which involve friction and external forcing.
In the first part, we study a family of dissipative standard maps of the cylinder for which the dissipation is a function of a small complex parameter of perturbation, $\varepsilon$. We compute perturbative expansions formally in $\varepsilon$ and use them to estimate the shape of the domains of analyticity of invariant circles as functions of $\varepsilon$. We also give evidence that the functions might belong to a Gevrey class. The numerical computations we perform support conjectures on the shape of the domains of analyticity.
In the second part, we study rigorously the(divergent) series of formal expansions of the torus obtained using Lindstedt method. We show that, for some systems in the literature, the series is Gevrey. We hope that the method of proof can be of independent interest: We develop KAM estimates for the divergent series. In contrast with the regular KAM method, we loose control of all the domains, so that there is no convergence, but we can generate enough control to show that the series is Gevrey.
https://bluejeans.com/417759047/0103
A link L in the 3-sphere is called chi-slice if it bounds a properly embedded surface F in the 4-ball with Euler characteristic 1. If L is a knot, then this definition coincides with the usual definition of sliceness. One feature of such a link L is that if the determinant of L is nonzero, then the double cover of the 3-sphere branched over L bounds a rational homology ball. In this talk, we will explore the chi-sliceness of 3-braid links. In particular, we will construct explicit families of chi-slice quasi-alternating 3-braids using band moves and we will obstruct the chi-sliceness of almost all other quasi-alternating 3-braid links by showing that their double branched covers do not bound rational homology 4-balls. This is a work in progress joint with Vitaly Brejevs.
Tutte paths have a critical role in the study of Hamiltonicity for 4-connected planar and other graph classes. We show quantitative Tutte path results in which we bound the number of bridges of the path. A corollary of this result is near optimal circumference bounds for essentially 4-connected planar and projective-planar graphs. Joint work with Xingxing Yu.
For any positive integer $t$, a $t$-broom is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $\chi(G) = o(\omega(G)^{t+1})$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. When $t=2$, this answers a question of Schiermeyer and Randerath. Moreover, for $t=2$, we strengthen the bound on $\chi(G)$ to $7.5\omega(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and {$t$-broom, $K_{t,t}$}-free graphs, we improve the bound to $o(\omega^{t-1+\frac{2}{t+1}})$. Joint work with Xiaonan Liu, Zhiyu Wang, and Xingxing Yu.
An equivalent formulation of the Fourier Extension (F.E.) conjecture for a compact piece of the paraboloid states that the F.E. operator maps $ L^{2+\frac{2}{d}}([0,1]^{d}) $ to $L^{2+\frac{2}{d}+\varepsilon}(\mathbb{R}^{d+1}) $ for every $\varepsilon>0 $. It has been fully solved only for $ d=1 $ and there are many partial results in higher dimensions regarding the range of $ (p,q) $ for which $L^{p}([0,1]^{d}) $ is mapped to $ L^{q}(\mathbb{R}^{d+1}) $. In this talk, we will take an alternative route to this problem: one can reduce matters to proving that a model operator satisfies the same mapping properties, and we will show that the conjecture holds in higher dimensions for tensor functions, meaning for all $ g $ of the form $ g(x_{1},\ldots,x_{d})=g_{1}(x_{1})\cdot\ldots\cdot g_{d}(x_{d}) $. Time permitting, we will also address multilinear versions of the statement above and get similar results, in which we will need only one of the many functions involved in each problem to be of such kind to obtain the desired conjectured bounds, as well as almost sharp bounds in the general case. This is joint work with Camil Muscalu.
While deep learning has been used for dynamics learning, limited physical accuracy and an inability to generalize under distributional shift limit its applicability to real world. In this talk, I will demonstrate how to incorporate symmetries into deep neural networks and significantly improve the physical consistency, sample efficiency, and generalization in learning dynamics. I will showcase the applications of these models to challenging problems such as turbulence forecasting and trajectory prediction for autonomous vehicles.
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Artin groups are a generalization of braid groups, first defined by Tits in the 1960s. While specific types of Artin groups have many of the same properties as braid groups, other examples of Artin groups are still very mysterious. Braid groups are can be thought of as the mapping class groups of a punctured disc. The combinatorial and geometric structure of the mapping class group is reflected in a Gromov-hyperbolic space called the curve graph of the mapping class group. Using the curve graph of the mapping class group of a punctured disc, we can define a graph associated to a given braid group. In this talk, I will discuss how to generalize this construction to more general classes of Artin groups. I will also discuss the current known properties of this graph and further open questions about what properties of the curve graph carry over to this new graph.
Primary decomposition is an indispensable tool in commutative algebra, both theoretically and computationally in practice. While primary decomposition of ideals is ubiquitous, the case for general modules is less well-known. I will give a comprehensive exposition of primary decomposition for modules, starting with a gentle review of practical symbolic algorithms, leading up to recent developments including differential primary decomposition and numerical primary decomposition. Based on joint works with Yairon Cid-Ruiz, Marc Harkonen, Robert Krone, and Anton Leykin.
Treewidth, introduced by Robertson and Seymour in the graph minors series, is a fundamental measure of the complexity of a graph. While their results give an answer to the question, “what minors occur in graphs of large treewidth?,” the same question for induced subgraphs is still open. I will talk about some conjectures and recent results in this area. Joint work with Tara Abrishami, Maria Chudnovsky, Cemil Dibek, Sepehr Hajebi, Pawel Rzazewski, Kristina Vuskovic.
Meeting Link: https://bluejeans.com/379561694/5031
This talk presents novel approaches to old techniques to forecast COVID-19: (i) a modeling framework that takes into consideration asymptomatic carriers and government interventions, (ii) a method to rectify daily case counts reported in public databases, and (iii) a method to study socioeconomic factors and propagation of disinformation. In the case of (i), results were obtained with a comprehensive data set of hospitalizations and cases in the metropolitan area of San Antonio through collaboration with local and regional government agencies, a level of data seldom studied in a disaggregated manner. In the case of (ii), results were obtained with a simple approach to data rectification that has not been exploited in the literature, resulting in a non-autonomous system that opens avenues of mathematical exploration. In the case of (iii), this talk presents a methodology to study the effect of socioeconomic and demographic factors, including the phenomenon of disinformation and its effect in public health; currently there are few mathematical results in this important area.
How do we build a knot table? We will discuss Conway’s paper “an enumeration of knots and links” and Hoste, Thistlethwaite and Weeks’ paper “the first 1701936 knots”.
Several constructions of the estimators of the mean of a random variable that admit sub-Gaussian deviation guarantees and are robust to adversarial contamination under minimal assumptions have been suggested in the literature. The goal of this talk is to discuss the size of constants appearing in the bounds, both asymptotic and non-asymptotic, satisfied by the median-of-means estimator and its analogues. We will describe a permutation-invariant version of the median-of-means estimator and show that it is asymptotically efficient, unlike its “standard" version. Finally, applications and extensions of these results to robust empirical risk minimization will be discussed.
Online link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b...
This talk is meant to be a gentle introduction to the algebraic theory of linear PDE with constant coefficients. We will present the connection between submodules of free modules of polynomial rings and solution sets of PDEs, and establish certain results relating analytical properties of solutions with algebraic properties of polynomial modules. We will also review classical spaces of functions in distribution theory and Fourier analysis.
Stream online at https://bluejeans.com/520769740/
In this talk, I will introduce some recent advances in designing stochastic primal-dual methods for bilinear saddle point problems, in the form of min_x max_y y^TAx under different geometries of x and y. These problems are prominent in economics, linear programming, machine learning and reinforcement learning. Specifically, our methods apply to Markov decision processes (MDPs), linear regression, and computational geometry tasks.
In our work, we propose a variance-reduced framework for solving convex-concave saddle-point problems, given a gradient estimator satisfying some local properties. Further, we show how to design such gradient estimators for bilinear objectives under different geometry including simplex (l_2), Euclidean ball (l_1) or box (l_inf) domains. For matrix A with larger dimension n, nonzero entries nnz and accuracy epsilon, our proposed variance-reduced primal dual methods obtain a runtime complexity of nnz+\sqrt{nnz*n}/epsilon, improving over the exact gradient methods and fully stochastic methods in the accuracy and/or the sparse regime (when epsilon < n/nnz). For finite-sum saddle-point problems sum_{k=1}^K f_k(x,y) where each f is 1-smooth, we show how to obtain an epsilon-optimal saddle point within gradient query complexity of K+\sqrt{K}/epsilon.
Moreover, we also provide a class of coordinate methods for solving bilinear saddle-point problems. These algorithms use either O(1)-sparse gradient estimators to obtain improved sublinear complexity over fully stochastic methods, or their variance-reduced counterparts for improved nearly-linear complexity, for sparse and numerically sparse instances A.
This talk is based on several joint works with Yair Carmon, Aaron Sidford and Kevin Tian, with links of papers below:
Variance Reduction for Matrix Games
Coordinate Methods for Matrix Games
Efficiently Solving MDPs using Stochastic Mirror Descent
Bio of the speaker: Yujia Jin is a fourth-year Ph.D. student in Department of Management Science and Engineering, Stanford University, working with Aaron Sidford. She is interested in designing efficient continuous optimization methods, which often run in nearly linear / sublinear time and find vast applications in machine learning, data analysis, reinforcement learning, and graph problems.
In this pair of talks I will survey some of the machinery developed by Conant, Schneiderman, and Teichner to study Whitney towers, and their applications to the study of knot and link concordance. Whitney towers can be thought of as measuring the failure of the Whitney trick in dimension 4 and can be used, in a sense, to approximate slice disks. The talks will be based on various papers of Schneiderman, Conant-Schneiderman-Teichner, Cochran-Orr-Teichner and lecture notes by those authors.
In 1995, Mitsumatsu constructed a large family of Liouville domains whose topology obstructs the existence of a Weinstein structure. Stabilizing these domains yields Liouville domains for which the topological obstruction is no longer in effect, and in 2019 Huang asked whether Mitsumatsu's Liouville domains were stably homotopic to Weinstein domains. We answer this question in the affirmative. This is joint work-in-progress with J. Breen.
Note the hybrid mode. The speaker will be in person in Skiles 005.
In this talk, we develop algorithms for numerical computation, based on ideas from competitive games and statistical inference.
In the first part, we propose competitive gradient descent (CGD) as a natural generalization of gradient descent to saddle point problems and general sum games. Whereas gradient descent minimizes a local linear approximation at each step, CGD uses the Nash equilibrium of a local bilinear approximation. Explicitly accounting for agent-interaction significantly improves the convergence properties, as demonstrated in applications to GANs, reinforcement learning, and computer graphics.
In the second part, we show that the conditional near-independence properties of smooth Gaussian processes imply the near-sparsity of Cholesky factors of their dense covariance matrices. We use this insight to derive simple, fast solvers with state-of-the-art complexity vs. accuracy guarantees for general elliptic differential- and integral equations. Our methods come with rigorous error estimates, are easy to parallelize, and show good performance in practice.
Each point x in Gr(r,n) corresponds to an r×n matrix A_x which gives rise to a matroid M_x on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets {y∈Gr(r,n)|M_y=M_x} form a stratification of Gr(r,n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich.
Fix a graph $F$. A classical problem in extremal graph theory asks about how many induced copies of $F$ can a graph with edge density $\rho$ have? The only case in which we know the asymptotic solution is when $F$ is a complete graph, and it was solved completely only recently by Reiher using the flag algebra machinery. We will consider the other cases and show some results when $F$ is a complete bipartite graph or a complete graph minus one edge. Many interesting related open problems will also be introduced. Joint work with Dhruv Mubayi and Christian Reiher.
Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.
One interesting question in low-dimensional topology is to understand the structure of mapping class group of a given manifold. In dimension 2, this topic is very well studied. The structure of this group is known for various 3-manifolds as well (ref- Hatcher's famous work on Smale's conjecture). But virtually nothing is known in dimension 4. In this talk I will try to motivate why this problem in dimension 4 is interesting and how it is different from dimension 2 and 3. I will demonstrate some "exotic" phenomena and if time permits, I will talk a few words on my upcoming work with Jianfeng Lin.
This is the first part of a two part introduction to tropical intersection theory. The first part will review some of the classical theory. We will mostly focus on the parts of the classical theory that have counterparts in the tropical theory but we may also cover some elements of the classical theory which do not have tropical analogues.
Stream online at https://bluejeans.com/520769740/
We present a mixed atomic matrix norm that, when used as regularization in optimization problems, promotes low-rank matrices with sparse factors. We show that in convex lifted formulations of sparse phase retrieval and sparse principal component analysis (PCA), this norm provides near-optimal sample complexity and error rate guarantees. Since statistically optimal sparse PCA is widely believed to be NP-hard, this leaves open questions about how practical it is to compute and optimize this atomic norm. Motivated by convex duality analysis, we present a heuristic algorithm in the case of sparse phase retrieval and show that it empirically matches existing state-of-the-art algorithms.
I will talk about introduction to mathematical image processing, and cover how numerical PDE can be used in data understanding. This talk will present some of variational/PDE-based methods for image processing, such as denoising, inpainting, colorization. If time permits, I will introduce identification of differential equation from given noisy data.
In this pair of talks I will survey some of the machinery developed by Conant, Schneiderman, and Teichner to study Whitney towers, and their applications to the study of knot and link concordance. Whitney towers can be thought of as measuring the failure of the Whitney trick in dimension 4 and can be used, in a sense, to approximate slice disks. The talks will be based on various papers of Schneiderman, Conant-Schneiderman-Teichner, Cochran-Orr-Teichner and lecture notes by those authors.
We introduce a nonlinear stochastic model reduction technique for high-dimensional stochastic dynamical systems that have a low-dimensional invariant effective manifold with slow dynamics, and high-dimensional, large fast modes. Given only access to a black box simulator from which short bursts of simulation can be obtained, we design an algorithm that outputs an estimate of the invariant manifold, a process of the effective stochastic dynamics on it, which has averaged out the fast modes, and a simulator thereof. This simulator is efficient in that it exploits of the low dimension of the invariant manifold, and takes time steps of size dependent on the regularity of the effective process, and therefore typically much larger than that of the original simulator, which had to resolve the fast modes. The algorithm and the estimation can be performed on-the-fly, leading to efficient exploration of the effective state space, without losing consistency with the underlying dynamics. This construction enables fast and efficient simulation of paths of the effective dynamics, together with estimation of crucial features and observables of such dynamics, including the stationary distribution, identification of metastable states, and residence times and transition rates between them.
Links of singularity and generalized algebraic links are ways of constructing three-manifolds and smooth links inside them from algebraic surfaces and curves inside them. Némethi created lattice homology as an invariant for links of normal surface singularities which developed out of computations for Heegaard Floer homology. Later Ozsváth, Stipsicz, and Szabó defined knot lattice homology for generalized algebraic knots in rational homology spheres, which is known to play a similar role to knot Floer homology and is known to compute knot Floer in some cases. I discuss a proof that knot lattice is an invariant of the smooth knot type, which had been previously suspected but not confirmed.
Tropical curves are piecewise linear objects arising as degenerations of algebraic curves. The close connection between algebraic curves and their tropical limits persists when considering moduli. This exhibits certain spaces of tropical curves as the tropicalizations of the moduli spaces of stable curves. It is, however, still unclear which properties of the algebraic moduli spaces of curves are reflected in their tropical counterparts. In my talk, I will report on joint work with Renzo Cavalieri and Hannah Markwig, in which we define tropical psi classes and study relations between them. I will explain how some of the expected identities cannot be recovered from a purely tropical perspective, whereas others can, revealing the tropical nature they have been of in the first place.
This talk is motivated by the Erdős–Szekeres theorem on monotone subsequences: given a sequence of $rs+1$ distinct numbers, there is either a subsequence of $r+1$ of them in increasing order, or a subsequence of $s+1$ of them in decreasing order.
We'll consider many related questions with an algorithmic flavor, such as: if we want to find one of the subsequences promised, how many comparisons do we need to make? What if we have to pre-register our comparisons ahead of time? Does it help if we search a longer sequence instead?
Some of these questions are still open; some of them have answers. The results I will discuss are joint work with Jozsef Balogh, Felix Clemen, and Emily Heath at UIUC.
Meeting Link: https://bluejeans.com/379561694/5031
Hybridization plays an important role during the evolutionary process of some species. In such cases, phylogenetic trees are sometimes insufficient to describe species-level relationships. We show that most topological features of a level-1 species network (a network with no interlocking cycles) are identifiable under the network multi-species coalescent model using the logDet distance between aligned DNA sequences of concatenated genes.
Different aspects of q-calculus are widely used in number theory, combinatorics, orthogonal polynomials, to name a few. In this talk we introduce q-calculus and consider its applications to the Stirling numbers.
A general theme in studying manifolds is understanding lower dimensional submanifolds that encode information. For contact manifolds, these are Legendrians. I will discuss some low and high dimensional examples of Legendrians, their invariants, and how they are applied to understand manifolds. I will also talk about the Legendrian Low Conjecture, which says that understanding linking of certain Legendrians is the key to understanding causal relations between events in a globally hyperbolic spacetime.
This talk is based on a chapter that I wrote for a book in honor of John Benedetto's 80th birthday. Years ago, John wrote a text "Real Variable and Integration", published in 1976. This was not the text that I first learned real analysis from, but it became an important reference for me. A later revision and expansion by John and Wojtek Czaja appeared in 2009. Eventually, I wrote my own real analysis text, aimed at students taking their first course in measure theory. My goal was that each proof was to be both rigorous and enlightening. I failed (in the chapters on differentiation and absolute continuity). I will discuss the real analysis theorem whose proof I find the most difficult and unenlightening. But I will also present the Banach-Zaretsky Theorem, which I first learned from John's text. This is an elegant but often overlooked result, and by using it I (re)discovered enlightening proofs of theorems whose standard proofs are technical and difficult. This talk will be a tour of the absolutely fundamental concept of absolute continuity from the viewpoint of the Banach-Zaretsky Theorem.
Zoom Link: https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Zoom link: https://us02web.zoom.us/j/84598656431?pwd=UGN5QmJZdnE2MktpM005bFZFK29Gdz09
By a theorem of Ben-Yaacov, Melleray, and Tsankov, whenever $G$ is a Polish group with metrizable universal minimal flow $M(G)$, then $M(G)$ must contain a comeager orbit. This has the following peculiar consequence: If $G$ is a Polish group and $X$ is some minimal metrizable $G$-flow with all orbits meager, then there must exist some non-metrizable minimal $G$-flow. So given such an $X$, can we use $X$ directly in order to construct a non-metrizable minimal $G$-flow? This talk will discuss such a construction, thus providing a new proof of the Generic Point Problem.
This is the second part of a two part introduction to tropical intersection theory. We will cover the tropical analogues of what was learned last week with some focus on what the speaker is currently researching.
Stream online at https://bluejeans.com/520769740/
Semidefinite programming is a useful type of convex optimization, which has applications in both graph theory and industrial engineering. Many semidefinite programs exhibit a kind of structured sparsity, which we can hope to exploit to reduce the memory requirements of solving such semidefinite programs. We will discuss an interesting relaxation of such sparse semidefinite programs, and a measurement of how well this relaxation approximates a true semidefinite program. We'll also discuss how these approximations relate to graph theory and the theory of sum-of-squares and nonnegative polynomials. This talk will not assume any background on semidefinite programming.
Donaldson’s Diagonalization Theorem has been used extensively over the past 15 years as an obstructive tool. For example, it has been used to obstruct: rational homology 3-spheres from bounding rational homology 4-balls; knots from being (smoothly) slice; and knots from bounding (smooth) Mobius bands in the 4-ball. In this multi-part series, we will see how this obstruction works, while getting into the weeds with concrete calculations that are usually swept under the rug during research talks.
To treat the multiple time scales of ocean dynamics in an efficient manner, the baroclinic-barotropic splitting technique has been widely used for solving the primitive equations for ocean modeling. In this paper, we propose second and third-order multirate explicit time-stepping schemes for such split systems based on the strong stability-preserving Runge-Kutta (SSPRK) framework. Our method allows for a large time step to be used for advancing the three-dimensional (slow) baroclinic mode and a small time step for the two-dimensional (fast) barotropic mode, so that each of the two mode solves only need satisfy their respective CFL condition to maintain numerical stability. It is well known that the SSPRK method achieves high-order temporal accuracy by utilizing a convex combination of forward-Euler steps. At each time step of our method, the baroclinic velocity is first computed by using the SSPRK scheme to advance the baroclinic-barotropic system with the large time step, then the barotropic velocity is specially corrected by using the same SSPRK scheme with the small time step to advance the barotropic subsystem with a barotropic forcing interpolated based on values from the preceding baroclinic solves. Finally, the fluid thickness and the sea surface height perturbation is updated by coupling the predicted baroclinic and barotropic velocities. Two benchmark tests drawn from the ``MPAS-Ocean" platform are used to numerically demonstrate the accuracy and parallel performance of the proposed schemes.
The bluejeans link for the seminar is https://bluejeans.com/457724603/4379
Unlike the integral case, given a prime number p, not all Z/p-homology 3-spheres can be constructed as a Heegaard splitting with a gluing map an element of mod p Torelli group, M[p]. Nevertheless, letting p vary we can get any rational homology 3-sphere. This motivated us to study invariants of rational homology 3-spheres that comes from M[p]. In this talk we present an algebraic tool to construct invariants of rational homology 3-spheres from a family of 2-cocycles on M[p]. Then we apply this tool to give all possible invariants that are induced by a lift to M[p] of a family of 2-cocycles on the abelianization of M[p], getting a family of invariants that we will describe precisely.
Each point x in Gr(r, n) corresponds to an r × n matrix A_x which gives rise to a matroid M_x on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|M_y = M_x} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals I_x of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the GrassmannCayley algebra may be used to derive non-trivial elements of I_x geometrically when the combinatorics of the matroid is sufficiently rich.
Note the unusual time!
Let $K^{(r)}_{s_1,s_2,\cdots,s_r}$ be the complete $r$-partite $r$-uniform hypergraph and $ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})$ be the maximum number of edges in any $n$-vertex $K^{(r)}_{s_1,s_2,\cdots,s_r}$-free $r$-uniform hypergraph. It is well-known in the graph case that $ex(n,K_{s,t})=\Theta(n^{2-1/s})$ when $t$ is sufficiently larger than $s$. We generalize the above to hypergraphs by showing that if $s_r$ is sufficiently larger than $s_1,s_2,\cdots,s_{r-1}$ then $$ex(n, K^{(r)}_{s_1,s_2,\cdots,s_r})=\Theta\left(n^{r-\frac{1}{s_1s_2\cdots s_{r-1}}}\right).$$ This is joint work with Jie Ma and Mingwei Zhang.
The mapping class group of a surface is well understood for surfaces of finite type. In contrast, the study of mapping class groups of infinite type surfaces is a new field with many opportunities to establish new results. In this talk, we will introduce infinite type surfaces and their mapping class groups.
https://bluejeans.com/506659049/8073
BlueJeans link: https://bluejeans.com/844708532/5458
I will revisit the relation between Anosov 3-flows and invariant volume forms, from a contact geometric point of view. Consequently, I will give a contact geometric characterization of when a flow with dominated splitting is Anosov based on its divergence, as well as a Reeb dynamical interpretation of when such flows are volume preserving. Moreover, I will discuss the implications of this study on the surgery theory of Anosov 3-flows. In particular, I will conclude that the Goodman-Fried surgery of Anosov flows can be reconstructed, using a bi-contact surgery of Salmoiraghi.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz... />
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This is a two-part talk- the continuation is to be given the following week.
Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng.
Donaldson’s Diagonalization Theorem has been used extensively over the past 15 years as an obstructive tool. For example, it has been used to obstruct: rational homology 3-spheres from bounding rational homology 4-balls; knots from being (smoothly) slice; and knots from bounding (smooth) Mobius bands in the 4-ball. In this multi-part series, we will see how this obstruction works, while getting into the weeds with concrete calculations that are usually swept under the rug during research talks.
In his works Carlitz defined and investigated a few generalizations of the Eulerian numbers and polynomials. For most of those generalizations he provided also a combinatorial interpretation. The classical Eulerian numbers and some of their generalizations are connected to combinatorial statistics on permutations. Carlitz intended to provide a combinatorial interpretation also to his degenerate Eulerian numbers. However since their introduction in 1979 these numbers had a pure analytic character. In this talk we consider a combinatorial model that generalizes the standard definition of permutations and show its relation to the degenerate Eulerian numbers.
Meeting Link: https://bluejeans.com/379561694/5031
Growth control establishes organism size, requiring mechanisms to sense and adjust growth. Studies of single cells revealed that size homeostasis uses distinct control methods: Size, Timer, and Adder. In multicellular organisms, mechanisms that regulate single cell growth must integrate control across organs and tissues during development to generate adult size and shape. We leveraged the roundworm Caenorhabditis elegans as a scalable and tractable model to collect precise growth measurements of thousands of individuals, measure feeding behavior, and quantify changes in animal size and shape. Using quantitative measurements and mathematical modeling, we propose two models of physical mechanisms by which C. elegans can control growth. First, constraints on cuticle stretch generate mechanical signals through which animals sense body size and initiate larval-stage transitions. Second, mechanical control of food intake drives growth rate within larval stages. These results suggest how physical constraints control developmental timing and growth rate in C. elegans.
https://www.biorxiv.org/content/10.1101/2021.04.01.438121v2
Recording link: https://bluejeans.com/s/9NyLSfq4tGD
The seminar will also be streamed live at https://bluejeans.com/787128769/7101 . Questions will be fielded by the organizer.
The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.
BlueJeans link: https://bluejeans.com/248767326/2767
Since Jones introduced his knot polynomial using representation theory, there has been a wide variety of invariants defined this way, e.g., HOMFLY-PT and Reshetikhin-Turaev. Recently, through the work of Bonahon-Wong and Constantino-Le, some of these invariants are reinterpreted as quantum matrices. In this talk, we will review the history of these representation theoretical knot invariants. Then we will discuss one particular connection to the quantum special linear group.
Motivated by toric geometry, Maclagan-Smith defined the multigraded Castelnuovo-Mumford regularity for sheaves on a simplicial toric variety. While this definition reduces to the usual definition on a projective space, other descriptions of regularity in terms of the Betti numbers, local cohomology, or resolutions of truncations of the corresponding graded module proven by Eisenbud and Goto are no longer equivalent. I will discuss recent joint work with Lauren Cranton Heller and Juliette Bruce on generalizing Eisenbud-Goto's conditions to the "easiest difficult" case, namely products of projective spaces, and our hopes and dreams for how to do the same for other toric varieties.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09 <br />
<br />
This is the continuation of last week's talk.
Breathers are temporally periodic and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables (in the so-called spatial dynamics framework), breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the original when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small splitting (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. This splitting leads to the transversal intersection of the center-stable and center-unstable manifolds which produces small amplitude generalized breathers with exponentially small tails. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work of O. Gomide, M. Guardia, T. Seara, and C. Zeng.
Donaldson’s Diagonalization Theorem has been used extensively over the past 15 years as an obstructive tool. For example, it has been used to obstruct: rational homology 3-spheres from bounding rational homology 4-balls; knots from being (smoothly) slice; and knots from bounding (smooth) Mobius bands in the 4-ball. In this multi-part series, we will see how this obstruction works, while getting into the weeds with concrete calculations that are usually swept under the rug during research talks.
Symmetric unions are an interesting class of knots. Although they have not been studied much for their own sake, they frequently appear in the literature. One such instance regards the question of whether there is a nontrivial knot with trivial Jones polynomial. In my talk, I will describe a class of symmetric unions, constructed by Tanaka, such that if any are amphichiral, they would have trivial Jones polynomial. Then I will show how such a knot not only answers the above question but also gives rise to a counterexample to the Cosmetic Surgery Conjecture. However, I will prove that such a knot is in fact trivial and hence cannot be used to answer any of these questions. Finally, I will discuss how the arguments that go into this proof can be generalized to study amphichiral symmetric unions.
The speaker will be in person, but there will also be a remote option https://bluejeans.com/457724603/4379
In recent years, various notions of capacity and complexity have been proposed for characterizing the generalization properties of stochastic gradient descent (SGD) in deep learning. Some of the popular notions that correlate well with the performance on unseen data are (i) the flatness of the local minimum found by SGD, which is related to the eigenvalues of the Hessian, (ii) the ratio of the stepsize to the batch-size, which essentially controls the magnitude of the stochastic gradient noise, and (iii) the tail-index, which measures the heaviness of the tails of the network weights at convergence. In this paper, we argue that these three seemingly unrelated perspectives for generalization are deeply linked to each other. We claim that depending on the structure of the Hessian of the loss at the minimum, and the choices of the algorithm parameters, the distribution of the SGD iterates will converge to a heavy-tailed stationary distribution. We rigorously prove this claim in the setting of quadratic optimization: we show that even in a simple linear regression problem with independent and identically distributed data whose distribution has finite moments of all order, the iterates can be heavy-tailed with infinite variance. We further characterize the behavior of the tails with respect to algorithm parameters, the dimension, and the curvature. We then translate our results into insights about the behavior of SGD in deep learning. We support our theory with experiments conducted on synthetic data, fully connected, and convolutional neural networks. This is based on the joint work with Mert Gurbuzbalaban and Umut Simsekli.
I’ll describe various aspects of my joint work with Oliver Lorscheid on the foundation of a matroid.
Note the unusual time!
In this talk, we will discuss the main results of our paper, Counting Colorings of Triangle-Free Graphs, in which we prove the Johansson-Molloy theorem for the upper bound on the chromatic number of a triangle free graph using a novel counting approach developed by Matthieu Rosenfeld, and also extend this result to obtain a lower bound on the number of proper q-colorings for a triangle free graph. The talk will go over the history of the problem, an outline of our approach, and a high-level sketch of the main proofs. This is joint work with Anton Bernshteyn, Tyler Brazelton, and Akum Kang.
Meeting Link: https://bluejeans.com/379561694/5031
The term cancer covers a multitude of bodily diseases, broadly categorized by having cells which do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modelling. In this talk I shall present a 3D individual-based force-based model for tumour growth and development in which we simulate the behavior of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent is fully realised, for example, cells are described as viscoelastic sphere with radius and centre given within the off-lattice model. Interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells, as well as intercellular aspects such as cell phenotypes.
Neural codes are inspired by John O'Keefe's discovery of the place cell, a neuron in the mammalian brain which fires if and only if its owner is in a particular region of physical space. Mathematically, a neural code $C$ on n neurons is a collection of subsets of $\{1,...,n\}$, with the subsets called codewords. The implication is that $C$ encodes how the members of some collection $\{U_i\}_{i=1}^n$ of subsets of $\mathbb{R}^d$ intersect one another.
The principal question driving the study of neural codes is that of convexity. Given just the codewords of $C$, can we determine if there is a collection of open convex subsets $ \{U_i\}_{i=1}^n$ of some $\mathbb{R}^d$ for which $C$ is the code? A convex code is a code for which there is such a realization of open convex sets. While the question of determining which codes are convex remains open, there has been significant progress as many large families of codes can now be ruled as convex or nonconvex. In this talk, I will give an overview of some of the results from this work. In particular, I will focus on a phenomenon called a local obstruction, which if found in a code forbids convexity.
BlueJeans link: https://bluejeans.com/936509442/0487
Given two knots K_1 and K_2, their 0-surgery manifolds S_0^3(K_1) and S_0^3(K_2) are homology cobordant rel meridian if they are homology cobordant preserving the homology class of the positively oriented meridian. It is known that if K_1 ∼ K_2, then S_0^3(K_1) and S_0^3(K_2) are homology cobordant rel meridian. The converse of this statement was first disproved by Cochran-Franklin-Hedden-Horn. In this talk we will provide a new counterexample, the pair of knots 4_1 and M(4_1) where M is the Mazur satellite operator. S_0^3(4_1) and S_0^3(M(4_1)) are homology cobordant rel meridian, but 4_1 and M(4_1) are non-concordant and have concordance orders 2 and infinity, respectively.
In this talk, we present an operator theoretic analogue of the F. and M. Riesz Theorem. We first recast the classical theorem in operator theoretic terms. We then establish an analogous result in the context of representations of the Cuntz algebra, highlighting notable differences from the classical setting. At the end, we will discuss some extensions of these ideas. This is joint work with R. Clouâtre and R. Martin.
Zoom Link:
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
The problem of classifying collections of objects (graphs, manifolds, operators, etc.) up to some notion of equivalence (isomorphism, diffeomorphism, conjugacy, etc.) is central in every domain of mathematical activity. Invariant descriptive set-theory provides a formal framework for measuring the intrinsic complexity of such classification problems and for deciding, in each case, which types of invariants are “too simple” to be used for a complete classification. It also provides a very interesting link between topological dynamics and the meta-mathematics of classification. In this talk I will discuss various forms of classification which naturally occur in mathematical practice (concrete classification, classification by countable structures, classification by cohomological invariants, etc.) and I will provide criteria for showing when some classification problem cannot be solved using these forms of classification.
Nonnegative polynomials are of fundamental interest in the field of real algebraic geometry. We will discuss a model of nonnegative polynomials over an interesting class of algebraic varieties which have potential applications in optimization theory. In particular, we will discuss connections between this subject and algebraic topology and the geometry of simplicial complexes.
We consider a service system consisting of parallel single server queues of infinite capacity. Work of different classes arrives as correlated Gaussian processes with known drifts but unknown covariances, and it is deterministically routed to the different queues according to some routing matrix. In this setting we show that, under some conditions, the covariance matrix of the arrival processes can be directly recovered from the large deviations behavior of the queue lengths. Also, we show that in some cases this covariance matrix cannot be directly recovered this way, as there is an inherent loss of information produced by the dynamics of the queues. Finally, we show how this can be used to quickly learn an optimal routing matrix with respect to some utility function.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Living and active systems exhibit various emergent dynamics necessary for system regulation, growth, and motility. However, how robust dynamics arises from stochastic components remains unclear. Towards understanding this, I develop topological theories that support robust edge states, effectively reducing the system dynamics to a lower-dimensional subspace. In particular, I will introduce stochastic networks in molecular configuration space that enable different phenomena from a global clock, stochastic growth and shrinkage, to synchronization. These out-of-equilibrium systems further possess uniquely non-Hermitian features such as exceptional points and vorticity. More broadly, my work provides a blueprint for the design and control of novel and robust function in correlated and active systems.
This is a public talk the School of Math is co-sponsoring with the Gathering 4 Gardner Foundation. I will be viewable both in the Clough Auditoria or by Bluejeans at https://primetime.bluejeans.com/a2m/live-event/wbxzuakh .
What are all the ways to draw the lines, when you're dividing up a state to get representation? If you can't find them all, can you choose a good sample? I'll discuss some surprisingly simple questions about graphs and geometry that can help us make advances in policy and civil rights.
Prompted by the definition for the Frobenius complexity of a local ring of positive characteristic, we examine generating functions that can be associated to the twisted construction of a graded ring of positive characteristic. There is a large class of graded rings for which this generating function is rational. We will discuss this class of rings. This work is joint with Yongwei Yao.
Zoom link: https://us04web.zoom.us/j/77238664391<br />
Password: graphs!
Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called geometric bijections. Recently we have extended the geometric bijections to subgraph-orientation correspondences. In this talk, I will introduce the bijections and the geometry behind them.
Talk will be presented live as well as streamed. Questions will be fielded by the organizer.
We'll discuss various operations which can be applied to a knot to "simplify" or "unknot" it. Study of these "unknotting operations" began in the 1800s and continues to be an active area of research in low-dimensional topology. Many of these operations have applications more broadly in topology including to 3- and 4-manifolds and even to DNA topology. I will define some of these operations and highlight a few open problems.
BlueJeans link: https://bluejeans.com/208969592/1051
In this talk, I will present a proof of Dyer-Grossman's description of Aut(B_n) inspired by Kordek-Margalit's work classifying homomorphisms B_n' to B_n. Time permitting, I will also discuss how these techniques can be used to classify homomorphisms B_n to B_m.
If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains. Examining the structure of nodal domains has been used for more than 150 years to deduce properties of eigenfunctions. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erdős-Rényi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava.
Delta-matroids are natural generalizations of matroids in which we replace each difference operator by the symmetric difference operator in the basis exchange axiom. I will briefly introduce (even) Delta-matroids and their representability. I will also discuss how they are related to the spinor varieties.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Let X_{N,n} be an iid product of N real Gaussian matrices of size n x n. In this talk, I will explain some recent joint work with G. Paouris
(arXiv:2005.08899) about a non-asymptotic analysis of the singular values of X_{N,n} . I will begin by giving some intuition and motivation for studying such matrix products. Then, I will explain two new results. The first gives a rate of convergence for the global distribution of singular values of X_{N,n} to the so-called Triangle Law in the limit where N,n tend to infinity. The second is a kind of quantitative version of the multiplicative ergodic theorem, giving estimates at finite but large N on the distance between the joint distribution of all Lyapunov exponents of X_{N,n} and appropriately normalized independent Gaussians in the near-ergodic regime (N >> n).
A central problem in low-dimensional topology asks which homology 3-spheres bound contractible 4-manifolds and homology 4-balls. In this talk, we address this problem for plumbed 3-manifolds and we present the classical and new results together. Our approach is based on Mazur’s famous argument and its generalization which provides a unification of all results.
Solving systems of polynomial equations is at the heart of algebraic geometry. In this talk I will discuss a new method that improves the efficiency of equation-by-equation algorithms for solving polynomial systems. Our approach uses fewer homotopy continuation paths than the current leading method based on regeneration. Moreover it is applicable in both projective and multiprojective settings. To motivate the approach I will also give some examples coming from applied algebraic geometry.
This is joint work with Tim Duff and Anton Leykin.
We prove that if a family of compact connected sets in the plane has the property that every three members of it are intersected by a line, then there are three lines intersecting all the sets in the family. This answers a question of Eckhoff from 1993, who proved that under the same condition there are four lines intersecting all the sets. We also prove a colorful version of this result under weakened conditions on the sets, improving results of Holmsen from 2013. Our proofs use the topological KKM theorem. Joint with Daniel McGinnis.
Semidefinite programming (SDP) is a very well behaved class of convex optimization problems. We will introduce this class of problems, illustrate how it allows to approximate many practical nonconvex optimization problems, and discuss the role of low rank structure.
BlueJeans link: https://bluejeans.com/473141052/9784
Morse theory is a standard concept used in the study of manifolds. PL-Morse theory is a variant of Morse theory developed by Bestvina and Brady that is used to study simplicial complexes. We develop an extension of PL-Morse theory to simplicial complexes equipped with an action of a group G. We will discuss some of the basic ideas in this theory and hopefully sketch proofs of some forthcoming results pertaining to the homology of the Torelli group.
Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a linear evolution operator. In Dynamical Sampling, both the signal, $f$, and the driving operator, $A$, may be unknown. For example, let $f \in l^2(I)$ where $I=\{1, \ldots, d\}$. Suppose for $\Omega \subset I$ we know $\{{ A^j f(i)} : j= 0, \ldots l_i, i\in \Omega\}$ for some $A: l^2(I) \to l^2(I)$. In this setting, we can obtain conditions on $\Omega, A, l_i$ that allow the stable reconstruction of $f$. Dynamical Sampling is closely related to frame theory and has applications to wireless sensor networks among other areas. In this talk, we will discuss the Dynamical Sampling problem, its motivation, related problems inspired by it, current/future work, and open problems.
The seminar will be held on Zoom and can be found at the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
The link for the talk is https://bluejeans.com/492736052/2047
Gibbs measures are ubiquitous in statistical mechanics and probability theory. In this talk I will discuss two types of classes of Gibbs measures – random line ensembles and triangular particle arrays, which have received considerable attention due, in part, to their occurrence in integrable probability.
Gibbsian line ensembles can be thought of as collections of finite or countably infinite independent random walkers whose distribution is reweighed by the sum of local interactions between the walkers. I will discuss some recent progress in the asymptotic study of Gibbsian line ensembles, summarizing some joint works with Barraquand, Corwin, Matetski, Wu and others.
Beta-corners processes are Gibbs measures on triangular arrays of interacting particles and can be thought of as analogues/extensions of multi-level spectral measures of random matrices. I will discuss some recent progress on establishing the global asymptotic behavior of beta-corners processes, summarizing some joint works with Das and Knizel.
One question addressed in the field of Diophantine approximation is precisely quantifying how many "good" approximations an algebraic number has by rational numbers. This is answered most soundly by a 1955 theorem of Klaus Roth. In this talk, I will cover this theorem, some related results and hint at how it can be used to bound the number of rational solutions to curves.
The minimum linear ordering problem (MLOP) asks to minimize the aggregated cost of a set function f with respect to some ordering \sigma of the base set. That is, MLOP asks to find a permutation \sigma that minimizes the sum \sum_{i = 1}^{|E|}f({e \in E : \sigma(e) \le i}). Many instances of MLOP have been studied in the literature, for example, minimum linear arrangement (MLA) or minimum sum vertex cover (MSVC). We will cover how graphic matroid MLOP, i.e. where f is taken to be the rank function of a graphic matroid, is NP-hard. This is achieved through a series of reductions beginning with MSVC. During these reductions, we will introduce a new problem, minimum latency vertex cover (MLVC) which we will also show has a 4/3 approximation. Finally, using the theory of principal partitions, we will show MLOP with monotone submodular function f : E \to \mathbb{R}^+ has a 2 - (1 + \ell_f)/(1 + |E|) approximation where \ell_f = f(E)/(\max_{x \in E}f({x})). As a corollary, we obtain a 2 - (1 + r(E))/(1 + |E|) approximation for matroid MLOP where r is the rank function of the matroid. We will also end with some interesting open questions.
Joint work with Majid Farhadi, Swati Gupta, Shengding Sun, and Prasad Tetali.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Traditional spacecraft trajectory optimization approaches focus on automatizing solution generation by capturing the solution space analytically, or numerically, in a single or few instances. However, critical human-computer interactions within optimization processes are almost always disregarded, and they are not well understood. In fact, human intervention spans across the entire optimization process, from the formulation of a problem that lands on known solution schemes, to the identification of an initial guess within the algorithm basin of convergence, to tuning the algorithm hyper-parameters, investigating anomalies, and parsing large databases of optimal solutions to gain insight. Vision-based interaction with sets of multi-dimensional information mitigates the complexity of several applications in astrodynamics. For example, visual-based processes are key to understanding solution space topology for orbit mechanics (e.g., Poincare’ maps), formulating higher quality initial trajectory guesses for early mission design studies, and investigating six-degree-of-freedom (6DOF) dynamics for proximity operations. The capillary diffusion of visual-based data interaction processes throughout astrodynamics has motivated the creation of virtual reality (VR) technology to facilitate scientific discovery since the advent of modern computers. The recent appearance of small, portable, and affordable devices may be a tipping point to advance astrodynamics applications via VR technology. Nonetheless, the tangible benefits for adoption of virtual reality frameworks are not yet fully characterized in the context of astrodynamics applications. What new opportunities virtual reality opens for astrodynamics? What applications benefits from virtual reality frameworks? To answer these and similar questions, our work focuses on a programmatic early assessment and exploration of VR technology for astrodynamics applications. The assessment is constructed by a review of VR literature with elements that are external to the astrodynamics community to facilitate cross-pollination of ideas. Next, the Johnson-Lindenstrauss lemma, together with a set of simplifying assumptions, is employed to analytically capture the value of projecting higher-dimensional information to a given lower dimensional space. Finally, two astrodynamics applications are presented to display solutions that are primarily enabled by virtual reality technology.
In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology. More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions. The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.
Random feature methods have been successful in various machine learning tasks, are easy to compute, and come with theoretical accuracy bounds. They serve as an alternative approach to standard neural networks since they can represent similar function spaces without a costly training phase. However, for accuracy, random feature methods require more measurements than trainable parameters, limiting their use for data-scarce applications or problems in scientific machine learning. This paper introduces the sparse random feature expansion to obtain parsimonious random feature models. Specifically, we leverage ideas from compressive sensing to generate random feature expansions with theoretical guarantees even in the data-scarce setting. We provide generalization bounds for functions in a certain class (that is dense in a reproducing kernel Hilbert space) depending on the number of samples and the distribution of features. The generalization bounds improve with additional structural conditions, such as coordinate sparsity, compact clusters of the spectrum, or rapid spectral decay. We show that the sparse random feature expansions outperform shallow networks in several scientific machine learning tasks. Applications to signal decompositions for music data, astronomical data, and various complicated signals are also provided.
A well known result of Fox and Milnor states that the Alexander polynomial of slice knots factors as f(t)f(t^{-1}), providing us with a useful obstruction to a knot being slice. In 1978 Kawauchi demonstrated this condition for the multivariable Alexander polynomial of slice links. In this talk, we will present an extension of this result for the multivariable Alexander polynomial of 1-solvable links. (Note: This talk will be in person)
For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is semistable -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities. When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the cluster data associated to $f$. When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated. I will give an overview of how to construct "nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.
For a planar graph $H$, let ${\bf N}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The case where $H$ is the path on $3$ vertices, $H=P_3$, was established by Alon and Caro. The case of $H=P_4$ was determined, also exactly, by Gy\H{o}ri, Paulos, Salia, Tompkins, and Zamora. In this talk, we will give the asymptotic values for $H$ equal to $P_5$ and $P_7$ as well as the cycles $C_6$, $C_8$, $C_{10}$ and $C_{12}$ and discuss the general approach which can be used to compute the asymptotic value for many other graphs $H$. This is joint work with Debarun Ghosh, Ervin Győri, Addisu Paulos, Nika Salia, Chuanqi Xiao, and Oscar Zamora and also joint work with Chris Cox.
Meeting Link: https://bluejeans.com/379561694/5031
Multiple cellular processes such as replication, recombination, and packing change the topology of nucleic acids. The genetic code of viruses and of living organisms is encoded in very long DNA or RNA molecules, which are tightly packaged in confined environments. Understanding the geometry and topology of nucleic acids is key to understanding the mechanisms of viral infection and the inner workings of a cell. We use techniques from knot theory and low-dimensional topology, aided by discrete methods and computational tools, to ask questions about the topological state of a genome. I will illustrate the use of these methods with examples drawn from recent work in my group.
Recording link: https://bluejeans.com/s/bQ3pI0YI2f5
Given a knot $K$ in the 3-sphere, one can ask: what kinds of surfaces in the 3-sphere are bounded by $K$? One can also ask: what kinds of surfaces in the 4-ball (which is bounded by the 3-sphere) are bounded by $K$? In this talk we will discuss how to construct surfaces in both the 3-sphere and in the 4-ball bounded by a given knot $K$, how to obstruct the existence of such surfaces, and explore what is known and unknown about surfaces bounded by so-called torus knots.
The Hot Spots conjecture (due to J. Rauch from the 1970s) is one of the most interesting open problems in elementary PDEs: it basically says that if we run the heat equation in an insulated domain for a long period of time, then the hottest and the coldest spot will be on the boundary. What makes things more difficult is that the statement is actually false but that it's extremely nontrivial to construct counterexamples. The statement is widely expected to be true for convex domains but even triangles in the plane were only proven recently. We discuss the problem, show some recent pictures of counterexample domains and discuss some philosophically related results: (1) the hottest and the coldest spots are at least very far away from each other and (2) whenever the hottest spot is inside the domain, it is not that much hotter than the hottest spot on the boundary. Many of these questions should have analogues on combinatorial graphs and we mention some results in that direction as well.
The seminar will be held on Zoom and can be found at the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Tropicalization is usually applied to algebraic or semi-algebraic sets, but I would like to introduce a different category of sets with well-behaved tropicalization: sets with the Hadamard property, i.e. subsets of the positive orthant closed under coordinate-wise (Hadamard) multiplication. Tropicalization (in the sense of logarithmic limit sets) of a set S with the Hadamard property is a convex cone, whose defining inequalities correspond to pure binomial inequalities valid on S.
I will do several examples of sets S with the Hadamard property coming from combinatorics, such as counts of independent sets in matroids, counts of faces in simplical complexes, and counts of graph homomorphisms. In all of our examples we observe a fascinating polyherdrality phenomenon: even though the sets S we are dealing with are not semilagebraic (they are infinite subsets of the integer lattice) the tropicalization is a rational polyhedral cone. Also, the pure binomial inequalities valid on S are often combinatorially interesting.
Joint work with Annie Raymond, Rekha Thomas and Mohit Singh.
I will define and discuss the tropicalization and analytification of semialgebraic sets. We show that the real analytification is homeomorphic to the inverse limit of real tropicalizations, analogously to a result of Payne. We also show a real analogue of the fundamental theorem of tropical geometry. This is based on joint work with Philipp Jell and Claus Scheiderer.
This talk will serve as an introduction to the algebra of hyperfields—fields with a multivalued addition. For example the sign hyperfield which is the arithmetic of real numbers modulo their absolute value (e.g. positive + positive = positive, positive + negative = any possibility). We will also introduce valued fields which capture the idea of how many times a fixed prime p divides the numerator or denominator of a rational number.
Using this arithmetic we will consider the combinatorial question of factoring a polynomial over a hyperfield. This will present a unified and conceptual way of looking at Descartes's rule of signs (how many positive roots does a real polynomial have) and the Newton polygon rule (how many roots are there which are divisible by p or p^2).
Statistical inference of stochastic processes based on high-frequency observations has been an active research area for more than a decade. The most studied problem is the estimation of the quadratic variation of an Itô semimartingale with jumps. Several rate- and variance-efficient estimators have been proposed when the jump component is of bounded variation. However, to date, very few methods can deal with jumps of unbounded variation. By developing new high-order expansions of truncated moments of Lévy processes, a new efficient estimator is developed for a class of Lévy processes of unbounded variation. The proposed method is based on an iterative debiasing procedure of truncated realized quadratic variations. This is joint work with Cooper Bonience and Yuchen Han.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
This talk will report some of our progress in showing how dynamics can be a useful mathematical tool for machine learning. Three demonstrations will be given, namely, how dynamics help design (and analyze) optimization algorithms, how dynamics help quantitatively understand nontrivial observations in deep learning practices, and how deep learning can in turn help dynamics (or more broadly put, AI for sciences). More precisely, in part 1 (dynamics for algorithm): I will talk about how to add momentum to gradient descent on a class of manifolds known as Lie groups. The treatment will be based on geometric mechanics and an interplay between continuous and discrete time dynamics. It will lead to accelerated optimization. Part 2 (dynamics for understanding deep learning) will be devoted to better understanding the nontrivial effects of large learning rates. I will describe how large learning rates could deterministically lead to chaotic escapes from local minima, which is an alternative mechanism to commonly known noisy escapes due to stochastic gradients. I will also mention another example, on an implicit regularization effect of large learning rates which is to favor flatter minimizers. Part 3 (AI for sciences) will be on data-driven prediction of mechanical dynamics, for which I will demonstrate one strong benefit of having physics hard-wired into deep learning models; more precisely, how to make symplectic predictions, and how that generically improves the accuracy of long-time predictions.
We design an O~(m)-time randomized algorithm for the l2-norm flow diffusion problem, a recently proposed diffusion model based on network flow with demonstrated graph clustering related applications both in theory and in practice. Examples include finding locally-biased low conductance cuts. Using a known connection between the optimal dual solution of the flow diffusion problem and the local cut structure, our algorithm gives an alternative approach for finding such cuts in nearly linear time.
From a technical point of view, our algorithm contributes a novel way of dealing with inequality constraints in graph optimization problems. It adapts the high-level algorithmic framework of nearly linear time Laplacian system solvers, but requires several new tools: vertex elimination under constraints, a new family of graph ultra-sparsifiers, and accelerated proximal gradient methods with inexact proximal mapping computation.
Joint work with Richard Peng and Di Wang.
In dimension three, Giroux developed the theory of convex surfaces in contact manifolds, and this theory has been used to prove many important results in contact geometry, as well as to establish deep connections with topology. More recently, Honda and Huang have reformulated the work of Giroux in order to extend the theory to higher dimensions. The purpose of this sequence of talks is to understand this reformulation and to see some of its applications.
We review some theoretical and computational results on locating eigenvalues coalescence for matrices smoothly depending on parameters. Focus is on the symmetric 2 parameter case, and Hermitian 3 parameter case. Full and banded matrices are of interest.
In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots. I will also show that stronger detection results can be obtained as the knot Floer homology can be shown to detect T(2,8) and T(2,10), and that link Floer homology detects (2,2n)-cables of trefoil and figure eight knot. This talk is based on a joint work with Fraser Binns (Boston College).
Large-scale machine learning models are trained by parallel (stochastic) gradient descent algorithms on distributed systems. The communications for gradient aggregation and model synchronization become the major obstacles for efficient learning as the number of nodes and the model's dimension scale up. In this talk, I will introduce several ways to compress the transferred data and reduce the overall communication such that the obstacles can be immensely mitigated. More specifically, I will introduce methods to reduce or eliminate the compression error without additional communication.
The $n$-th ordered configuration space of a graph parametrizes ways of placing $n$ distinct and labelled particles on that graph. The homology of the one-point compactification of such configuration space is equipped with commuting actions of a symmetric group and the outer automorphism group of a free group. We give a cellular decomposition of these configuration spaces on which the actions are realized cellularly and thus construct an efficient free resolution for their homology representations. As our main application, we obtain computer calculations of the top weight rational cohomology of the moduli spaces $\mathcal{M}_{2,n}$, equivalently the rational homology of the tropical moduli spaces $\Delta_{2,n}$, as a representation of $S_n$ acting by permuting point labels for all $n\leq 10$. This is joint work with Christin Bibby, Melody Chan, and Nir Gadish. Our paper can be found on arXiv with ID 2109.03302.
For a graphic degree sequence $\mathbf{d_n}= (d_1 , . . . , d_n)$ of graphs with vertices $v_1 , . . . , v_n$, $\mathbf{d_n}$-process is the random graph process that inserts one edge at a time at random with the restriction that the degree of $v_i$ is at most $d_i$ . In 1999, N. Wormald asked whether the final graph of random $\mathbf{d_n}$-process is "similar" to the uniform random graph with degree sequence $\mathbf{d_n}$ when $\mathbf{d_n}=(d,\dots, d)$. We answer this question for the $\mathbf{d_n}$-process when the degree sequence $\mathbf{d_n}$ that is not close to being regular. We used the method of switching for stochastic processes; this allows us to track the edge statistics of the $\mathbf{d_n}$-process. Joint work with Mike Molloy and Lutz Warnke.
Meeting Link: https://bluejeans.com/379561694/5031
In close collaboration with experimentalists and clinicians, mathematical models that are parameterized with experimental and clinical data can help estimate patient-specific disease dynamics and treatment success. This positions us at the forefront of the advent of ‘virtual trials’ that predict personalized optimized treatment protocols. I will discuss a couple of different projects to demonstrate how to integrate calculus into clinical decision making. I will present a variety of mathematical model that can be calibrated from early treatment response dynamics to forecast responses to subsequent treatment. This may help us to identify patient candidates for treatment escalation when needed, and treatment de-escalation without jeopardizing outcomes.
Recording link: https://bluejeans.com/s/dcDrDQuxm2W
BlueJeans link: https://bluejeans.com/575457754/6776
Given a surface S, the Alexander method is a combinatorial tool used to determine whether two self-homeomorphisms of S are isotopic. This statement was formalized in the case of finite-type surfaces, which are surfaces with finitely generated fundamental groups. A version of the Alexander method was extended to infinite-type surfaces by Hernández-Morales-Valdez and Hernández-Hidber. We extend the remainder of the Alexander method to include infinite-type surfaces.
In this talk, we will talk about several applications of the Alexander method. Then, we will discuss a technique useful in proofs dealing with infinite-type surfaces and provide a "proof by example" of an infinite-type analogue of the Alexander method.
This will be practice for a future talk and comments and suggestions are appreciated.
I will discuss a general framework for studying what can be said about the rank of a matrix A over a field K if we only know certain crude features of A. For example, what can we say about rank(A) if we only know which entries are zero and which are nonzero? Or if K = R, what if we only know the signs of the entries of A? Or K is a normed field and we only know the absolute values? Or K=C and we only know the arguments? There are many partial answers to questions like this scattered throughout the literature, and I will explain how at least some of these results can be unified through a theory of ranks of matrices over hyperfields. This is work in progress with Tianyi Zhang.
From a perspective of an applied algebraic geometer, I will address several use cases of algorithmic machinery that goes hand in hand with the language of tropical geometry. One of the examples originates in dynamical systems and may shed (tropical) light on a long-standing conjecture in celestial mechanics.
In their seminal paper on combinatorial Hodge theory, Adiprasito, Huh, and Katz showed, among other things, that a very specific set of toric varieties has Chow rings that satisfy Poincaré duality, even though the varieties are not compact. In joint work with Farbod Shokrieh, we generalize this statement to all toric varieties whose fans are supported on a tropically smooth set. This has several consequences in tropical intersection theory; most notably it allows us to prove the long-suspected duality between tropical cycles and cocycles.
In my talk I will assume no prior knowledge of tropical intersection theory. I will define tropical cycles and cocycles explicitly and explain how they are connected to the intersection theory of toric varieties and the Chow rings of fans appearing in combinatorial Hodge theory. Finally, we will see how to use the duality statement mentioned above to define the tropical intersection product.
Recall that, for a variety $X$ in a projective space $\mathbb{P}^d$, the $X$-rank of a point $p\in \mathbb{P}^d$ is the least number of points of $X$ whose span contains the point $p$. Studies about $X$-ranks include some well-known and important results about various tensor ranks. For example,
In this talk, we focus on ranks with respect to Veronese embeddings of a projective line $\mathbb{P}^1$. i.e. symmetric tensor ranks of binary forms. We will discuss how to associate points in $\mathbb{P}^d$ with binary forms and I will introduce apolarity for binary forms which gives an effective method to study Waring ranks of binary forms. We will discuss various ranks on the Veronese embedding and some results on the ranks.
Talk will be held in-person in Skiles 005 and streamed synchronously. <br />
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Zoom link-- https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
My goal is to present a computer assisted proof of a non-trivial theorem in nonlinear dynamics, in full detail. My (quite biased) definition of non-trivial is that there should be some infinite dimensional complications. However, since I want to go through all the details, I need these complications to be as simple as possible. So, I'll consider the Henon map, and prove that some 1 dimensional stable and unstable manifolds attached to a hyperbolic fixed point intersect transversally. By Smale's theorem, this implies the existence of chaotic motions. Recall that one can prove the existence chaotic dynamics for the Henon map more or less by hand using topological methods. Yet transverse intersection of the manifolds is a stronger statement, and moreover the method I'll discuss generalizes to much more sophisticated examples where pen-and-paper fail.
The idea of the proof is to develop a high order polynomial expansion of the stable/unstable manifolds of the fixed point, to prove an a-posteriori theorem about the convergence and truncation error bounds for this expansion, and to check the hypotheses of this theorem using the computer. All of this relies on the parameterization method of Cabre, Fontich, and de la Llave, and on finite numerical calculations using interval arithmetic to manage the inevitable roundoff errors. Once global enough representations of the local invariant manifolds are obtained and equipped with mathematically rigorous error bounds, it is a finite dimensional problem to establish that the manifolds intersect transversally.
remote
In this talk we’ll discuss strong 4-colourings of graphs and prove two new cases of the Strong Colouring Conjecture. Let H be a graph with maximum degree at most 2, and let G be obtained from H by gluing in vertex-disjoint copies of K_4. We’ll show that if H contains at most one odd cycle of length exceeding 3, or if H contains at most 3 triangles, then G is 4-colourable. This is joint work with Greg Puleo.
This paper proposes a model-free and data-adaptive feature screening method for ultra-high dimensional data. The proposed method is based on the projection correlation which measures the dependence between two random vectors. This projection correlation based method does not require specifying a regression model, and applies to data in the presence of heavy tails and multivariate responses. It enjoys both sure screening and rank consistency properties under weak assumptions. A two-step approach, with the help of knockoff features, is advocated to specify the threshold for feature screening such that the false discovery rate (FDR) is controlled under a pre-specified level. The proposed two-step approach enjoys both sure screening and FDR control simultaneously if the pre-specified FDR level is greater or equal to 1/s, where s is the number of active features. The superior empirical performance of the proposed method is illustrated by simulation examples and real data applications. This is a joint work with Wanjun Liu, Jingyuan Liu and Runze Li.
Meeting link: https://bluejeans.com/722836372/4781?src=join_info
Anosov flows are an important class of dynamical systems due to their ergodic properties and structural stability. Geometrically, they are defined by two transverse invariant foliations with expanding and contracting behaviors. Much of our understanding of the structure of an Anosov flow relies on the study of the leaves space of the invariant foliations. In this talk we adopt a different approach: in the early 90s Mitsumatsu first noticed that and Anosov vector field also belongs to the intersection of two transverse contact structures rotating towards each other. After giving the necessary background I will use this point of view to address questions in surgery theory on Anosov flows and contact structures.
If Z is a set of points in projective space, we can ask which polynomials of degree d vanish at every point in Z. If P is one point of Z, the vanishing of a polynomial at P imposes one linear condition on the coefficients. Thus, the vanishing of a polynomial on all of Z imposes |Z| linear conditions on the coefficients. A classical question in algebraic geometry, dating back to at least the 4th century, is how many of those linear conditions are independent? For instance, if we look at the space of lines through three collinear points in the plane, the unique line through two of the points is exactly the one through all three; i.e. the conditions imposed by any two of the points imply those of the third. In this talk, I will survey several classical results including the original Cayley-Bacharach Theorem and Castelnuovo’s Lemma about points on rational curves. I’ll then describe some recent results and conjectures about points satisfying the so-called Cayley-Bacharach condition and show how they connect to several seemingly unrelated questions in contemporary algebraic geometry relating to the gonality of curves and measures of irrationality of higher dimensional varieties.
Note the unusual time!
Recently, significant progress has been made in the area of machine learning algorithms, and they have quickly become some of the most exciting tools in a scientist’s toolbox. In particular, recent advances in the field of reinforcement learning have led computers to reach superhuman level play in Atari games and Go, purely through self-play. In this talk I will give a basic introduction to neural networks and reinforcement learning algorithms. I will also indicate how these methods can be adapted to the "game" of trying to find a counterexample to a mathematical conjecture, and show some examples where this approach was successful.
BlueJeans link: https://bluejeans.com/609527728/0740
The main goal of manifold theory is to classify all n-dimensional topological manifolds. For a smooth 4-manifold X, we aim to understand all of the exotic smooth structures there are to the smooth structure on X. Exotic smooth structures are homeomorphic but not diffeomorphic. Cork twists, Gluck twists, and Log transforms are all ways to construct possible exotic pairs by re-gluing embedded surfaces in the 4-manifold. In this talk, we define these three constructions.
Zoom link-- https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Using techniques from dynamical systems theory, we rigorously study an experimentally validated model by [Barkley et al., Nature, 526:550-553, 2015], which describes the rise of turbulent pipe flow via a PDE system of reduced complexity. The fast evolution of turbulence is governed by reaction-diffusion dynamics coupled to the centerline velocity, which evolves with advection of Burgers' type and a slow relaminarization term. Applying to this model a spatial dynamics ansatz, we prove the existence of a heteroclinic loop between a turbulent and a laminar steady state and establish a cascade of bifurcations of traveling waves mediating the transition to turbulence, with a focus on an intermediate Reynolds number regime.
This is joint work with Björn de Rijk and Christian Kuehn.
Dynamical systems model the way that real-world systems evolve in time. While the time-asymptotic behavior of many systems can be characterized by “simple” dynamical features such as equilibria and periodic orbits, some systems evolve in a chaotic, seemingly random way. For such systems it is no longer meaningful to track one trajectory at a time individually- instead, a natural approach is to treat the initial condition as random and to observe how its probabilistic law evolves in time. This is the core idea of ergodic theory, the topic of this talk. I will not assume much beyond some basics of probability theory, e.g., random variables.
Structural graph theory has usually focused on classes of graphs that are 'sparse' rather than 'dense' (that is, have few edges rather than many edges). We discuss this paradigm, focusing on classes with a forbidden vertex-minor. In particular, we discuss progress on a conjecture of Geelen that would totally characterize classes with a forbidden vertex-minor. This is joint work with Jim Geelen and Paul Wollan.
In this talk, we survey known results and open problems tied to the dual graph of a projective algebraic F-scheme over a field F, a construction that apparently Janos Kollar is familiar with. In particular one can use this construction to answer the following question: if you consider the 27 lines on a cubic surface in P^3, how many lines meet a given line? The dual graph can answer this and more questions in enumerative geometry and intersection theory easily, based on work of Benedetti -- Varbaro and others.
Given a multigraph $G=(V,E)$, the chromatic index $\chi'(G)$ is the minimum number of colors needed to color the edges of $G$ such that no two incident edges receive the same color. Let $\Delta(G)$ be the maximum degree of $G$ and let $\Gamma(G):=\max \big\{\frac{2|E(U)|}{|U|-1}:\,\, U \subseteq V, \,\, |U|\ge 3 \hskip 2mm
{\rm and \hskip 2mm odd} \big\}$. $\Gamma(G)$ is called the density of $G$. Clearly, the density is a lower bound for the chromatic index $\chi'(G)$. Moreover, this value can be computed in polynomial time. Goldberg and Seymour in the 1970s conjectured that $\chi'(G)=\lceil\Gamma(G)\rceil$ for any multigraph $G$ with $\chi'(G)\geq\Delta(G)+2$, known as the Goldberg-Seymour conjecture. In this talk we will discuss this conjecture and some related open problems. This is joint work with Guantao Chen and Wenan Zang.
This talk is about the arithmetic of points of small canonical height relative to dynamical systems over number fields, particularly those aspects amenable to the use of equidistribution techniques. Past milestones in the subject include the proof of the Bogomolov Conjecture given by Ullmo and Zhang, and Baker-DeMarco's work on the finiteness of common preperiodic points of unicritical maps. Recently, quantitative equidistribution techniques have emerged both as a way of improving upon some of these old results, and as an avenue to studying previously inaccessible problems, such as the Uniform Boundedness Conjecture of Morton and Silverman. I will describe the key ideas behind these developments, and raise related questions for future research.
https://bluejeans.com/788895268/8348
Graph alignment refers to recovering the underlying vertex correspondence between two random graphs with correlated edges. This problem can be viewed as an average-case and noisy version of the well-known graph isomorphism problem. For correlated Erdős-Rényi random graphs, we will give insights on the fundamental limits for the planted formulation of this problem, establishing statistical thresholds for partial recovery. From the computational point of view, we are interested in designing and analyzing efficient (polynomial-time) algorithms to recover efficiently the underlying alignment: in a sparse regime, we exhibit an local rephrasing of the planted alignment problem as the correlation detection problem in trees. Analyzing this related problem enables to derive a message-passing algorithm for our initial task and gives insights on the existence of a hard phase.
Based on joint works with Laurent Massoulié and Marc Lelarge:
https://arxiv.org/abs/2002.01258
https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
For a dynamical system, a physical measure is an ergodic invariant measure that captures the asymptotic statistical behavior of the orbits of a set with positive Lebesgue measure. A natural question in the theory is to know when such measures exist.
It is expected that a "typical" system with enough hyperbolicity (such as partial hyperbolicity) should have such measures. A special type of physical measure is the so-called hyperbolic SRB (Sinai-Ruelle-Bowen) measure. Since the 70`s the study of SRB measures has been a very active topic of research.
In this talk, we will see a new example of open sets of partially hyperbolic systems with two dimensional center having a unique SRB measure. One of the key features for these examples is a rigidity result for a special type of measure (the so-called u-Gibbs measure) which allows us to conclude the existence of the SRB measures.
Szemerédi's regularity lemma is a game-changer in extremal combinatorics and provides a global perspective to study large combinatorial objects. It has connections to number theory, discrete geometry, and theoretical computer science. One of its classical applications, the removal lemma, is the essence for many property testing problems, an active field in theoretical computer science. Unfortunately, the bound on the sample size from the regularity method typically is either not explicit or enormous. For testing natural permutation properties, we show one can avoid the regularity proof and yield a tester with polynomial sample size. For graphs, we prove a stronger, "L_\infty'' version of the graph removal lemma, where we conjecture that the essence of this new removal lemma for cliques is indeed the regularity-type proof. The analytic interpretation of the regularity lemma also plays an important role in graph limits, a recently developed powerful theory in studying graphs from a continuous perspective. Based on graph limits, we developed a method combining with both analytic and spectral methods, to answer and make advances towards some famous conjectures on a common theme in extremal combinatorics: when does randomness give nearly optimal bounds?
These works are based on joint works with Jacob Fox, Dan Kral', Jonathan Noel, Sergey Norin, and Jan Volec.
Note the unusual time!
The famous Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In this talk, I will briefly sketch a proof of this conjecture for every large n. If time permits, I will also talk about our solution to a problem of Erdős from 1977 about chromatic index of hypergraphs with bounded codegree. Joint work with D. Kang, T. Kelly, D.Kuhn and D. Osthus.
Thresholds for increasing properties of random structures are a central concern in probabilistic combinatorics and related areas. In 2006, Kahn and Kalai conjectured that for any nontrivial increasing property on a finite set, its threshold is never far from its "expectation-threshold," which is a natural (and often easy to calculate) lower bound on the threshold. In this talk, I will first introduce the Kahn-Kalai Conjecture with some motivating examples and then talk about the recent resolution of a fractional version of the Kahn-Kalai Conjecture due to Frankston, Kahn, Narayanan, and myself. Some follow-up work, along with open questions, will also be discussed.
We present results on the number of linear regions of the functions that can be represented by artificial feedforward neural networks with maxout units. A rank-k maxout unit is a function computing the maximum of k linear functions. For networks with a single layer of maxout units, the linear regions correspond to the regions of an arrangement of tropical hypersurfaces and to the (upper) vertices of a Minkowski sum of polytopes. This is joint work with Guido Montufar and Leon Zhang.
Meeting link: https://bluejeans.com/912860268/9947
The Navier-Stokes and Euler equations are the fundamental models for describing viscous and inviscid fluids, respectively. Based on ideas which date back to Kolmogorov and Onsager, solutions to these equations are expected to dissipate energy, which in turn suggests that such solutions are somewhat rough and thus only weak solutions. At these low regularity levels, however, one may construct wild weak solutions using convex integration methods. In this talk, I will discuss the motivation and methodology behind joint work with Tristan Buckmaster, Nader Masmoudi, and Vlad Vicol in which we construct wild solutions to the Euler equations which deviate from the predictions of Kolmogorov's classical K41 phenomenological theory of turbulence.
Random graphs with latent geometric structure, where the edges are generated depending on some hidden random vectors, find broad applications in the real world, including social networks, wireless communications, and biological networks. As a first step to understand these models, the question of when they are different from random graphs with independent edges, i.e., Erd\H{o}s--R\'enyi graphs, has been studied recently. It was shown that geometry in these graphs is lost when the dimension of the latent space becomes large. In this talk, we focus on the case when there exist different notions of noise in the geometric graphs, and we show that there is a trade-off between dimensionality and noise in detecting geometry in the random graphs.
Link: https://bluejeans.com/520769740/3630
Designing mechanisms to maximize revenue is a fundamental problem in mathematical economics and has various applications like online ad auctions and spectrum auctions. Unfortunately, optimal auctions for selling multiple items can be unreasonably complex and computationally intractable. In this talk, we consider a revenue-maximizing seller with n items facing a single unit-demand buyer. Our work shows that simple mechanisms can achieve almost optimal revenue. We approached the tradeoffs of simplicity formally through the lens of computation and menu size. Our main result provides a mechanism that gets a (1 − ε)-approximation to the optimal revenue in time quasi-polynomial in n and has quasi polynomial (symmetric) menu complexity.
Joint work with Pravesh Kothari, Ariel Schvartzman, Sahil Singla, and Matt Weinberg.
https://bluejeans.com/910698769/4854
Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”. Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations. After an informal introduction to dispersive equations, I will survey some of my recent results towards understanding the long-time behavior of dispersive waves and the soliton resolution using techniques from both partial differential equations and inverse scattering transforms.
Meeting Link: https://bluejeans.com/426529046/8775
Two years after the beginning of the pandemic, we are still working to understand the mechanisms of immunopathology in COVID-19. Immune responses following SARS-CoV-2 infections are heterogeneous, and biomarkers of this variability remain to be elucidated. In collaboration with experimentalists and clinicians, we have deployed various mathematical and computational approaches to understand longitudinal immunological data from patients, and to generate new hypotheses about the factors determining COVID-19 severity and disease dynamics.
To answer foundational questions about immunopathology and heterogeneity in COVID-19, we have developed a multi-scale, mechanistic mathematical model of the immune response to SARS-CoV-2 that includes several innate and adaptive immune cells and their communication via signalling networks. By generating a population of virtual patients, we identified dysregulated rates of monocyte-to-macrophage differentiation that distinguishes disease severity in these in silico patients. Further, our results suggest that maximal IL-6 concentrations can be used as a predictive biomarker of CD8+ T cell lymphopenia. Using the same cohort of virtual patients, we have also studied the influence of variant on immunopathology by combining our model with data of intra-host viral evolution. We predicted that the combined effects of mutations affecting the spike proteins and interferon evasion on the severity of COVID-19 are mostly determined by the innate host immune response. Our approaches can be used to study the factors regulated immunopathology during SARS-CoV-2 infections, and represent a quantitative framework for the study of COVID-19 and other viral diseases.
Recording link: https://bluejeans.com/s/6CmKwHWWc2O
Meeting link: https://bluejeans.com/961048334/8189
A long-standing problem in the field of graph coloring is the Erdős–Faber–Lovász conjecture (posed in 1972), which states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$, or equivalently, that a nearly disjoint union of $n$ complete graphs on at most $n$ vertices has chromatic number at most $n$. In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$. Recently, we also solved a related problem of Erdős from 1977 on the chromatic index of hypergraphs of small codegree. In this talk, I will survey the history behind these results and discuss some aspects of the proofs.
Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group. The lowest dimension for such classical phenomena is 5.
I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)—parameter gauge theory. The construction uses a topological technique. I’ll mention some other applications to embeddings of surfaces and 3-manifolds in 4-manifolds.
Zoom Link- https://brandeis.zoom.us/j/99772088777 (password- hyperbolic)
Here is alternative link where the password is embedded- https://brandeis.zoom.us/j/99772088777?pwd=WHpFQk1Fem5jZVRNRUwzVmpmck4xdz09
The first part of the talk is devoted to robust algorithms for the k-means clustering problem where a quantizer is constructed based on N independent observations. I will present recent sharp non-asymptotic performance guarantees for k-means that hold under the two bounded moments assumption in a general Hilbert space. These bounds extend the asymptotic result of D. Pollard (Annals of Stats, 1981) who showed that the existence of two moments is sufficient for strong consistency of an empirically optimal quantizer. In the second part of the talk I discuss a dimension-free version of the result of Adamczak, Litvak, Pajor, Tomczak-Jaegermann (Journal of Amer. Math. Soc, 2010) for the sample-covariance matrix in log-concave ensembles. The proof of the dimension-free result is based on a duality formula between entropy and moment generating functions. Finally, I will briefly discuss a recent bound on an empirical risk minimization strategy in stochastic convex optimization with strongly convex and Lipschitz losses.
Link to the online talk: https://bluejeans.com/958288541/0675
Given a system of analytic functions and an approximate zero, we introduce inflation to transform this system into one with a regular quadratic zero. This leads to a method for isolating a cluster of zeros of the given system.
(This is joint work with Michael Burr.)
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In this talk, we show an adaptation of the Kohler-Jobin rearrangement technique to the setting of the Gauss space. As a result, we present the Gaussian analogue of the Kohler-Jobin's resolution of a conjecture of Polya-Szego: when the Gaussian torsional rigidity of a (convex) domain is fixed, the Gaussian principal frequency is minimized for the half-space. At the core of this rearrangement technique is the idea of considering a ``modified'' torsional rigidity, with respect to a given function, and rearranging its layers to half-spaces, in a particular way; the Rayleigh quotient decreases with this procedure.
We emphasize that the analogy of the Gaussian case with the Lebesgue case is not to be expected here, as in addition to some soft symmetrization ideas, the argument relies on the properties of some special functions; the fact that this analogy does hold is somewhat of a miracle.
The seminar will be held on Zoom via the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
The Schinzel-Zassenhaus conjecture describes the narrowest collar width around the unit circle that contains a full set of conjugate algebraic integers of a given degree, at least one of which lies off the unit circle. I will explain what this conjecture precisely says and how it is proved. The method involved in this solution turns out to yield some other new results whose ideas I will describe, including to the closest interlacing of Frobenius eigenvalues for abelian varieties over finite fields, the closest separation of Salem numbers in a fixed interval, and the distribution of the short Kobayashi geodesics in the Siegel modular variety.
The linear stability of a family of Kelvin-Stuart Cat's eyes flows of 2D Euler equation was studied both analytically and numerically. We proved linear stability under co-periodic perturbations and linear instability under multi-periodic perturbations. These results were first obtained numerically using spectral methods and then proved analytically.
The Bluejeans link is: https://bluejeans.com/353383769/0224
This talk is a primer on solving certain kinds of counting problems through regular languages, finite automata and transfer matrices. Example problems: count the number of binary strings that contain "0110", count the number of binary strings that contain 0, 1, 2,... copies of "0110," a derivation of the negative binomial distribution function.
The only requirements for this talk is a basic familiarity with directed graphs, matrices and generating functions.
Link: https://bluejeans.com/520769740/
The equivalence of realizable and agnostic learnability is a fundamental phenomenon in learning theory. With variants ranging from classical settings like PAC learning and regression to recent trends such as adversarially robust and private learning, it’s surprising we still lack a unifying theory explaining these results.
In this talk, we'll introduce exactly such a framework: a simple, model-independent blackbox reduction between agnostic and realizable learnability that explains their equivalence across a wide host of classical models. We’ll discuss how this reduction extends our understanding to traditionally difficult settings such as learning with arbitrary distributional assumptions and general loss, and look at some applications beyond agnostic learning as well (e.g. to privacy). Finally, we'll end by surveying a few nice open problems in the area.
Based on joint work with Daniel Kane, Shachar Lovett, and Gaurav Mahajan.
Quantum spin systems are many-body physical models where particles are bound to the sites of a lattice. These are widely used throughout condensed matter physics and quantum information theory, and are of particular interest in the classification of quantum phases of matter. By pinning down the properties of new exotic phases of matter, researchers have opened the door to developing new quantum technologies. One of the fundamental quantitites for this classification is whether or not the Hamiltonian has a spectral gap above its ground state energy in the thermodynamic limit. Mathematically, the Hamiltonian is a self-adjoint operator and the set of possible energies is given by its spectrum, which is bounded from below. While the importance of the spectral gap is well known, very few methods exist for establishing if a model is gapped, and the majority of known results are for one-dimensional systems. Moreover, the existence of a non-vanishing gap is generically undecidable which makes it necessary to develop new techniques for estimating spectral gaps. In this talk, I will discuss my work proving non-vanishing spectral gaps for key quantum spin models, and developing new techniques for producing lower bound estimates on the gap. Two important models with longstanding spectral gap questions that I recently contributed progress to are the AKLT model on the hexagonal lattice, and Haldane's pseudo-potentials for the fractional quantum Hall effect. Once a gap has been proved, a natural next question is whether it is typical of a gapped phase. This can be positively answered by showing that the gap is robust in the presence of perturbations. Ensuring the gap remains open in the presence of perturbations is also of interest, e.g., for the development of robust quantum memory. A second topic I will discuss is my research studying spectral gap stability.
URL for the talk: https://bluejeans.com/602513114/7767
We first review the problem of the curse of dimensionality when approximating multi-dimensional functions. Several approximation results from Barron, Petrushev, Bach, and etc . will be explained.
Then we present two approaches to break the curse of the dimensionality: one is based on probability approach explained in Barron, 1993 and the other one is based on a deterministic approach using the Kolmogorov superposition theorem. As the Kolmogorov superposition theorem has been used to explain the approximation of neural network computation, I will use it to explain why the deep learning algorithm works for image classification.
In addition, I will introduce the neural network approximation based on higher order ReLU functions to explain the powerful approximation of multivariate functions using deep learning algorithms with multiple layers.
Let M be a complex-hyperbolic n-manifold, i.e. a quotient of the complex-hyperbolic n-space $\mathbb{H}^n_\mathbb{C}$ by a torsion-free discrete group of isometries, $\Gamma = \pi_1(M)$. Suppose that M is convex-cocompact, i.e. the convex core of M is a nonempty compact subset. In this talk, we will discuss a sufficient condition on $\Gamma$ in terms of the growth-rate of its orbits in $\mathbb{H}^n_\mathbb{C}$ for which M is a Stein manifold. We will also talk about some interesting questions related to this result. This is a joint work with Misha Kapovich.
https://bluejeans.com/196544719/9518
The goal of the meeting is to decide what paper(s) we will be reading and make a rough plan going forward. The following two possibilities were suggested:
Other suggestions are also welcome!
Meeting Link: https://bluejeans.com/426529046/8775
In this talk, I will discuss recent advances and challenges in modelling complex dynamics of pedestrian-bridge interactions, These challenges include a proper understanding of the biomechanics of walking on a moving structure and of the psychology of walking in crowds. I will explain the fundamental mechanism behind pedestrian-induced lateral instability of bridges due to some positive feedback from uncorrelated walkers whose foot forces do not cancel each other but create a bias. I will also present the results of our past and ongoing work that reveal the role of foot placement strategies and social force dynamics in initiating bridge instabilities. In particular, I will show that (i) paradoxically, depending on the human balance law (and the frequency of bridge motion), larger crowds can stabilize bridge motions and (ii) crowd heterogeneity can promote large vibrations of bridges.
Recording link: https://bluejeans.com/s/h0TpdyBRatJ
A lot of the algebraic and arithmetic information of a curve is contained in its interaction with the Galois group. This draws inspiration from topology, where given a family of curves over a base B, the fundamental group of B acts on the cohomology of the fiber. As an arithmetic analogue, given an algebraic curve C defined over a non-algebraically closed field K, the absolute Galois group of K acts on the etale cohomology of the geometric fiber and this action gives rise to various Galois cohomology classes. In this talk, we discuss how to use these classes to detect algebraic/arithmetic properties of the curve, such as the rational points (following Grothendieck's section conjecture), whether the curve is hyperelliptic, and the set of ``supersingular'' primes.
Quantum Teichmüller space was first introduced by Chekhov and Fock as a version of 2+1d quantum gravity. The definition was translated over time into an algebra of curves on surfaces, which coincides with an extension of the Kauffman bracket skein algebra. In this talk, we will discuss the relation between the Teichmüller space and the Kauffman bracket, and time permitting, the quantized version of this correspondence.
Meeting URL: https://bluejeans.com/106460449/5822
Let f be a real-valued Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.
What is the probability that f remains above a certain fixed line for a long period of time?
We give simple spectral(and almost tight) conditions under which this probability is asymptotically exponential, that is, that the limit of log P(f>a on [0,T])/ T, as T approaches infinity, exists.
This limit defines "the persistence exponent", and we further show it is continuous in the level a, in the spectral measure corresponding to f (in an appropriate sense), and is unaffected by the singular part of the spectral measure.
Proofs rely on tools from harmonic analysis.
Joint work with Ohad Feldheim and Sumit Mukherjee, arXiv:2112.04820.
The talk will be on Zoom via the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
A central goal in modern data science is to design algorithms for statistical inference tasks such as community detection, high-dimensional clustering, sparse PCA, and many others. Ideally these algorithms would be both statistically optimal and computationally efficient. However, it often seems impossible to achieve both these goals simultaneously: for many problems, the optimal statistical procedure involves a brute force search while all known polynomial-time algorithms are statistically sub-optimal (requiring more data or higher signal strength than is information-theoretically necessary). In the quest for optimal algorithms, it is therefore important to understand the fundamental statistical limitations of computationally efficient algorithms.
I will discuss an emerging theoretical framework for understanding these questions, based on studying the class of "low-degree polynomial algorithms." This is a powerful class of algorithms that captures the best known poly-time algorithms for a wide variety of statistical tasks. This perspective has led to the discovery of many new and improved algorithms, and also many matching lower bounds: we now have tools to prove failure of all low-degree algorithms, which provides concrete evidence for inherent computational hardness of statistical problems. This line of work illustrates that low-degree polynomials provide a unifying framework for understanding the computational complexity of a wide variety of statistical tasks, encompassing hypothesis testing, estimation, and optimization.
We will introduce the machinery of hyperbolic polynomial, and see how it can help us generalize classical linear algebra theorems and inequalities on symmetric matrices, including Hadamard-Fischer inequality, Koteljanskii's inequality and Schur-Horn theorem (last one is conjectured but not proved). Joint work with Greg Blekherman, Mario Kummer, Raman Sanyal and Kevin Shu.
The stable commutator length function measures the growth rate of the commutator length of powers of elements in the commutator subgroup of a group. In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will show that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed. Along the way to proving our main results, we will discuss certain topological properties of a class of infinite-type surfaces and their end spaces which may be of independent interest. This talk represents joint work with Priyam Patel and Alexander Rasmussen.
Efficient simulation of SDEs is essential in many applications, particularly for ergodic
systems that demand efficient simulation of both short-time dynamics and large-time
statistics. To achieve the efficiency, dimension reduction is often required in both space
and time. In this talk, I will talk about our recent work on both spatial and temporal
reductions.
For spatial dimension reduction, the Mori-Zwanzig formalism is applied to derive
equations for the evolution of linear observables of the Langevin dynamics for both
overdamped and general cases.
For temporal dimension reduction, we introduce a framework to construct inference-
based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in
time by several orders of magnitudes.
This is a joint work with Dr. Thomas Hudson from the University of Warwick, UK; Dr. Fei
Lu from the Johns Hopkins University and Dr Xiaofeng Felix Ye from SUNY at Albany.
In the first talk of this seminar series, we follow the manuscript of Carl Miller and introduce the concept of elusive graph properties—those properties for which any edge-querying algorithm requires all possible queries in the worst case. Karp conjectured in 1973 that all nontrivial monotonic graph properties are elusive, and a celebrated theorem by Kahn in 1984 used topological fixed-point methods to show the conjecture is true in the case of graphs with order equal to a prime power. To set the stage for the proof of this result in later talks, we introduce monotone graph properties and their connection to collapsible simplicial complexes.
This is an expository talk about the slice-ribbon conjecture. A knot is slice if it bounds a disk in the four ball. We call a slice knot ribbon if it bounds a slice disk with no local maxima. The slice-ribbon conjecture asserts all slice knots arise in this way. We also give a very brief introduction to Greene, Jabuka and Lecuona's works on the slice-ribbon conjecture for 3-stranded pretzel knots.
This talk will detail two recent papers concerning Rogers-Shephard inequalities and Zhang inequalities for various classes of measures, the first of which is a reverse form of the Brunn-Minkowsk inequality, and the second of which can be seen to be a reverse affine isoperimetric inequality; the feature of both inequalities is that they each provide a classification of the n-dimensional simplex in the volume case. The covariogram of a measure plays an essential role in the proofs of each of these inequalities. In particular, we will discuss a variational formula concerning the covariogram resulting in a measure theoretic version of the projection body, an object which has recently gained a lot of attention--these objects were previously studied by Livshyts in her analysis of the Shephard problem for general measure.
The talk will be on Zoom via the link
https://us02web.zoom.us/j/71579248210?pwd=d2VPck1CbjltZStURWRWUUgwTFVLZz09
Many modern problems in data science aim to efficiently and accurately extract important features and make predictions from high dimensional and large data sets. While there are many empirically successful methods to achieve these goals, large gaps between theory and practice remain. A geometric viewpoint is often useful to address these challenges as it provides a unifying perspective of structure in data, complexity of statistical models, and tractability of computational methods. As a consequence, an understanding of problem geometry leads both to new insights on existing methods as well as new models and algorithms that address drawbacks in existing methodology.
In this talk, I will present recent progress on two problems where the relevant model can be viewed as the projection of a lifted formulation with a simple stochastic or convex geometric description. In particular, I will first describe how the theory of stationary random tessellations in stochastic geometry can address computational and theoretical challenges of random decision forests with non-axis-aligned splits. Second, I will present a new approach to convex regression that returns non-polyhedral convex estimators compatible with semidefinite programming. These works open a number of future research directions at the intersection of stochastic and convex geometry, statistical learning theory, and optimization.
The k-cap (or k-winners-take-all) process on a graph works as follows: in each
iteration, exactly k vertices of the graph are in the cap (i.e., winners); the next round
winners are the vertices that have the highest total degree to the current winners,
with ties broken randomly. This natural process is a simple model of firing activity
in the brain. We study its convergence on geometric random graphs revealing rather
surprising behavior
Link: https://bluejeans.com/398474745/0225
In 1943, Hadwiger conjectured that every graph with no $K_t$ minor is $(t-1)$-colorable for every $t \ge 1$. Hadwiger's Conjecture is a vast generalization of the Four Color Theorem and one of the most important open problems in graph theory. Only the cases when $t$ is at most 6 are known. In the 1980s, Kostochka and Thomason independently proved that every graph with no $K_t$ minor has average degree $O(t (\log t)^{0.5})$ and hence is $O(t (\log t)^{0.5})$-colorable. In a recent breakthrough, Norin, Song, and I proved that every graph with no $K_t$ minor is $O(t (\log t)^c)$-colorable for every $c > 0.25$, Subsequently I showed that every graph with no $K_t$ minor is $O(t (\log \log t)^6)$-colorable. Delcourt and I improved upon this further by showing that every graph with no $K_t$ minor is $O(t \log \log t)$-colorable. Our main technical result yields this as well as a number of other interesting corollaries. A natural weakening of Hadwiger's Conjecture is the so-called Linear Hadwiger's Conjecture that every graph with no $K_t$ minor is $O(t)$-colorable. We prove that Linear Hadwiger's Conjecture reduces to small graphs. In 2005, Kühn and Osthus proved that Hadwiger's Conjecture for the class of $K_{s,s}$-free graphs for any fixed positive integer $s \ge 2$. Along this line, we show that Linear Hadwiger's Conjecture holds for the class of $K_r$-free graphs for every fixed $r$.
The mapping class group of a compact and orientable surface of genus g has an important subgroup called the Torelli group, which is the kernel of the action on the homology of the surface. In this talk we will discuss the stable rational homology of the Torelli group of a surface with a boundary component, about which very little is known in general. These homology groups are representations of the arithmetic group Sp_{2g}(Z) and we study them using an Sp_{2g}(Z)-equivariant map induced on homology by the so-called Johnson homomorphism. The image of this map is a finite dimensional and algebraic representation of Sp_{2g}(Z). By considering a type of homology classes called abelian cycles, which are easy to write down for Torelli groups and for which we can derive an explicit formula for the map in question, we may use classical representation theory of symplectic groups to describe a large part of the image.
Let A be a subset of the integers of size n. In 1983, Erdos and Szemeredi conjectured that either A+A or A*A must have size nearly n^2. We discuss ideas towards this conjecture, such as an older connection to incidence geometry as well as somewhat newer breakthroughs in additive combinatorics. We further highlight applications of the sum-product phenomenon.
We prove that every simple 2-connected subcubic graph on $n$ vertices with $n_2$ vertices of degree 2 has a TSP walk of length at most $\frac{5n+n_2}{4}-1$, confirming a conjecture of Dvořák, Král', and Mohar. This bound is best possible; there are infinitely many subcubic and cubic graphs whose minimum TSP walks have lengths $\frac{5n+n_2}{4}-1$ and $\frac{5n}{4} - 2$ respectively. We characterize the extremal subcubic examples meeting this bound. We also give a quadratic-time combinatorial algorithm for finding such a TSP walk. In particular, we obtain a $\frac{5}{4}$-approximation algorithm for the graphic TSP on simple cubic graphs, improving on the previously best known approximation ratio of $\frac{9}{7}$.
Meeting Link: https://bluejeans.com/426529046/8775
Cell cycle duration changes dramatically during development, starting out fast to generate cells quickly and slowing down over time as the organism matures. The cell cycle can also act as a transcriptional filter to control the expression of long gene transcripts which are partially transcribed in short cycles. Using mathematical simulations of cell proliferation, we identify an emergent property, that this filter can act as a tuning knob to control gene transcript expression, cell diversity and the number and proportion of different cell types in a tissue. Our predictions are supported by comparison to single-cell RNA-seq data captured over embryonic development. Additionally, evolutionary genome analysis shows that fast developing organisms have a narrow genomic distribution of gene lengths while slower developers have an expanded number of long genes. Our results support the idea that cell cycle dynamics may be important across multicellular animals for controlling gene transcript expression and cell fate.
Recording link: https://bluejeans.com/s/QhCWmELH6AC
Topological methods have had a rich history of use in convex optimization, including for instance the famous Pataki-Barvinok bound on the ranks of solutions to semidefinite programs, which involves the Borsuk-Ulam theorem. We will give two proofs of a similar sort involving the use of some basic homotopy theory. One is a new proof of Brickman's theorem, stating that the image of a sphere into R^2 under a quadratic map is convex, and the other is an original theorem stating that the image of certain matrix groups under linear maps into R^2 is convex. We will also conjecture some higher dimensional analogues.
Gibbs measures describing directed polymers in random potential are tightly related to the stochastic Burgers/KPZ/heat equations. One of the basic questions is: do the local interactions of the polymer chain with the random environment and with itself define the macroscopic state uniquely for these models? We establish and explore the connection of this problem with ergodic properties of an infinite-dimensional stochastic gradient flow. Joint work with Hong-Bin Chen and Liying Li.
We introduce the notion of braided monoidal categories and fusion categories, which are one way of reframing algebraic structures in a categorical context. After discussing various examples and analogies with the theory of finite groups, we build up to a classification of pointed fusion categories.
https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
In this talk, we discuss ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles over mixing subshifts of finite type and that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.
We consider a subadditive potential $\Phi$. We obtain that for $t \to \infty$ any accumulation point of a family of equilibrium states of $t\Phi$ is a maximizing measure and that the Lyapunov exponent and entropy of equilibrium states for $t\Phi$ converge in the limit $t\to \infty$ to the maximal Lyapunov exponent and entropy of maximizing measures. Moreover, we show that if a $SL(2, \mathbb{R})$ one-step cocycle satisfies pinching and twisting conditions and there exist strictly invariant cones whose images do not overlap on the Mather set then the Lyapunov-maximizing measures have zero entropy.
Fiber sums and the rational blowdown have been very useful tools in constructing smooth, closed, oriented 4-manifolds. Applying these tools to genus g>1 Lefschetz fibrations with clustered nodal fibers, we will construct symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line, providing a symplectic extension of classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Moreover, Lefschetz fibrations with certain clustered nodal fibers provide rational blowdown configurations that yield new constructions of small symplectic exotic 4-manifolds. We will present an example of a construction of a minimal symplectic exotic CP^2#-5CP^2 through this procedure applied to a genus-3 fibration. This work is joint with Inanc Baykur and Mustafa Korkmaz.
In modern data analysis, the datasets are often represented by large-scale matrices or tensors (the generalization of matrices to higher dimensions). To have a better understanding or extract values effectively from these data, an important step is to construct a low-dimensional/compressed representation of the data that may be better to analyze and interpret in light of a corpus of field-specific information. To implement the goal, a primary tool is the matrix/tensor decomposition. In this talk, I will talk about novel matrix/tensor decompositions, CUR decompositions, which are memory efficient and computationally cheap. Besides, I will also discuss the applications of CUR decompositions on developing efficient algorithms or models to robust decompositions or data completion problems. Additionally, some simulation results will be provided on real and synthetic datasets.
Graph burning is a simplified model for the spread of influence in a network. Associated with the process is the burning number, which quantifies the speed at which the influence spreads to every vertex. The Burning Number Conjecture claims that for every connected graph $G$ of order $n,$ its burning number satisfies $b(G) \le \lceil \sqrt{n} \rceil$. While the conjecture remains open, we prove the best-known bound on the burning number of a connected graph $G$ of order $n,$ given by $b(G) \le \sqrt{4n/3} + 1$, improving on the previously known $\sqrt{3n/2}+O(1)$ bound.
Meeting Link: https://bluejeans.com/426529046/8775
We describe a spatial Moran model that captures mechanical interactions and directional growth in spatially extended populations. The model is analytically tractable and completely solvable under a mean-field approximation and can elucidate the mechanisms that drive the formation of population-level patterns. As an example, we model a population of E. coli growing in a rectangular microfluidic trap. We show that spatial patterns can arise because of a tug-of-war between boundary effects and growth rate modulations due to cell-cell interactions: Cells align parallel to the long side of the trap when boundary effects dominate. However, when cell-cell interactions exceed a critical value, cells align orthogonally to the trap’s long side. This modeling approach and analysis can be extended to directionally growing cells in a variety of domains to provide insight into how local and global interactions shape collective behavior. As an example, we discuss how our model reveals how changes to a cell-shape describing parameter may manifest at the population level of the microbial collective. Specifically, we discuss mechanisms revealed by our model on how we may be able to control spatiotemporal patterning by modifying cell shape of a given strain in a multi-strain microbial consortium.
Recording Link: https://bluejeans.com/s/0g6lBzbf0XT
We will discuss Akbulut's construction of two smooth, contractible four-manifolds whose boundaries are diffeomorphic and extend to a homeomorphism but not to a diffeomorphism of the manifolds.
We will review how divisors on abstract algebraic curves are connected with projective embeddings and then see how that language translates to tropical curves and tropicalization. This talk aims to explain some of the connections between tropical curves and algebraic curves that was not discussed during the seminar on tropical Brill-Noether theory.
Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Suppose that $M$ is a closed isotropic Riemannian manifold and that $R_1,...,R_m$ generate the isometry group of $M$. Let $f_1,...,f_m$ be smooth perturbations of these isometries. We show that the $f_i$ are simultaneously conjugate to isometries if and only if their associated uniform Bernoulli random walk has all Lyapunov exponents zero. This extends a linearization result of Dolgopyat and Krikorian from $S^n$ to real, complex, and quaternionic projective spaces.
Recent research on solving partial differential equations with deep neural networks (DNNs) has demonstrated that spatiotemporal-function approximators defined by auto-differentiation are effective for approximating nonlinear problems. However, it remains a challenge to resolve discontinuities in nonlinear conservation laws using forward methods with DNNs without beginning with part of the solution. In this study, we incorporate first-order numerical schemes into DNNs to set up the loss function approximator instead of auto-differentiation from traditional deep learning framework such as the TensorFlow package, thereby improving the effectiveness of capturing discontinuities in Riemann problems. We introduce a novel neural network method. A local low-cost solution is first used as the input of a neural network to predict the high-fidelity solution at a space-time location. The challenge lies in the fact that there is no way to distinguish a smeared discontinuity from a steep smooth solution in the input, thus resulting in “multiple predictions” of the neural network. To overcome the difficulty, two solutions of the conservation laws from a converging sequence, computed from low-cost numerical schemes, and in a local domain of dependence of the space-time location, serve as the input. Despite smeared input solutions, the output provides sharp approximations to solutions containing shocks and contact surfaces, and the method is efficient to use, once trained. It works not only for discontinuities, but also for smooth areas of the solution, implying broader applications for other differential equations.
Let $M$ be the underlying smooth $4$-manifold of a degree $d$ del Pezzo surface. In this talk, we will discuss two related results about finite subgroups of the mapping class group $\text{Mod}(M) := \pi_0(\text{Homeo}^+(M))$. A motivating question for both results is the Nielsen realization problem for $M$: which finite subgroups $G$ of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$? For del Pezzo surfaces $M$ of degree $d \geq 7$, we will give a complete classification of such finite subgroups. Furthermore, we will give a classification of, and a structure theorem for, all involutions in $\text{Mod}(M)$ for all del Pezzo surfaces $M$. This yields a positive solution to the Nielsen realization problem for involutions on $M$ and a connection to Bertini's classification of birational involutions of $\mathbb{CP}^2$ (up to conjugation by birational automorphisms of $\mathbb{CP}^2$).
In the second talk of this seminar series, we continue to follow the manuscript of Carl Miller and building up concepts from algebraic topology. In particular, we will introduce chain complexes to define homology groups and provide some of the standard theory for them.
Meeting Link: https://bluejeans.com/426529046/8775
The type VI secretion system (T6SS) is a bacterial subcellular structure that has been likened to a molecular syringe, capable of directly injecting toxins into neighboring cells. Bacteria use T6SS to kill competitor cells, gaining limited space and resources, such as a niche in a host. T6SS has been found in about 25% of Gram negative bacteria, including some human pathogens. Thus, understanding regulation, control, and function of T6SS, as well as the role of T6SS in interbacterial competition, has far-reaching ramifications. However, there are many open questions in this active research area, especially since bacteria have evolved diverse ways in producing and engaging this lethal weapon.
In a multidisciplinary collaboration, we combine experiments and applied mathematics to address a central question about T6SS’s role in interbacterial competition: what is the connection between the subcellular dynamics of T6SS and the competitive strength of the population as a whole? Based on detailed microscopy data, we develop a model on the scale of individual T6SS structures, which is then integrated with an agent-based model (ABM) to enable multi-scale simulations. In this talk, we present the experimental data, the subcellular T6SS model, and findings about T6SS-dependent competitions obtained by simulating the ABM.
Recording link: https://bluejeans.com/s/6fzcqvzTQ5m
Generative networks have made it possible to generate meaningful signals such as images and texts. They were also extended to graphs and applied, for example, to generate molecules. However, the mathematical properties of generative methods are unclear, and training good generative models is difficult. Moreover, some basic and intuitive ideas of generative networks for signals and images do not apply to graphs and we thus focus on this talk on graph generation. An earlier joint work of the speaker generalized Mallat's scattering transform to graphs and later used it as an encoder within an autoencoder for graph generation (while applying a simple Gaussianization procedure to the output of the encoder) . For the graph scattering component, this work proved asymptotic invariance to permutations and stability to graph manipulations. The issue is that the decoder of this graph generation component used two fully connected networks and was not adapted to the graph structure. In fact, many other graph generation methods do not sufficiently utilize the graph structure. In order to address this issue, I will present a new recent joint work that develops a novel and trainable graph unpooling layer for effective graph generation. Given a graph with features, the unpooling layer enlarges this graph and learns its desired new structure and features. Since this unpooling layer is trainable, it can be applied to graph generation either in the decoder of a variational autoencoder or in the generator of a generative adversarial network (GAN). We establish connectivity and expressivity. That is, we prove that the unpooled graph remains connected and any connected graph can be sequentially unpooled from a 3-nodes graph. We apply the unpooling layer within the GAN generator and address the specific task of molecular generation. This is a joint work with Yinglong Guo and Dongmian Zou.
The problem of partial permutation synchronization (PPS) provides a global mathematical formulation for the multiple image matching problem. In this matching problem, one is provided with possibly corrupted matches (i.e., partial permutations) between keypoints in pairs of images and the underlying task is to match keypoints in each image to universal 3D scene points (resulting in other partial permutations). For structure-from-motion (SfM) common datasets, previous PPS algorithms for image matching often become computationally intractable and demand an exceedingly large amount of memory. We address this issue by extending the recent framework of Cycle-Edge Message Passing (CEMP) to the setting of PPS despite the fact that partial permutations do not have a full group structure. We emphasize mathematical difficulties that arise when extending CEMP to PPS and also explain the mathematical guarantees for the performance of the modified CEMP algorithm in the setting of adversarial corruption and sufficiently small noise. This is a joint work with Shaohan Li and Yunpeng Shi.
Restricting to symmetric homogeneous polynomials of degree 2d we compare the cones of nonnegative polynomials with the cone of sums of squares when the number of variables goes to infinity. We consider two natural notions of limit and for each we completely characterize the degrees for which the limit cones are equal. To distinguish these limit cones we tropicalize their duals, which we compute via tropicalizing spectrahedra and tropical convexity. This gives us convex polyhedral cones which we can completely describe and from them obtain explicit examples of nonnegative symmetric polynomials that are not sums of squares (in some cases for any number >=4 of variables).
This is joint work with Grigoriy Blekherman, Sebastian Debus, and Cordian Riener.
Different ways have been introduced to define intermittency in the theory of turbulence, like for example the non-gaussianity, the lack of self-similarity or the deviation of the theory of turbulence by Kolmogorov from 1941.
The usual tool to measure intermittency is the flatness, a measure of the variation of the velocity at small scale, using structure functions in the spatial domain, or high-pass filters in the frequency domain. However, these two approaches give different results in some experiences.
Note the talk will be hosted by Zoom, not Bluejeans any more.
Symmetry is ubiquitous in machine learning and scientific computing. Robust incorporation of symmetry prior into the learning process has shown to achieve significant model improvement for various learning tasks, especially in the small data regime.
The contact invariant, introduced by Kronheimer and Mrowka,
is an element in the monopole Floer homology of a 3-manifold which is
canonically attached to a contact structure. I will describe an
application of monopole Floer homology and the contact invariant to
study the topology of spaces of contact structures and
contactomorphisms on 3-manifolds. The main new tool is a version of
the contact invariant for families of contact structures.
Graphs are central objects of study in Discrete Mathematics. A graph consists of a set of vertices, some of which are connected by edges. Their elementary structure makes graphs widely applicable, but the theoretical understanding of graphs is far from complete. Extremal graph theory aims to find connections between global parameters and substructure. A key topic is how a large average or minimum degree of a graph can force certain subgraphs (where the degree is the number of edges at a vertex). For instance, Erdős and Gallai proved in the 1960's that any graph of average degree at least $k$ contains a path of length $k$. Some of the most intriguing open questions in this area concern trees (connected graphs without cycles) as subgraphs. For instance, can one substitute the path from the previous paragraph with a tree? We will give an overview of open problems and recent results in this area, as well as their possible extensions to hypergraphs.
Hilbert’s 17th problem asked whether every nonnegative polynomial is a sum of squares of rational functions. This problem was solved affirmatively by Artin in the 1920’s, but very little is known about degree bounds (on the degrees of numerators and denominators) in such a representation. Artin’s original proof does not yield any upper bounds, and making such techniques quantitative results in bounds that are likely to be far from optimal, and very far away from currently known lower bounds. Before stating the 17th problem Hilbert was able to prove that any globally nonnegative polynomial in two variables is a sum of squares of rational functions, and the degree bounds in his proof have been best known for that two variable case since 1893. Taking inspiration from Hilbert’s proof we study degree bounds for nonnegative polynomials on surfaces. We are able to improve Hilbert’s bounds and also give degree bounds for some non-rational surfaces. I will present the history of the problem and outline our approach. Joint work with Rainer Sinn, Greg Smith and Mauricio Velasco.
In the third talk of this seminar series, we continue to follow the manuscript of Carl Miller. We will begin with a quick review of chain complexes and simplicial isomorphisms and then we will detour to discuss the geometric interpretation of homology groups in lower dimensions. This work can help us understand the structure of simplicial complexes with boundary maps and their homology groups. Then we go back to abstract homological algebra which is the study of homology groups without reference to simplicial complexes. We will introduce the Snake Lemma without proof. Finally, we will apply this lemma to prove the goal of this chapter: collapsibility for a simplicial complex implies its homology groups are trivial which is called acyclicity.
Meeting Link: https://bluejeans.com/426529046/8775
Prion proteins are responsible for a variety of fatal neurodegenerative diseases in mammals but are harmless to Baker's yeast (S. cerevisiae)- making it an ideal system for investigating the protein dynamics associated with prion diseases. Most mathematical frameworks for modeling prion aggregate dynamics either focus on protein dynamics in isolation, absent from a changing cellular environment, or modeling prion aggregate dynamics in a population of cells by considering the "average" behavior. However, such models are unable to reproduce in vivo properties of different yeast prion strains.
In this talk, I will show some results from recent individual-based simulations where we study how the organization of a yeast population depends on the division and growth properties of the colonies. Each individual cell has their own configuration of prion aggregates, and we study how the population level phenotypes are a natural consequence of the interplay between the cell cycle, budding cell division and aggregate dynamics. We quantify how common experimentally observed outcomes depend on population heterogeneity.
Recording link: https://bluejeans.com/s/lbpACr_YZ0N
We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix $A$ with real, bounded, and possibly non-symmetric coefficients. If a proper {$L^1$-mean oscillation} of the coefficients of $A$ satisfies suitable Dini-type assumptions, we prove the following: if $\mu$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$, and
$$T_\mu f(x)=\int \nabla_x\Gamma_A (x,y)f(y)\, d\mu(y)$$
denotes the gradient of the single layer potential associated with $L_A$, then
$$1+ \|T_\mu\|_{L^2(\mu)\to L^2(\mu)}\approx 1+ \|\mathcal R_\mu\|_{L^2(\mu)\to L^2(\mu)},$$
where $\mathcal R_\mu$ indicates the $n$-dimensional Riesz transform. This makes possible to obtain direct generalization of some deep geometric results, initially obtained for $\mathcal R_\mu$, which were recently extended to $T_\mu$ under a H\"older continuity assumption on the coefficients of the matrix $A$.
This is a joint work with Alejandro Molero, Mihalis Mourgoglou, and Xavier Tolsa.
Classical matrix concentration inequalities are sharp up to a logarithmic factor. This logarithmic factor is necessary in the commutative case but unnecessary in many classical noncommutative cases. We will present some matrix concentration results that are sharp in many cases, where we overcome this logarithmic factor by using an easily computable quantity that captures noncommutativity. Joint work with Afonso Bandeira and Ramon van Handel.
Due to privacy, access to real data is often restricted. Data that are not completely real but resemble certain properties of real data become natural substitutes. Data of this type are called synthetic data. I will talk about the extent to which synthetic data may resemble real data under privacy and computational complexity restrictions. Joint work with Thomas Strohmer and Roman Vershynin.
The link to the online talk: https://bluejeans.com/405947238/3475
We will motivate this talk by exhibiting recent progress on (either general or symmetric anisotropic) bootstrap percolation models in $d$-dimensions. Then, we will discuss our intention to start a deeper study of non-symmetric models for $d\ge 3$. It looks like some proportion of them could be related to first passage percolation models (in lower dimensions).
This talk will be online at https://bluejeans.com/216376580/6460
There are many interesting classes of polynomials in real algebraic geometry that are of modern interest. A polynomial is nonnegative if it only takes nonnegative values on R^n. A univariate polynomial is real-rooted if all of its complex roots are real, and a hyperbolic polynomial is a multivariate generalization of a real-rooted polynomial. We will discuss connections between these two classes of polynomials. In particular, we will discuss recent ideas of Saunderson giving new ways of proving that a polynomial is nonnegative beyond showing that it is sum-of-squares.
Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1646885419648?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d
Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Consider any area-preserving map of R2 which has an elliptic periodic orbit. We show that arbitrarily close to this map (in the C-infinity topology) there exists an area-preserving map which has a "chaotic island" - an open set where every point has positive maximal Lyapunov exponent. The result implies that the naturally sound conjectures that relate the observed chaotic behavior in non-hyperbolic conservative systems with the positivity of the metric entropy need a rethinking.
Given an n-by-d matrix A and a vector of size n, the p-norm problem asks for a vector x that minimizes the following
\sum_{i=1}^n (a_i^T x - b_i)^p,
where a_i is the i’th row of A. The study of p=2 and p=1 cases dates back to Legendre, Gauss, and Laplace. Other cases of p have also been used in statistics and machine learning as robust estimators for decades. In this talk, we present the following improvements in the running time of solving p-norm regression problems.
For the case of 1
For 1
The talk is based on a joint work with Richard Peng and Santosh Vempala.
The math-themed show at the Atlanta Science Festival will be less elaborate than in the last few years, but we are back to apearing live on stage! We are also hoping to arrange for live-streaming. Mathematics in Motion will use dance and circus arts to engage the public. (Dan and Evans and several GT students are involved, but don't worry, mathematicians won't be doing the dancing!)
There will be two shows on Sunday the 13th, begininng at 2:00 and 5:00 pm.
In this talk, I will be defining the grand arc graph for infinite-type surfaces. This simplicial graph is motivated by the works of Fanoni-Ghaswala-McLeay, Bavard, and Bavard-Walker to define an infinite-type analogue of the curve graph. As in these earlier works, the grand arc graph is connected, (oftentimes) infinite-diameter, and (sometimes) delta hyperbolic. Moreover, the mapping class group acts on it by isometries, and the action is continuous on the visible boundary. If there's time, this talk will degenerate into open speculation about what the boundary looks like and what we can do with it.
In the past decade, the revival of deep neural networks has led to dramatic success in numerous applications ranging from computer vision to natural language processing to scientific discovery and beyond. Nevertheless, the practice of deep networks has been shrouded with mystery as our theoretical understanding of the success of deep learning remains elusive.
In this talk, we will exploit low-dimensional modeling to help understand and improve deep learning performance. We will first provide a geometric analysis for understanding neural collapse, an intriguing empirical phenomenon that persists across different neural network architectures and a variety of standard datasets. We will utilize our understanding of neural collapse to improve training efficiency. We will then exploit principled methods for dealing with sparsity and sparse corruptions to address the challenges of overfitting for modern deep networks in the presence of training data corruptions. We will introduce a principled approach for robustly training deep networks with noisy labels and robustly recovering natural images by deep image prior.
Given an affine space of matrices L and a matrix Θ ∈ L, consider the problem of computing the closest rank deficient matrix to Θ on L with respect to the Frobenius norm. This is a nonconvex problem with several applications in control theory, computer algebra, and computer vision. We introduce a novel semidefinite programming (SDP) relaxation, and prove that it always gives the global minimizer of the nonconvex problem in the low noise regime, i.e., when Θ is close to be rank deficient. Our SDP is the first convex relaxation for this problem with provable guarantees. We evaluate the performance of our SDP relaxation in examples from system identification, approximate GCD, triangulation, and camera resectioning. Our relaxation reliably obtains the global minimizer under non-adversarial noise, and its noise tolerance is significantly better than state of the art methods.
Suppose you have a subset $S$ of the vertices of a polytope which contains at least one vertex from every face. How large must $S$ be? We believe, in the worst case, about half of the number of vertices of the polytope. But we don’t really know why. We have found some situational evidence, but also some situational counter-evidence. This is based on joint work with Michael Dobbins and Seunghun Lee.
Meeting Link: https://bluejeans.com/426529046/8775
Wild-type zebrafish are named for their dark and light stripes, but mutant zebrafish feature variable skin patterns, including spots and labyrinth curves. All of these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells in the skin. This leads to the question: how do cell interactions change to create mutant patterns? The longterm biological motivation for my work is to shed light on this question — I strive to help link genes, cell behavior, and visible animal characteristics. Toward this goal, I build agent-based models to describe cell behavior in growing fish body and fin-shaped domains. However, my models are stochastic and have many parameters, and comparing simulated patterns, alternative models, and fish images is often a qualitative process. This, in turn, drives my mathematical goal: I am interested in developing methods for quantifying variable cell-based patterns and linking computational and analytically tractable models. In this talk, I will overview our agent-based models for body and fin pattern formation, share how topological data analysis can be used to quantify cell-based patterns and models, and discuss ongoing work on relating agent-based and continuum models for zebrafish patterns.
From networks to genomics, large amounts of data are abundant and play critical roles in helping us understand complex systems. In many such settings, these data take the form of large discrete structures with important combinatorial properties. The interplay between structure and randomness in these systems presents unique mathematical and statistical challenges. In this talk I will highlight these through two vignettes: (1) inference problems on networks, and (2) DNA data storage.
First, I will discuss statistical inference problems on edge-correlated stochastic block models. We determine the information-theoretic threshold for exact recovery of the latent vertex correspondence between two correlated block models, a task known as graph matching. As an application, we show how one can exactly recover the latent communities using multiple correlated graphs in parameter regimes where it is information-theoretically impossible to do so using just a single graph. Furthermore, we obtain the precise threshold for exact community recovery using multiple correlated graphs, which captures the interplay between the community recovery and graph matching tasks.
Next, I will give an overview of DNA data storage. Storing data in synthetic DNA is an exciting emerging technology which has the potential to revolutionize data storage. Realizing this goal requires innovation across a multidisciplinary pipeline. I will explain this pipeline, focusing on our work on statistical error correction algorithms and optimizing DNA synthesis, highlighting the intimate interplay between statistical foundations and practice.
We will prove lower bounds for energy functionals of mappings of the real, complex and quaternionic projective spaces with their canonical Riemannian metrics. For real and complex projective spaces, these results are sharp, and we will characterize the family of energy-minimizing mappings which occur in these results. For complex projective spaces, these results extend to all Kahler metrics. We will discuss the connections between these results and several theorems and questions in systolic geometry.
For each element $z$ of the symmetric group algebra we define a symmetric generating function
$Y(z) = \sum_\lambda \epsilon^\lambda(z) m_\lambda$, where $\epsilon^\lambda$ is the induced sign
character indexed by $\lambda$. Expanding $Y(z)$ in other symmetric function bases, we obtain
other trace evaluations as coefficients. We show that we show that all symmetric functions in
$\span_Z \{m_\lambda \}$ are $Y(z)$ for some $z$ in $Q[S_n]$. Using this fact and chromatic symmetric functions, we give new interpretations of permanents of totally nonnegative matrices.
For the full paper, see https://arxiv.org/abs/2010.00458v2.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Two quantitative notions of mixing are the decay of correlations and the decay of a mix-norm --- a negative Sobolev norm --- and the intensity of mixing can be measured by the rates of decay of these quantities. From duality, correlations are uniformly dominated by a mix-norm; but can they decay asymptotically faster than the mix-norm? We answer this question by constructing an observable with correlation that comes arbitrarily close to achieving the decay rate of the mix-norm. Therefore the mix-norm is the sharpest rate of decay of correlations in both the uniform sense and the asymptotic sense. Moreover, there exists an observable with correlation that decays at the same rate as the mix-norm if and only if the rate of decay of the mix-norm is achieved by its projection onto low-frequency Fourier modes. In this case, the function being mixed is called q-recurrent; otherwise it is q-transient. We use this classification to study several examples and raise questions for future investigations.
We discuss a few new results concerning the descriptive combinatorics of bounded degree hyperfinite Borel graphs. In particular, we show that the Baire measurable edge chromatic number of $G$ is at most $\lceil\frac{3}{2}\Delta(G)\rceil+6$ when G is a multigraph, and for bipartite graphs we improve this bound to $\Delta(G)+1$ and show that degree regular one-ended bipartite graphs have Borel perfect matchings generically. Similar results hold in the measure setting assuming some hyperfiniteness conditions. This talk is based on joint work with Kun and Sabok, Weilacher, and upcoming work with Poulin and Zomback.
Abstract:
In this talk, we mainly focus on the applications of optimal transport theory from the following two aspects:
(1)Based on the theory of Wasserstein gradient flows, we develop and analyze a numerical method proposed for solving high-dimensional Fokker-Planck equations (FPE). The gradient flow structure of FPE allows us to derive a finite-dimensional ODE by projecting the dynamics of FPE onto a finite-dimensional parameter space whose parameters are inherited from certain generative model such as normalizing flow. We design a bi-level minimization scheme for time discretization of the proposed ODE. Such algorithm is sampling-based, which can readily handle computations in high-dimensional space. Moreover, we establish theoretical bounds for the asymptotic convergence analysis as well as the error analysis for our proposed method.
(2)Inspired by the theory of Wasserstein Hamiltonian flow, we present a novel definition of stochastic Hamiltonian process on graphs as certain kinds of inhomogeneous Markov process. Such definition is motivated by lifting to the probability space of the graph and considering the Hamiltonian dynamics on this probability space. We demonstrate some examples of the stochastic Hamiltonian process in classical discrete problems, such as the optimal transport problems and Schrödinger bridge problems (SBP).
The Bluejeans link: https://bluejeans.com/982835213/2740
We discuss the problem of constructing quasi-morphisms on the group of diffeomorphisms of a surface that are isotopic to the identity, thereby resolving a problem of Burago-Ivanov-Polterovich from the mid 2000’s. This is achieved by considering a new kind of curve graph, in analogy to the classical curve graph first studied by Harvey in the 70’s, on which the full diffeomorphism group acts isometrically. Joint work with S. Hensel and R. Webb.
We will present a new class of continuous-depth deep neural networks that were motivated by the ODE limit of the classical momentum method, named heavy-ball neural ODEs (HBNODEs). HBNODEs enjoy two properties that imply practical advantages over NODEs: (i) The adjoint state of an HBNODE also satisfies an HBNODE, accelerating both forward and backward ODE solvers, thus significantly accelerate learning and improve the utility of the trained models. (ii) The spectrum of HBNODEs is well structured, enabling effective learning of long-term dependencies from complex sequential data.
Second, we will extend HBNODE to graph learning leveraging diffusion on graphs, resulting in new algorithms for deep graph learning. The new algorithms are more accurate than existing deep graph learning algorithms and more scalable to deep architectures, and also suitable for learning at low labeling rate regimes. Moreover, we will present a fast multipole method-based efficient attention mechanism for modeling graph nodes interactions.
Third, if time permits, we will discuss proximal algorithms for accelerating learning continuous-depth neural networks.
Projective space, rational maps, and other notions from algebraic geometry appear naturally in the study of image formation and various camera models in computer vision. Considerable attention has been paid to multiview ideals, which collect all polynomial constraints on images that must be satisfied by a given camera arrangement. We extend past work on multiview ideals to settings where the camera arrangement is unknown. We characterize various "image formation ideals", which are interesting objects in their own right. Some nice previous results about multiview ideals also fall out from our framework. We give a new proof of a result by Aholt, Sturmfels, and Thomas that the multiview ideal has a universal Groebner basis consisting of k-focals (also known as k-linearities in the vision literature) for k in {2,3,4}. (Preliminary report based on ongoing joint projects with Sameer Agarwal, Max Lieblich, Jessie Loucks Tavitas, and Rekha Thomas.)
Note this talk is at a different time and day
We first construct a complex surface with positive signature, which is a ball quotient. We obtain it as an abelian Galois cover of CP^2 branched over the Hesse arrangement. Then we analyze its fibration structure, and by using it we build new symplectic and also non-symplectic exotic 4-manifolds with positive signatures.
In the second part of the talk, we discuss Cartwright-Steger surfaces, which are also ball quotients. Next, we present our constructions of new symplectic and non-symplectic exotic 4-manifolds with non-negative signatures that have the smallest Euler characteristics in the so-called ‘arctic region’ on the geography chart.
More precisely, we prove that there exist infinite families of irreducible symplectic and infinite families of irreducible non-symplectic, exotic 4-manifolds that have the smallest Euler characteristics among the all known simply connected 4-manifolds with nonnegative signatures and with more than one smooth structures. This is a joint work with A. Akhmedov and S.-K. Yeung.
Ramsey theory studies the paradigm that every sufficiently large system contains a well-structured subsystem. Within graph theory, this translates to the following statement: for every positive integer $s$, there exists a positive integer $n$ such that for every partition of the edges of the complete graph on $n$ vertices into two classes, one of the classes must contain a complete subgraph on $s$ vertices. Beginning with the foundational work of Ramsey in 1928, the main question in the area is to determine the smallest $n$ that satisfies this property.
For many decades, randomness has proved to be the central idea used to address this question. Very recently, we proved a theorem which suggests that "pseudo-randomness" and not complete randomness may in fact be a more important concept in this area. This new connection opens the possibility to use tools from algebra, geometry, and number theory to address the most fundamental questions in Ramsey theory. This is joint work with Jacques Verstraete.
Please Note: Meeting Link: https://bluejeans.com/426529046/8775
Why is a species’ geographic range where it is? Immediate thoughts such as penguins cannot climb steep cliffs or colonize deserts are often not the answer. In fact, identifying causes of species’ range limits is a fundamental problem in evolutionary ecology that has crucial implications in conservation biology and understanding mechanisms of speciation.
In this talk, I will briefly introduce some of the biotic, genetic, and environmental processes that can determine a species’ range. I will then focus on two of such processes, competition and (mal)adaptation to heterogeneous environments, that are commonly thought to halt species’ range expansion and stabilize their range boundary. I will present a model of species range dynamics that incorporates these eco-evolutionary processes in a community of biologically related species. I will discuss biologically plausible ranges of values for the parameters of this model, and will demonstrate its dynamic behavior in a number of different evolutionary regimes.
Meeting also available online: https://gatech.zoom.us/j/92742811112
Wave turbulence is the theory of nonequilibrium statistical mechanics for wave systems. Initially formulated in pioneering works of Peierls, Hasselman, and Zakharov early in the past century, wave turbulence is widely used across several areas of physics to describe the statistical behavior of various interacting wave systems. We shall be interested in the mathematical foundation of this theory, which for the longest time had not been established.
The central objects in this theory are: the "wave kinetic equation" (WKE), which stands as the wave analog of Boltzmann’s kinetic equation describing interacting particle systems, and the "propagation of chaos” hypothesis, which is a fundamental postulate in the field that lacks mathematical justification. Mathematically, the aim is to provide a rigorous justification and derivation of those two central objects; This is Hilbert’s Sixth Problem for waves. The problem attracted considerable interest in the mathematical community over the past decade or so. This culminated in recent joint works with Yu Deng (University of Southern California), which provided the first rigorous derivation of the wave kinetic equation, and justified the propagation of chaos hypothesis in the same setting.
Meeting also available online: https://gatech.zoom.us/j/92742811112
In first-passage percolation (FPP), we let \tau_v be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If F is the distribution function of \tau_v, there are different regimes: if F(0) is small, this weight typically grows like a linear function of the distance, and when F(0) is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge, but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results which show that if sum of F^{-1}(1/2+1/2^k) diverges, then a.s. there are exceptional times at which the weight grows atypically, but if sum of k^{7/8} F^{-1}(1/2+1/2^k) converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. These results show a wider range of dynamical behavior than one sees in subcritical (usual) FPP. This is a joint work with M. Damron, J. Hanson, W.-K. Lam.
This talk will be given on Bluejeans at the link https://bluejeans.com/547955982/2367
Baker and Lorscheid have developed a theory of foundation that characterize the representability of matroids. Justin Chen and I are developing a computer program that computes representations of matroids based on the theory of foundation. In this talk, I will introduce backgrounds on matroids and the foundation, then I will talk about the key algorithms in computing the morphisms of pastures. If possible, I will also show some examples of the program.
Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1648750292956?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d
Many real life processes that we would like to model have a self-exciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time. Examples of processes that have this property are earthquakes, crime in a neighborhood, or emails within a company. In 1971, Alan Hawkes first used what is now known as the Hawkes process to model such processes. Since then much work has been done on estimating the parameters of a Hawkes process given a data set and creating variants of the process for different applications.
In this talk, we propose a new variant of a Hawkes process, called a self-limiting Hawkes process, that takes into account the effect of police activity on the underlying crime rate and an algorithm for estimating its parameters given a crime data set. We show that the self-limiting Hawkes process fits real crime data just as well, if not better, than the standard Hawkes model. We also show that the self-limiting Hawkes process fits real financial data at least as well as the standard Hawkes model.
We present in this talk some results concerning the metric measure spaces with lower Ricci curvature bounds.
Firstly, we extend the technique of smoothing Riemannian metric by heat kernel pull back metrics to non-compact setting, and use it to solve a conjecture of De Philippis-Gigli. This is joint work with Brena-Gigli-Honda. Secondly, we study the second term in the short time expansion of the heat kernel pull back metrics and the connection with non-collapsed spaces. This is joint work with Honda. Finally, we use the 1D localization technique to extend some convexity results on the regular set and in the interior of such metric measure spaces.
Link: https://gatech.zoom.us/j/5491403383?pwd=Um1NM05MeWJMRnNuVHViQ1NWdHFaZz09
We discuss the relationship between the Borel/Baire measurable/measurable combinatorics of the action of a finitely generated group on its Bernoulli shift and the discrete combinatorics of the multiplication action of that group on itself. Our focus is on various chromatic numbers of graphs generated by these actions. We show that marked groups with isomorphic Cayley graphs can have Borel/Baire measurable/measurable chromatic numbers which differ by arbitrarily much. In the Borel two-ended, Baire measurable, and measurable hyperfinite settings, we show our constructions are nearly best possible (up to only a single additional color). Along the way, we get tightness of some bounds of Conley and Miller on Baire measurable and measurable chromatic numbers of locally finite Borel graphs.
Given a collection of functions in a Banach space, typically called a dictionary in machine learning, we study the approximation properties of its convex hull. Specifically, we develop techniques for bounding the metric entropy and n-widths, which are fundamental quantities in approximation theory that control the limits of linear and non-linear approximation. Our results generalize existing methods by taking the smoothness of the dictionary into account, and in particular give sharp estimates for shallow neural networks. Consequences of these results include: the optimal approximation rates which can be attained for shallow neural networks, that shallow neural networks dramatically outperform linear methods of approximation, and indeed that shallow neural networks outperform all continuous methods of approximation on the associated convex hull. Next, we discuss greedy algorithms for constructing approximations by non-linear dictionary expansions. Specifically, we give sharp rates for the orthogonal greedy algorithm for dictionaries with small metric entropy, and for the pure greedy algorithm. Finally, we give numerical examples showing that greedy algorithms can be used to solve PDEs with shallow neural networks.
The contact connected sum is a well-understood operation for contact manifolds. I will discuss work in progress on how pseudo-holomorphic curves behave in the symplectization of the 3-dimensional contact connected sum, and as a result the connected sum formula of embedded contact homology.
Numerical algebraic geometry revolves around the study of solutions to polynomial systems via numerical method. Two of the fundamental tools in this field are the polyhedral homotopy of Huber and Sturmfels for computing isolated solutions and the concept of witness sets put forth by Sommese and Wampler as numerical representations for non-isolated solution components. In this talk, we will describe a stratified polyhedral homotopy method that will bridge the gap between these two largely independent area. Such a homotopy method will discover numerical representations of non-isolated solution components as by-products from the process of computing isolated solutions. We will also outline the pipeline of numerical algorithms necessary to implement this homotopy method on modern massively parallel computing architecture.
https://bluejeans.com/421317143/2787<br />
We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth x2 \in (-h,0) linearized at a uniformly monotonic shear flow U(x2). Our main results consist of two aspects, eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity wave, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes, we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain the stronger decay for the remainders.
The rank of a point $p$ with respect to a non-degenerate variety is the smallest number of the points in the variety that spans the point $p$. Studies about the ranks of points are interesting and important in various areas of applied mathematics and engineering in the sense that they are the shortest sizes of the decompositions of vectors into combinations of simple vectors.
In this talk, we focus on the ranks of points with respect to the rational normal curves, i.e. Waring ranks of binary forms. We introduce an algorithm that produces random points of given rank r. (Note that if we choose points randomly, we expect the rank of the points is just the generic rank.) Moreover, we check some known facts by Macaulay 2 computations. Lastly, we discuss the maximal and minimal rank of points in linear spaces.
Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1649360107625?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d
Link: https://bluejeans.com/520769740/3630
Parallel server systems have received a lot of attention since their introduction about 20 years ago. They are commonly used to model a situation where different type of jobs can be treated by servers with different specialties like data and call centers. Exact optimal policies are often not tractable for those systems. Instead, part of the literature was focused on finding policies that are asymptotically optimal as the load of the network approaches a value critical for stability (heavy traffic approximations). This is done by obtaining a weak convergence to a brownian control problem that is linked to a non-linear differential equation (Hamilton-Jacobi-Bellman). Asymptotically optimal policies have been analyzed for a long time under a restrictive assumption that is not natural for practical applications. This talk will present recent developments that allow for a more general asymptotic optimality result by focusing on the simplest non-trivial example.
In this talk, we focus on designing computational methods supported by theoretical properties for optimal motion planning and optimal transport (OT).
Over the past decades, motion planning has attracted large amount of attention in robotics applications. Given certain
configurations in the environment, the objective is to find trajectories which move the robot from one position to the other while satisfying given constraints. We introduce a new method to produce smooth and collision-free trajectories for motion planning task. The proposed model leads to short and smooth trajectories with advantages in numerical computation. We design an efficient algorithm which can be generalized to robotics applications with multiple robots.
The idea of optimal transport naturally arises from many application scenarios and provides powerful tools for comparing probability measures in various types. However, obtaining the optimal plan is generally a computationally-expensive task, sometimes even intractable. We start with the entropy transport problem as a relaxed version of original optimal transport problem with soft marginals, and propose an efficient algorithm to obtain the sample approximation for the optimal plan. We also study an inverse problem of OT and present the computational methods for learning the cost function from the given optimal transport plan.
Link: https://us06web.zoom.us/j/2782194473?pwd=L1Nnc0c1SXFFYkZqSkVGUGpEd2E4dz09
Consider two volume-preserving, smooth diffeomorphisms f and g of a compact manifold M. Define the random walk on M by selecting either f or g (i.i.d.) at each iterate. A number of questions arise in this setting:
Conjecturally, for a generic pair of f and g we should be able to answer the above. I will describe one sufficient criteria on f and g underwhich we can give some partial answers to the above questions. Such a criteria is expected to be generic amoung pairs of (volume-preserving) diffeomorphisms and should be able to be verified in a number of naturally occurring geometric settings where the above questions are not fully answered.
We define a graph process based on a discrete branching process with deletions and mergers, which is inspired by the 4-cycle structure of the hypercube $\mathcal{Q}_d$ for large $d$. We prove survival and extinction under certain conditions on $p$ and $q$ that heuristically match the known expansions of the critical probabilities for bond percolation on the hypercube. Joint work with Laura Eslava and Sarah Penington. Based on https://arxiv.org/abs/2104.04407.
Supervised operator learning is an emerging machine learning paradigm with applications to modeling the evolution of spatio-temporal dynamical systems and approximating general black-box relationships between functional data. We propose a novel operator learning method, LOCA (Learning Operators with Coupled Attention), motivated from the recent success of the attention mechanism. In our architecture, the input functions are mapped to a finite set of features which are then averaged with attention weights that depend on the output query locations. By coupling these attention weights together with an integral transform, LOCA is able to explicitly learn correlations in the target output functions, enabling us to approximate nonlinear operators even when the number of output function measurementsin the training set is very small. Our formulation is accompanied by rigorous approximation theoretic guarantees on the universal expressiveness of the proposed model. Empirically, we evaluate the performance of LOCA on several operator learning scenarios involving systems governed by ordinary and partial differential equations, as well as a black-box climate prediction problem. Through these scenarios we demonstrate state of the art accuracy, robustness with respect to noisy input data, and a consistently small spread of errors over testing data sets, even for out-of-distribution prediction tasks.
Upsilon is an invariant of knots defined using knot Floer homology by Ozsváth, Szabó and Stipsicz. In this talk, we discuss a generalization of their invariant for embedded graphs in rational homology spheres satisfying specific properties. Our construction will use a generalization of Heegaard Floer homology for “generalized tangles” called tangle Floer homology. As a result, we get a family of homomorphisms from the homology cobordism group of homology cylinders (over a surface of genus 0), which is an enlargement of the mapping class group defined by Graoufaldis and Levine.
For polynomials in 1 variable, Matt Baker and Oliver Lorschied were able to connect results about roots of polynomials over valued
fields (Newton polygons) and over real fields (Descartes's rule) by looking at factorization of polynomials over the tropical and signed
hyperfields respectively. In this talk, I will describe some ongoing work with Andreas Gross about extending these ideas to two or more
variables. Our main tool is the use of resultants to transform questions about 0-dimensional systems of equations to factoring a single
homogeneous polynomial.
Vizing's Theorem states that simple graphs can be edge-colored using $\Delta+1$ colors. The problem of developing efficient $(\Delta+1)$-edge-coloring algorithms has been a major challenge. The algorithms involve iteratively finding small subgraphs $H$ such that one can extend a partial coloring by modifying the colors of the edges in $H$. In a recent paper, Bernshteyn showed one can find $H$ such that $e(H) = \mathrm{poly}(\Delta)(\log n)^2$. With this result, he defines a $(\Delta+1)$-edge-coloring algorithm which runs in $\mathrm{poly}(\Delta, \log n)$ rounds. We improve on this by showing we can find $H$ such that $e(H) = \mathrm{poly}(\Delta)\log n$. As a result, we define a distributed algorithm that improves on Bernshteyn's by a factor of $\mathrm{poly}(\log n)$. We further apply the idea to define a randomized sequential algorithm which computes a proper $(\Delta+1)$-edge-coloring in $\mathrm{poly}(\Delta)n$ time. Under the assumption that $\Delta$ is a constant, the previous best bound is $O(n\log n)$ due to Sinnamon.
Meeting Link: https://gatech.zoom.us/j/94882290086 (Meeting ID: 948 8229 0086, Passcode: 264830)
Abstract: R-loops are three-stranded structures formed by a DNA:RNA hybrid and a single strand of DNA, often appearing during transcription. Although R-loops can threaten genome integrity, recent studies have shown that they also play regulatory roles in physiological processes. However, little is known about their structure and formation. In this talk, we introduce a model for R-loops based on formal grammars, that are systems to generate words widely applied in molecular biology. In this framework, R-loops are described as strings of symbols representing the braiding of the strands in the structure, where each symbol corresponds to a different state of the braided structure. We discuss approaches to develop a stochastic grammar for R-loop prediction using experimental data, as well as refinements of the model by incorporating the effect of DNA topology on R-loop formation.
Note the unusual time!
When do two balls in a metric space have small intersection? We give some natural conditions to guarantee an exponential decay on the volume of such intersections. Our proof is conceptually simple, making use of concentration of measure on a "slice." We will discuss a couple of applications of this volume estimate in coding theory. This is joint work with Jaehoon Kim and Tuan Tran.
In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a one-to-one correspondence between primary ideals whose variety is a rational point, and finite dimensional vector spaces of differential operators with constant coefficients. The question for general ideals was left open. Significant progress was made in the 1960's by analysts, culminating in a deep result known as the Ehrenpreis-Palamodov fundamental principle, connecting polynomial ideals and modules to solution sets of linear, homogeneous partial differential equations with constant coefficients.
This talk aims to survey classical results, and provide new constructions, applications, and insights, merging concepts from analysis and nonlinear algebra. We offer a new formulation generalizing Gröbner's duality for arbitrary polynomial ideals and modules and connect it to the analysis of PDEs. This framework is amenable to the development of symbolic and numerical algorithms. We also study some applications of algebraic methods in problems from analysis.
Link: https://gatech.zoom.us/j/95997197594?pwd=RDN2T01oR2JlaEcyQXJCN1c4dnZaUT09
Link: https://gatech.zoom.us/j/91232113113?pwd=MDhteEdtcENuME9kdXJmcUY0eWlSUT09
In this talk, we will look into the two most widely studied settings of the stochastic multi-armed bandit problems - regret minimization and pure exploration. The algorithm is presented with a finite set of unknown distributions from which it can generate samples. In the regret-minimization setting, its aim is to sample sequentially so as to maximize the total average reward accumulated. In the pure exploration setting, we are interested in algorithms that identify the arm with the maximum mean in a small number of samples on an average while keeping the probability of false selection to at most a pre-specified and small value. Both of these problems are well studied in literature and tight lower bounds and optimal algorithms exist when the arm distributions are known to belong to simple classes of distributions such as single-parameter exponential family, distributions that have bounded support, etc. However, in practice, the distributions may not satisfy these assumptions and may even be heavy-tailed. In this talk, we will look at techniques and algorithms for optimally solving these two problems with minimal assumptions on the arm distributions. These ideas can be extended to a more general objective of identifying the distribution with the minimum linear combination of risk and reward, which captures the risk-reward trade-off that is popular in many practical settings, including in finance.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
In which cases and ways can one perturb the action on the torus of a commuting pair of $SL(n, \mathbb Z)$ matrices?
Two famous manifestations of local rigidity in this context are: 1) KAM-rigidity of simultaneously Diophantine torus translations (Moser) and 2) smooth rigidity of hyperbolic or partially hyperbolic higher rank actions (Damjanovic and Katok). To complete the study of local rigidity of affine $\mathbb Z^k$ actions on the torus one needs to address the case of actions with parabolic generators. In this talk, I will review the two different mechanisms behind the rigidity phenomena in 1) and 2) above, and show how blending them with parabolic cohomological stability and polynomial growth allows to address the rigidity problem in the parabolic case.
This is joint work with Danijela Damjanovic and Maria Saprykina.
The untwisting number of a knot K is the minimum number of null-homologous full twists required to unknot K. The surgery description number of K can be defined similarly, allowing for multiple full twists in a single twisting region. We can find no examples of knots in the literature where these two invariants are not equal. In this talk, I will provide the first known example where untwisting number and surgery description number are not equal and discuss challenges to distinguishing these invariants in general. This will involve an exploration of the existing obstructions (often Heegaard-Floer theoretic) as well as the algebraic versions of these invariants. In addition, we show the surprising result that the untwisting number of a knot is at most three times its surgery description number. This work is joint with Kenan Ince, Seungwon Kim, Benjamin Ruppik, and Hannah Turner.
MCMC and variational inference are two competing paradigms for the problem of sampling from a given probability distribution. In this talk, I'll show how they can work together to give the first polynomial-time sampling algorithm for approximately low-rank Ising models. Sampling was previously known when all eigenvalues of the interaction matrix fit in an interval of length 1; however, a single outlier can cause Glauber dynamics to mix torpidly. Our result covers the case when all but O(1) eigenvalues lie in an interval of length 1. To deal with positive eigenvalues, we use a temperature-based heuristic for MCMC called simulated tempering, while to deal with negative eigenvalues, we define a nonconvex variational problem over Ising models, solved using SGD. Our result has applications to sampling Hopfield networks with a fixed number of patterns, Bayesian clustering models with low-dimensional contexts, and antiferromagnetic/ferromagnetic Ising model on expander graphs.
Tensegrities are mechanical structures that include cable-like elements that are strong and lightweight relative to rigid rods yet support only extensile stress. From suspension bridges to the musculoskeletal system to individual biological cells, humanity makes excellent use of tensegrities, yet the sharply nonlinear response of cables presents serious challenges to analytical theory. Here we consider large tensegrity structures with randomly placed cables (and struts) overlaid on a regular rigid backbone whose corresponding system of inequalities is reduced via analytic theory to an exact graph theory. We identify a novel coordination number that controls two rigidity percolation transitions: one in which global interactions between cables first support external loads and one in which the structure becomes fully rigid. We show that even the addition of a few cables strongly modifies conventional rigidity percolation, both by modifying the sharpness of the transition and by introducing avalanche effects in which a single constraint can eliminate multiple floppy modes.
Also ONLINE: https://gatech.zoom.us/j/99313032175
This talk focuses on Eulerian graphs whose arcs are directed and labelled in a group. Each circuit yields a word over the group, and we say that a circuit is non-zero if this word does not evaluate to 0. We give a precise min-max theorem for the following problem. Given a vertex $v$, what is the maximum number of non-zero circuits in a circuit decomposition where each circuit begins and ends at $v$? This is joint work with Jim Geelen and Paul Wollan. Our main motivation is a surprising connection with vertex-minors which is due to Bouchet and Kotzig.
Meeting Link: https://gatech.zoom.us/j/94882290086 (Meeting ID: 948 8229 0086, Passcode: 264830)
Modeling is essential in the design of genetic circuits with desired properties. I will review several examples where mathematical models have been central to the development and understanding of the dynamic of synthetic organisms. I will start with a discussion of synthetic bacterial consortia that exhibit emergent oscillatory behavior - when co-cultured, the interaction between two bacterial strains results in population-level transcriptional oscillations. The spatio-temporal dynamics of such consortia, including synchrony between distant parts of the population, depend sensitively on the architecture of the underlying genetic circuits. I will then describe how oscillations, and other spatiotemporal patterns can arise in consortia of cells that individually exhibit bistable dynamics. I will show how simplified mathematical models can help us understand how order emerges in these system, how robust oscillations and other patterns can arise, and how they are maintained.
Online link: https://gatech.zoom.us/j/93504092832?pwd=V29FVVFlcEtwNWhkTnUyMnFqbVYyUT09
Many proposed interplanetary space missions, including Europa Lander and Dragonfly, involve trajectory design in environments where multiple large bodies exert gravitational influence on the spacecraft, such as the Jovian and Saturnian systems as well as cislu- nar space. In these contexts, an analysis based on the mathematical theory of dynamical systems provides both better insight as well as new tools to use for the mission design compared to classic two-body Keplerian methods. Indeed, a rich variety of dynamical phenomena manifest themselves in such systems, including libration point dynamics, stable and unstable mean-motion resonances, and chaos. To understand the previously mentioned dynamical behaviors, invariant manifolds such as periodic orbits, quasi-periodic invariant tori, and stable/unstable manifolds are the major objects whose interactions govern the local and global dynamics of relevant celestial systems.
This work is focused on the development of numerical methodologies for computing such invariant manifolds and investigating their interactions. After a study of persistence of mean-motion resonances in the planar circular restricted 3-body problem (PCRTBP), techniques for computing the stable/unstable manifolds attached to resonant periodic orbits and heteroclinics corresponding to resonance transitions are presented. Next, I will focus on the development of accurate and efficient parameterization methods for numerical calculation of whiskered quasi-periodic tori and their attached stable/unstable manifolds, for periodically-forced PCRTBP models. As part of this, a method for Levi- Civita regularization of such periodically-forced systems is introduced. Finally, I present methods for combining the previously mentioned parameterizations with knowl- edge of the objects’ internal dynamics, collision detection algorithms, and GPU computing to very rapidly compute propellant-free heteroclinic connecting trajectories between them, even in higher dimensional models. Such heteroclinics are key to the generation of chaos and large scale transport in astrodynamical systems.
In 1996, C.~Heil, J.~Ramanatha, and P.~Topiwala conjectured that the (finite) set $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N$ is linearly independent for any non-zero square integrable function $g$ and subset $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ This problem is now known as the HRT Conjecture, and is still largely unresolved.
In this talk, I will then introduce an inductive approach to investigate the conjecture, by attempting to answer the following question. Suppose the HRT conjecture is true for a function $g$ and a fixed set of $N$ points $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2.$ For what other point $(a, b)\in \mathbb{R}^2\setminus \Lambda$ will the HRT remain true for the same function $g$ and the new set of $N+1$ points $\Lambda'=\Lambda \cup \{(a, b)\}$? I will report on a recent joint work with V.~Oussa in which we use this approach to prove the conjecture when the initial configuration $\Lambda=\{(a_k, b_k)\}_{k=1}^N $ is either a subset of the unit lattice $\mathbb{Z}^2$ or a subset of a line $L$.
Ivanov’s metaconjecture says that every object naturally associated to a surface S with a sufficiently rich structure has the mapping class group as its group of automorphisms. In this talk, I will present several cases of curve graphs that satisfy this metaconjecture and some extensions to even richer structures.
The rank of a point $p$ with respect to a non-degenerate variety is the smallest number of the points in the variety that spans the point $p$. Studies about the ranks of points are important in various areas of applied mathematics and engineering in the sense that they are the smallest number of summands in the decompositions of vectors into combinations of simple vectors.
In the last talk, we discussed how to generate points of given ranks with respect to the rational normal curves. We continue to discuss some known facts via Macaulay 2 and how to find the list of all ranks of points in linear spaces.
Links to Teams: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1650576543136?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%221269007f-fe20-4c2c-b6fa-a7e0eff0131e%22%7d
Streaming online at https://gatech.zoom.us/j/91232113113?pwd=MDhteEdtcENuME9kdXJmcUY0eWlSUT09
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no inclusion-maximal clique is monochromatic (ignoring isolated vertices).
For the binomial random graph G_{n,p} the clique chromatic number has been studied in a number of works since 2016, but for sparse edge-probabilities in the range n^{-2/5} \ll p \ll 1 even the order of magnitude remained a technical challenge.
Resolving open problems of Alon and Krivelevich as well as Lichev, Mitsche and Warnke, we determine the clique chromatic number of the binomial random graph G_{n,p} in most of the missing regime: we show that it is of order (\log n)/p for edge-probabilities n^{-2/5+\eps} \ll p \ll n^{-1/3} and n^{-1/3+\eps} \ll p \ll 1, for any constant \eps > 0.
Perhaps surprisingly for a result about random graphs, a key ingredient in the proof is an application of the probabilistic method (that hinges on careful counting and density arguments).
This talk is based on joint work with Lutz Warnke.
Link for streaming: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
In this talk, we study the existence of quasi periodic solutions to the generalized Surface Quasi-Geostropic (gSQG) equations. Despite its similar structure with the 2D Euler equation, the global existence/finite time singularity formation of gSQG equations have been open for a long time. Exploiting its Hamiltonian structure, we are able to construct a quasi periodic solutions with the initial date that are sufficiently close to its steady states. This is a joint work with Javier Gomez-Serrano and Alex Ionescu.
A cubic graph is one where every vertex has degree three. A linear forest is a disjoint union of paths. It is known that the edge set of every cubic graph can be partitioned into two linear forests where each path is short (of constant size). A conjecture of Wormald asks for such a partition where the two forests are isomorphic (we no longer insist on having short paths, although that is also an open question). Note that this also can be phrased as an edge-colouring question. Is it possible to colour the edge set of a cubic graph by red and blue such that the two monochromatic components induce isomorphic linear forests? Recently we proved this for all connected graphs on a sufficiently large number of vertices. I will talk about the result and give some idea of the proof method. This is joint work with Gal Kronenberg, Shoham Letzter and Alexey Pokrovskiy.
Abstract: Gromov introduced some “hyperbolization” procedures, i.e. some procedures that turn a given polyhedron into a space of non-positive curvature. Charney and Davis developed a refined “strict hyperbolization” procedure that outputs a space of strictly negative curvature. Their procedure has been used to construct new examples of manifolds and groups with negative curvature, and other prescribed features. We construct actions of the resulting groups on CAT(0) cube complexes. As an application, we obtain that they are virtually special, hence linear over the integers and residually finite. This is joint work with J. Lafont.
Note the unusual time!
For given graphs $G$ and $H$ and a graph $F$, we say that $F$ is Ramsey for $(G, H)$ and we write $F \longrightarrow (G,H)$, if for every $2$-edge coloring of $F$, with colors red and blue, the graph $F$ contains either a red copy of $G$ or a blue copy of $H$. A natural question is how few vertices can a graph $F$ have, such that $F \longrightarrow (G,H)$? Frank P. Ramsey studied this question and proved that for given graphs $G$ and $H$, there exists a positive integer $n$ such that for the complete graph $K_n$ we have $ K_n \longrightarrow (G,H)$. The smallest such $n$ is known as the Ramsey number of $G$, $H$ and is denoted by $R(G, H)$. Instead of minimizing the number of vertices, one can ask for the minimum number of edges of such a graph, i.e. can we find a graph which possibly has more vertices than $R(G, H)$, but has fewer edges and still is Ramsey for $(G,H)$? How many edges suffice to construct a graph which is Ramsey for $(G,H)$? The attempts at answering the last question give rise to the notion of size-Ramsey number of graphs. In 1978, Erdős, Faudree, Rousseau and Schelp pioneered the study of the size-Ramsey number to be the minimum number of edges in a graph $F$, such that $F$ is Ramsey for $(G,H)$. In this talk, first I will give a short history about the size Ramsey number of graphs with a special focus on sparse graphs. Moreover, I will talk about the multicolor case of the size Ramsey number of cycles with more details.
This is joint work with Aaron Landesman. There are a number of difficult open questions around representations of free and surface groups, which it turns out are accessible to methods from Hodge theory and arithmetic geometry. For example, I'll discuss applications of these methods to the following concrete theorem about surface groups, whose proof relies on non-abelian Hodge theory and the Langlands program:
Theorem. Let $\rho: \pi_1(\Sigma_{g,n})\to GL_r(\mathbb{C})$ be a representation of the fundamental group of a compact orientable surface of genus $g$ with $n$ punctures, with $r<\sqrt{g+1}$. If the conjugacy class of $\rho$ has finite orbit under the mapping class group of $\Sigma_{g,n}$, then $\rho$ has finite image.
This answers a question of Peter Whang. I'll also discuss closely related applications to the Putman-Wieland conjecture on homological representations of mapping class groups.
Meeting Link: https://gatech.zoom.us/j/94882290086 (Meeting ID: 948 8229 0086, Passcode: 264830)
A central goal of biological experiments that generate omics time-course data is the discovery of patterns, or signatures, of response. A natural representation of such data is in the form of a third-order tensor. For example, if the dataset is from a bulk RNASeq experiment, which measures tissue-level gene expression collected at multiple time points, the data can be structured into a gene-by-subject-by-time tensor. We consider the use of a non-negative CANDECOMP/PARAFAC (CP) decomposition (NCPD) on the tensor to derive rank-one components that correspond to biologically meaningful signatures. To assess whether over-factoring has occurred in a model, we develop the maximum internal n-similarity score (mINS) score. We use the mINS as well as other metrics to choose a model rank for downstream analysis. We show that on time-course data profiling vaccination responses against the Influenza and Bordetella Pertussis pathogens, our NCPD pipeline yields novel and informative signatures of response. We finish with outstanding research challenges in the application of tensor decomposition to modern biological datasets.
MS Thesis Defense
Meeting link: https://bluejeans.com/865908583/9834
Statistical consistency in phylogenetics has traditionally referred to the accuracy of estimating mutation rates and phylogenies for a fixed number of species as we increase the amount of data within their signatures, such as DNA and protein sequences. Analyzing sequences undergoing indel mutations (insertions and deletions of sites) has provided a venue for understanding what power can be provided by a lot of data. In this talk, we discuss some of the failings of this approach. For instance, it will be shown that phylogeny estimation is impossible for infinitely long sequences, even with infinite data. This motivates a dual type of statistical consistency, where the number of species is taken to infinity rather than the size of each signature. Here, we give polynomial-time algorithms for ancestral sequence estimation and sequence alignment for reference phylogenies with so many species that they are sufficiently dense. Based on joint work with Louis Fan and Sebastien Roch.
This is an expanded version of a 10-minute presentation in MATH 6422. I'll explain what matroids and their characteristic polynomials as well as log-concavity mean, and then sketch a proof due to Petter Brändén and Jonathan Leake (arXiv:2110.00487). If time permits, I'll describe several consequences of this and/or other existing yet different proofs.
Teams link: https://teams.microsoft.com/l/meetup-join/19%3a3a9d7f9d1fca4f5b991b4029b09c69a1%40thread.tacv2/1651153648881?context=%7b%22Tid%22%3a%22482198bb-ae7b-4b25-8b7a-6d7f32faa083%22%2c%22Oid%22%3a%2206706002-23ff-4989-8721-b078835bae91%22%7d
Abstract: This talk has 4 or 5 parts
This is a joint work with Prof. Leonid Bunimovich.
Given a graph $G$ let $\lambda_1$ and $\lambda_n$ be the maximum and minimum eigenvalues of its adjacency matrix and define the spread of $G$ to be $\lambda_1 - \lambda_n$. In this talk we discuss solutions to a pair of 20-year-old conjectures of Gregory, Hershkowitz, and Kirkland regarding the spread of graphs.
The first, referred to as the spread conjecture, states that over all graphs on $n$ vertices the join of a clique of order $\lfloor 2n/3 \rfloor$ and an independent set of order $\lceil n/3 \rceil$ is the unique graph with maximum spread. The second, referred to as the bipartite spread conjecture, says that for any fixed $e\leq n^2/4$, if $G$ has maximum spread over all $n$-vertex graphs with $e$ edges, then $G$ must be bipartite.
We show that the spread conjecture is true for all sufficiently large $n$, and we prove an asymptotic version of the bipartite spread conjecture. Furthermore, we exhibit an infinite family of counterexamples to the bipartite spread conjecture which shows that our asymptotic solution is tight up to a multiplicative factor in the error term. This is joint work with Jane Breen, Alex Riasanovsky, and John Urschel.
Link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
Symmetries are fundamental concepts in modern physics and biology. Spontaneous symmetry breaking often leads to fascinating dynamical patterns such as chimera states in which structurally and dynamically identical oscillators split into coherent and incoherent clusters. Solitary states in which one oscillator separates from the coherent cluster and oscillates with a different frequency represent “weak” chimeras. While a rigorous stability analysis of a “strong” chimera with a multi-oscillator incoherent cluster is typically out of reach for finite-size networks, solitary states offer a unique test bed for the development of stability approaches to large chimeras. In this talk, we will present such an approach and study the stability of solitary states in Kuramoto networks of identical 2D phase oscillators with inertia and a phase-lagged coupling. We will derive asymptotic stability conditions for such solitary states as a function of inertia, network size, and phase lag that may yield either attractive or repulsive coupling. Counterintuitively, our analysis demonstrates that (i) increasing the size of the coherent cluster can promote the stability of the solitary state in the attractive coupling case and (ii) the solitary state can be stable in small-size networks with all repulsive coupling. We also discuss the implications of our analysis for the emergence of rotatory chimeras and splay states. This is a joint work with V. Munyaev, M. Bolotov, L. Smirnov, and G. Osipov.
Zoom link: https://gatech.zoom.us/j/92161924238
Random matrix has been found useful in many real world applications. The celebrated Johnson-Lindenstrauss lemma states that certain geometric structure of deterministic vectors is preserved when projecting high dimensional space $R^n$ to a lower dimensional space $R^m$. However, when random vectors are concerned, it is still unclear how the distribution of the geometry is affected by random matrices. Since random projection or embedding introduces dependence to independent random vectors, does it imply random matrices are inferior for transforming random vectors?
We will start with establishing a new central limit theorem for random variables with certain product dependence structure. At the same time, we obtain its Berry-Esseen type rate of convergence. Then we apply this general central limit theorem to random projections and embeddings of two independent random vectors $X, Z$. In particular, we show the distribution of inner product structure is preserved by random matrices. Roughly speaking, two independent random vectors remain "independent" in the randomly projected lower dimensional space or randomly embedded high dimensional space. More importantly, we also quantitatively characterize the distortion of distribution introduced by random matrices. The error term has a bound at most $O(\frac{1}{\sqrt{m}} + \frac{1}{\sqrt{n}})$.
Then we also establish the fact that random matrices have low distortion on the norm of a random vector. It is first justified by establishing concentration of the projected or embedded norm under sub-Gaussian assumptions. A central limit theorem for the randomly projected norm is established as well similar to the CLT for inner product.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
The computation of invariant manifolds for parabolic PDE is an important problem due to its many applications. One of the main difficulties is dealing with irregular high dimensional domains when the classical Fourier methods are not applicable, and it is necessary to employ more sophisticated numerical methods. This work combines the parameterization method based on an invariance equation for the invariant manifold, with the finite element method. We implement the method for a variety of example problems having both polynomial and non-polynomial nonlinearities, on non-convex two-dimensional polygonal domains (not necessary simply connected), for equilibrium solutions. We implement a-posteriori error indicators which provide numerical evidence of the accuracy of the computations. This is a joint work with J.D Mireles-James, and Necibe Tuncer.
An important recent topic is matrix completion, which is trying to recover a matrix from a small set of observed entries, subject to particular requirements. In this talk, we discuss results on symmetric tropical and symmetric Kapranov rank 2 matrices, and establish a technique of examining the phylogenetic tree structure obtained from the tropical convex hulls of their columns to construct the algebraic matroid of symmetric tropical rank 2 $n \times n$ matrices. This matroid directly answers the question of what entries of a symmetric $n \times n$ matrix needs to be specified generically to be completable to a symmetric tropical rank 2 matrix, as well as to a symmetric classical rank 2 matrix.
This is based on joint work with Cvetelina Hill and Kisun Lee.
Zoom link : https://gatech.zoom.us/j/98171168149
Since their introduction in the early 1960s, Anosov flows have defined an important class of dynamics, thanks to their many interesting chaotic features and rigidity properties. Moreover, their topological aspects have been deeply explored, in particular in low dimensions, thanks to the use of foliation theory in their study. Although the connection of Anosov flows to contact and symplectic geometry was noted in the mid 1990s by Mitsumatsu and Eliashberg-Thurston, such interplay has been left mostly unexplored. I will present some recent results on the contact and symplectic geometric aspects of Anosov flows in dimension 3, including in the presence of an invariant volume form, which is known to have grave consequences for the dynamics of these flows. Time permitting, the interplay of Anosov flows with Reeb dynamics, Liouville geometry and surgery theory will be briefly discussed as well.
Zoom Link- https://gatech.zoom.us/j/97563537012?pwd=dlBVUVh2ZDNwdDRrajdQcDltMmRaUT09 (Meeting ID: 975 6353 7012 Passcode: 525012)
In this talk I will discuss the conjecture that every 3 manifolds can be smoothly embedded in symplectic 4 manifolds. I will give some motivation on why is this an interesting conjecture. As an evidence for the conjecture, I will prove that every 3 manifolds can be embedded in a topological way and such an embedding can be made a smooth one after a single stabilization. As a corollary of the proof, I will prove that integer/rational cobordism group is generated by Stein fillable 3 manifolds. And if time permits, I will give some idea on how one can try to obstruct smooth embeddings of 3 manifolds in symplectic 4 manifolds.
Zoom link: https://gatech.zoom.us/j/4561289292
Abstract: In recent years we have seen the popularity of optimal transport and deep learning. Optimal transport theory works well in studying differences among distributions, while deep learning is powerful to analyze high dimensional data. In this presentation we will discuss some of our recent work that combine both optimal transport and deep learning on data-driven problems. We will cover four parts in this presentation. The first part is studying stochastic behavior from aggregate data where we recover the drift term in an SDE, via the weak form of Fokker-Planck equation. The second part is applying Wasserstein distance on the optimal density control problem where we parametrize the control strategy by a neural network. In the third part we will show a novel form of computing Wasserstein distance, geometric and map all together in a scalable way. And in the final part, we consider an inverse OT problem where we recover cost function when an observed policy is given.
Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph on n vertices, where n is a multiple of 3 and large, and the minimum vertex degree of H is greater than {(n-1) choose 2} - {2n/3 choose 2}, then H contains a perfect matching.
We show that for sufficiently large n divisible by 3, if F_1, ..., F_{n/3} are 3-uniform hypergraphs with a common vertex set and the minimum vertex degree in each F_i is greater than {(n-1) choose 2} - {2n/3 choose 2} for i = 1, ..., n/3, then the family {F_1, ..., F_{n/3}} admits a rainbow matching, i.e., a matching consisting of one edge from each F_i. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.
We also prove that, for any integers k, l with k >= 3 and k/2 < l <= k-1, there exists a positive real μ such that, for all sufficiently large integers m, n satisfying n/k - μn <= m <= n/k - 1 - (1 - l/k){ceil of (k - l)/(2l - k)}, if H is a k-uniform hypergraph on n vertices and the minimum l-degree of H is greater than {(n-l) choose (k-l)} - {(n-l-m) choose (k-l)}, then H has a matching of size m+1. This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l <= k-1. In many cases, our result gives tight bound on the minimum l-degree of H for near perfect matchings. For example, when l >= 2k/3, n ≡ r (mod k), 0 <= r < k, and r + l >= k, we can take m to be the minimum integer at least n/k - 2.
Zoom link: https://gatech.zoom.us/j/91659544858?pwd=SWZtVG15dGFiWEFXSHR1U0JNbVVBZz09
Erdos and Posa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part to its algorithmic applications. In this thesis we investigate the Erdos-Posa property of various families of constrained cycles and paths by developing new structural tools for undirected group-labelled graphs.
Our first result is a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs. This structure theorem is then used to prove the Erdos-Posa property of A-paths of length 0 modulo p for a fixed odd prime p, answering a question of Bruhn and Ulmer. Further, we obtain a characterization of the abelian groups G and elements g for which A-paths of weight g satisfy the Erdos-Posa property. These results are from joint work with Robin Thomas.
We extend our structural tools to graphs labelled by multiple abelian groups and consider the Erdos-Posa property of cycles whose weights avoid a fixed finite subset in each group. We find three types of topological obstructions and show that they are the only obstructions to the Erdos-Posa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdos-Posa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdos-Posa property holds for such cycles, unifying a large number of results in this area into a general framework. As a special case, we characterize the integer pairs (L,M) for which cycles of length L mod M satisfy the Erdos-Posa property. This resolves a question of Dejter and Neumann-Lara from 1987. Further, our description of the obstructions allows us to obtain an analogous characterization of the Erdos-Posa property of cycles in graphs embeddable on a fixed compact orientable surface. This is joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.
Zoom link: https://gatech.zoom.us/j/96860495360?pwd=cktMRVVqMDRtVnJsb3ZLRll1bFRJQT09
I will be defending my thesis on the shortest closed curve to inspect a sphere.<br />
<br />
Time: 11am EST<br />
Location: Skiles 005, also on Zoom at https://gatech.zoom.us/j/97708515339<br />
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Committee:<br />
<br />
Dr. Mohammad Ghomi, Advisor<br />
School of Mathematics<br />
Georgia Institute of Technology<br />
<br />
Dr. Igor Belegradek<br />
School of Mathematics<br />
Georgia Institute of Technology<br />
<br />
Dr. Jason Cantarella<br />
Department of Mathematics<br />
University of Georgia<br />
<br />
Dr. Rob Kusner<br />
Department of Mathematics<br />
University of Massachusetts<br />
<br />
Dr. Galyna Livshyts<br />
School of Mathematics<br />
Georgia Institute of Technology<br />
<br />
Dr. Michael Loss<br />
School of Mathematics<br />
Georgia Institute of Technology<br />
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This thesis consists of three applications of the circle method in number theory problems. In the first part, we study the $p-$divisibility of the central binomial coefficients. For a certain set of large prime numbers, we prove that there are infinitely many integers $n$, which $\binom{2n}{n}$ has these primes with unexpectedly small multiplicity in its prime factorization. This result is related to an open problem conjectured by Graham, stating that there are infinitely many integers $n$ which the binomial coefficients $\binom{2n}{n}$ is coprime with $105$. The proof consists of the Fourier analysis method, as well as geometrically bypassing an old conjecture about the primes.
In the second part, we discover an unexpected cancellation on the sums involving the exponential functions. Applying this theorem on the first terms of the Ramanujan-Hardy-Rademacher expansion gives us a natural proof of a ``weak" pentagonal number theorem. We find several similar upper bounds for many different partition functions. Additionally, we prove another set of ``weak" pentagonal number theorems for the primes, which allows us to count the number of primes in certain intervals with small error. Finally, we show an approximate solution to the Prouhet-Tarry-Escott problem using a similar technique. The core of the proofs is an involved circle method argument.
The third part of this thesis is about finding an endpoint $\ell^p-$improving inequality for an ergodic sum involving the primes. As the set of the prime is almost full-dimensional, the question on the endpoint becomes more interesting, because we want to bound $\ell^{\infty}$ to $\ell^{1}$ operator. The weak-type inequality we propose depends on the assumption of the Generalized Riemann Hypothesis. Assuming GRH, we prove the sharpest possible bound up to a constant. Unconditionally, we prove the same inequality up to a $\log $ factor. The proof is based on a circle method argument and careful use of the Ramanujan sums.
Dissertation defense information
Date and Time: July 22, 2022, 08:30 am - 10:30 am (EST)
Location:
Summary
This dissertation consists of two topics concerning algebraic and semi-algebraic invariants on quadrics.
The ranks of the minimal graded free resolution of square-free quadratic monomial ideals can be investigated combinatorially. We study the bounds on the algebraic invariant, Castelnuovo-Mumford regularity, of the quadratic ideals in terms of properties on the corresponding simple graphs. Our main theorem is the graph decomposition theorem that provides a bound on the regularity of a quadratic monomial ideal. By combining the main theorem with results in structural graph theory, we proved, improved, and generalized many of the known bounds on the regularity of square-free quadratic monomial ideals.
The Hankel index of a real variety is a semi-algebraic invariant that quantifies the (structural) difference between nonnegative quadrics and sums of squares on the variety. This project is motivated by an intriguing (lower) bound of the Hankel index of a variety by an algebraic invariant, the Green-Lazarsfeld index, of the variety. We study the Hankel index of the image of the projection of rational normal curves away from a point. As a result, we found a new rank of the center of the projection which detects the Hankel index of the rational curves. It turns out that the rational curves are the first class of examples that the lower bound of the Hankel index by the Green-Lazarsfeld index is strict.
Advisor: Dr. Grigoriy Blekherman, School of Mathematics, Georgia Institute of Technology
Committee:
Let T be a set of n triangles in 3-space, and let \Gamma be a family of
algebraic arcs of constant complexity in 3-space. We show how to preprocess T
into a data structure that supports various "intersection queries" for
query arcs \gamma \in \Gamma, such as detecting whether \gamma intersects any
triangle of T, reporting all such triangles, counting the number of
intersection points between \gamma and the triangles of T, or returning the
first triangle intersected by a directed arc \gamma, if any (i.e., answering
arc-shooting queries). Our technique is based on polynomial partitioning and
other tools from real algebraic geometry, among which is the cylindrical
algebraic decomposition.
Our approach can be extended to many other intersection-searching problems in
three and higher dimensions. We exemplify this versatility by giving an
efficient data structure for answering segment-intersection queries amid a set
of spherical caps in 3-space, and we lay a roadmap for extending our approach
to other intersection-searching problems.
Joint work with Pankaj Agarwal, Boris Aronov, Matya Katz, and Micha Sharir.
Compact, graphical representations of error-correcting codes called trellises are a crucial tool in classical coding theory, establishing both theoretical properties and performance metrics for practical use. The idea was extended to quantum error-correcting codes by Ollivier and Tillich in 2005. Here, we use their foundation to establish a practical decoder able to compute the maximum-likely error for any stabilizer code over a finite field of prime dimension. We define a canonical form for the stabilizer group and use it to classify the internal structure of the graph. Similarities and differences between the classical and quantum theories are discussed throughout. Numerical results are presented which match or outperform current state-of-the-art decoding techniques. New construction techniques for large trellises are developed and practical implementations discussed. We then define a dual trellis and use algebraic graph theory to solve the maximum-likely coset problem for any stabilizer code over a finite field of prime dimension at minimum added cost.
Classical trellis theory makes occasional theoretical use of a graph product called the trellis product. We establish the relationship between the trellis product and the standard graph products and use it to provide a closed form expression for the resulting graph, allowing it to be used in practice. We explore its properties and classify all idempotents. The special structure of the trellis allows us to present a factorization procedure for the product, which is much simpler than that of the standard products.
Finally, we turn to an algorithmic study of the trellis and explore what coding-theoretic information can be extracted assuming no other information about the code is available. In the process, we present a state-of-the-art algorithm for computing the minimum distance for any stabilizer code over a finite field of prime dimension. We also define a new weight enumerator for stabilizer codes over F_2 incorporating the phases of each stabilizer and provide a trellis-based algorithm to compute it.
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Advisor: Dr. Evans Harrell, School of Mathematics, Georgia Institute of Technology
Committee:
Dr. Evans Harrell, School of Mathematics, Georgia Institute of Technology
Dr. Matthew Baker, School of Mathematics, Georgia Institute of Technology
Dr. Martin Short, School of Mathematics, Georgia Institute of Technology
Dr. Moinuddin Qureshi, School of Computer Science, Georgia Institute of Technology
Dr. Kenneth Brown, Pratt School of Engineering, Duke University
Reader: Dr. Kenneth Brown, Pratt School of Engineering, Duke University
Link: https://gatech.zoom.us/j/98306382257
Oral Comprehensive Exam
One application of the immersed-curve technique, introduced by Hanselman-Rasmussen-Watson, is to study rank inequalities for Heegaard Floer homology in the presence of certain degree-one maps. Another application, discovered by Chen, is to describe the knot Floer homology of satellite knots with (1,1)-patterns. We will discuss similar rank inequalities for the knot Floer homology of (1,1)-satellites.
This talk will focus on surfaces (orientable connected 2-manifolds) with noncompact boundary. Since a general surface with noncompact boundary can be extremely complicated, we will first consider a particular class called Sliced Loch Ness Monsters. We will discuss how to show the mapping class group of any Sliced Loch Ness Monster is uniformly perfect and automatically continuous. Depending on the time remaining, we will also discuss the classification of surfaces with noncompact boundary due to Brown and Messer, and how Sliced Loch Ness Monsters are used to prove results about the mapping class groups of general surfaces.
Cyclic orbit codes are subspace codes generated by the action of the Singer subgroup F_{q^n}^* on an F_q-subspace U of F_{q^n}. The weight distribution of a code is the vector whose ith entry is the number of codewords with distance i to a fixed reference generator of the code. We will investigate the weight distribution for a few categories of cyclic orbit codes, including optimal codes. Further, we want to know when two cyclic orbit codes with the same weight distribution are isometric. To answer this question, we determine the possible automorphism groups for cyclic orbit codes.
In this talk I will present some of their theoretical guarantees with an emphasis on their behavior under the so-called manifold hypothesis. Such theoretical guarantees are non-vacuous and provide insight on the empirical behavior of these models. I will show how these results imply generalization bounds on denoising diffusion models. This presentation is based on https://arxiv.org/abs/2208.05314
A map (respectively, a unicellular map) on a genus g surface Sg is the Homeo+(Sg)-orbit of a graph G embedded on Sg such that Sg-G is a collection of finitely many disks (respectively, a single disk). The study of maps was initiated by W. Tutte, who was interested in counting the number of planar maps. However, we will consider maps from a more graph theoretic perspective in this talk. We will introduce a topological operation called surgery, which turns one unicellular map into another. Then, we will address natural questions (such as connectedness and diameter) about surgery graphs on unicellular maps, which are graphs whose vertices are unicellular maps and where two vertices share an edge if they are related by a single surgery. We will see that these problems translate to a well-known combinatorial problem: the card shuffling problem.
In a seminal work, Leray demonstrated the existence of global weak solutions to the Navier-Stokes equations in three dimensions. Are Leray's solutions unique? This is a fundamental question in mathematical hydrodynamics, which we answer in the negative, within the `forced' category, by exhibiting two distinct Leray solutions with zero initial velocity and identical body force. This is joint work with Elia Brué and Maria Colombo.
An order-n Latin square is an $n \times n$ matrix with entries from a set of $n$ symbols, such that each row and each column contains each symbol exactly once. Suppose that $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k \in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i,j\in[n]$. How likely does there exist an order-$n$ Latin square where the entry in the $i$th row and $j$th column lies in $L_{i,j}$? This question was initially raised by Johansson in 2006, and later Casselgren and H{\"a}ggkvist and independently Luria and Simkin conjectured that $\log n / n$ is the threshold for this property. In joint work with Dong-yeap Kang, Daniela K\"{u}hn, Abhishek Methuku, and Deryk Osthus, we proved that for some absolute constant $C$, if $p > C \log^2 n / n$, then asymptotically almost surely there exists such a Latin square. We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs.
Two of the aims in using mathematics in real world applications are: (1) understanding the mechanisms responsible for different effects and phenomena, and (2) predicting the future state of the system under study. Dynamical systems provides a perspective and a lens for addressing these two questions. The system under study is formulated as an evolving set of state variables and the set of trajectories with different initializations are viewed geometrically.
I will use this lens to look at a pressing problem in climate science: how a climate subsystem might abruptly “tip” from its current state into a completely different state. This is a problem that requires dynamical systems to understand, and I will show how we can decode different ways in which the tipping might happen.
Dynamical systems models tend to be simplified; extraneous forces are ignored to produce models which attempt to capture the key mechanisms. The inclusion of data from observations is a way to connect these models with reality and I will discuss the area of data assimilation that achieves a balance between data and physical models in a systematic way.
The Prophet Inequality and Pandora's Box problems are fundamental stochastic problems. A usual assumption for both problems is that the probability distributions of the n underlying random variables are given as input to the algorithm. In this talk, we assume the distributions are unknown, and study them in the Multi-Armed Bandits model: We interact with the unknown distributions over T rounds. In each round we play a policy and receive only bandit feedback. The goal is to minimize the regret, which is the difference in the total value of the optimal algorithm that knows the distributions vs. the total value of our algorithm that learns the distributions from the bandit feedback. Our main results give near-optimal O(poly(n)sqrt{T}) total regret algorithms for both Prophet Inequality and Pandora's Box.
Beginning in Spring 2020, we stepped away from traditional exams and collaboratively developed the concept portfolio assessment with the aim of creating a more equitable learning experience for students. Since then, we have implemented this model of assessment in courses from Pre-Calculus through Number Theory as faculty at a community college and a small liberal arts college. For the concept portfolio, students choose a subset of the topics covered in the course and synthesize the topics by providing a summary and annotated examples. The portfolio is completed iteratively where students submit rough drafts and engage in peer review. During this talk, we will share our motivation to design an equitable alternative to exams, compare and contrast our implementations of the concept portfolio assessment, and discuss student feedback.
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The talk is delivered in a hybrid format Everyone is welcome to join via zoom
https://gatech.zoom.us/j/94287395719?pwd=U216WTlIZHdMNVErZlFWUGlleDBiQT09
but we have also reserved 005 to attend the talk all together, hoping discussion will be easier.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz... />
Given a divergence-free vector field on the torus, we consider the mixing properties of the associated flow. There is a rich body of work studying the dependence of the mixing scale on various norms of the vector field. We will discuss some interesting examples of vector fields that mix at the optimal rate, and an improved bound on the mixing scale under the extra assumption that the vector field is a shear at each time.
Thomassen famously showed that every planar graph is 5-choosable, and that every planar graph of girth at least five is 3-choosable. These theorems are best possible for uniform list assignments: Voigt gave a construction of a planar graph that is not 4-choosable, and of a planar graph of girth four that is not 3-choosable. In this talk, I will introduce the concept of a local girth list assignment: a list assignment wherein the list size of each vertex depends not on the girth of the graph, but only on the length of the shortest cycle in which the vertex is contained. I will present a local list colouring theorem that unifies the two theorems of Thomassen mentioned above and discuss some of the highlights and difficulties of the proof. This is joint work with Luke Postle.
I will give an introduction to tropical geometry which arises when you take the coordinate-wise logarithm of points in a curve and then take the base of the logarithm to infinity. This gives a combinatorial curve which is basically a bunch of rays starting at the origin. I will also talk a bit about polygons, number theory and geometry.
(Based on paper by Fawzi, Saunderson and Parrilo in 2015) The space of complex-valued functions on a fixed abelian group has an orthonormal basis of group homomorphisms, via the well-known Discrete Fourier Transform. Given any nonnegative function with sparse Fourier support, it turns out that it’s possible to write it as a sum of squares, where the common Fourier support for all squares is not big. This can be used to prove results for the usual degree-based sum-of-squares hierarchy.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
This talk concentrates on the study of stability of floating objects through mathematical modeling and experimentation. The models are based on standard ideas of center of gravity, center of buoyancy, and Archimedes’ Principle. There will be a discussion of a variety of floating shapes with two-dimensional cross sections for which it is possible to analytically and/or computationally a potential energy landscape in order to identify stable and unstable floating orientations. I then will compare the analysis and computations to experiments on floating objects designed and created through 3D printing. The talk includes a demonstration of code we have developed for testing the floating configurations for new shapes. I will give a brief overview of the methods involved in 3D printing the objects.
This research is joint work with Dr. Dan Anderson at GMU and undergraduate students Brandon G. Barreto-Rosa, Joshua D. Calvano, and Lujain Nsair, all of whom who were part of an undergraduate research program run by the MEGL at GMU.
While gerrymandering has been widely suspected in Georgia for years, it has been difficult to quantify. We generate a large ensemble of randomly generated non-partisan maps that are sampled from a probability distribution which respects the geographical constraints of the redistricting process. Using a Markov chain Monte Carlo process and techniques involving spanning trees, we can quickly generate a robust set of plans.
Based on historical voting data, we compare the Georgia congressional redistricting plan enacted in 2021 with the non-partisan maps. We find that the 2021 plan will likely be highly non-responsive to changing opinions of the electorate, unlike the plans in the ensemble. Using additional spatial analysis, we highlight areas where the map has been redrawn to weaken the influence of Democratic voters.
This talk is based on joint work with Swati Gupta, Gregory Herschlag, Jonathan Mattingly, Dana Randall, and Zhanzhan Zhao.
One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.
The cluster expansion is a classical tool from statistical physics used to understand systems of weakly interacting particles in the high temperature regime of statistical physics models. It can also be a very useful tool in probabilistic, extremal, and enumerative combinatorics and in the study of large deviations in probability theory. I will give an introduction to the cluster expansion, present some examples of combinatorial applications, and try to provide some intuition about when the cluster expansion should or should not be a useful tool for a particular problem.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz... />
In this talk, I will present the analysis of two astrophysical systems. First, exoplanets (planets orbiting a star that is not our Sun) are thought to sometimes naturally evolve into a state such that its spin axis is significantly tilted from its orbital axis. The most well-known examples of such tilts come from our own Solar System: Uranus with its 98 degree tilt is spinning entirely on its side, while Venus with its 177 degree tilt spins in the opposite direction to its orbit. I show that tilted exoplanets form probabilistically due to encountering a separatrix, and this probability can be analytically calculated using Melnikov's Method. Second, the origin of the binary black holes (BBHs) whose gravitational wave radiation has been detected by the LIGO/VIRGO Collaboration is currently not well-understood. Towards disambiguating among many proposed formation mechanisms, certain studies have computed the distributions of various physical parameters when BBHs form via certain mechanisms. A curious result shows that one such formation mechanism commonly results in black holes tilted on their sides. I show that this can be easily understood by identifying a hidden adiabatic invariant that links the initial and final spin orientations of the BBHs. No astrophysical knowledge is expected; please stop by!
In this talk, I will present a simple proof of the matrix Spencer conjecture up to poly-logarithmic rank: given symmetric d by d matrices A_1,...,A_n each with operator norm at most 1 and rank at most n/\log^3 n, one can efficiently find \pm 1 signs x_1,... ,x_n such that their signed sum has spectral norm \|\sum_{i=1}^n x_i A_i\|_op= O(\sqrt{n}). This result also implies a (\log n - 3 \log \log n) qubit lower bound for quantum random access codes encoding n classical bits with advantage >> 1/\sqrt{n}. Our proof uses the recent refinement of the non-commutative Khintchine inequality in [Bandeira, Boedihardjo, van Handel, 2022] for random matrices with correlated Gaussian entries.
Let G be a graph. Backman, Baker, and Yuen have constructed a family of bijections between spanning trees of G and the equivalence classes of orientations up to cycle-cocycle reversal, called the geometric bijections. Their proof makes use of zonotopal subdivisions. Recently we have extended the geometric bijections to subgraph-orientation correspondences. Moreover, we have also constructed a larger family of bijections, which contains the geometric bijections and the Bernardi bijections. Most of our work is inspired by geometry but proved combinatorially.
In this talk, we propose a Neural Oracle Search(NOS) model in Automatic Speech Recognition(ASR) to select the most likely hypothesis using a sequence of acoustic representations and multiple hypotheses as input. The model provides a sequence level score for each audio-hypothesis pair that is obtained by integrating information from multiple sources, such as the input acoustic representations, N-best hypotheses, additional 1st-pass statistics, and unpaired textual information through an external language model. These scores are then used to map the search problem of identifying the most likely hypothesis to a sequence classification problem. The definition of the proposed model is broad enough to allow its use as an alternative to beam search in the 1st-pass or as a 2nd-pass, rescoring step. This model achieves up to 12% relative reductions in Word Error Rate (WER) across several languages over state-of-the-art baselines with relatively few additional parameters. In addition, we investigate the use of the NOS model on a 1st-pass multilingual model and show that similar to the 1st-pass model, the NOS model can be made multilingual.
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I will introduce the concept of an ordered blueprint and a tract and discuss some algebraic and categorical properties. I will then discuss the notion of a "tropical extension" and discuss the theory of polynomials in these contexts.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Abstract: The interaction of machine learning and dynamics can lead to both new methodology for dynamics, and deepened understanding and/or efficacious algorithms for machine learning. This talk will give examples in both directions. Specifically, I will first discuss data-driven learning and prediction of mechanical dynamics, for which I will demonstrate one strong benefit of having physics hard-wired into deep learning models; more precisely, how to make symplectic predictions, and how that probably improves the accuracy of long-time predictions. Then I will discuss how dynamics can be used to better understand the implicit biases of large learning rates in the training of machine learning models. They could lead to quantitative escapes from local minima via chaos, which is an alternative mechanism to commonly known noisy escapes due to stochastic gradients. I will also report how large learning rates bias toward flatter minimizers, which arguably generalize better.
One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.
I will discuss an old conjecture of Ron Graham on whether there are infinitely many integers $n$ so that $\mathrm{gcd}({{2n} \choose n}, 105)=1$, as well as recent progress on a version of this problem where 105 is replaced with a product of $r$ distinct primes. This is joint work with Hamed Mousavi and Maxie Schmidt.
A geodesic metric space is said to be CAT(0) if triangles are at most as fat as triangles in the Euclidean plane. A CAT(0) cube complex is a CAT(0) space which is built by gluing Euclidean cubes isometrically along faces. Due to their fundamental role in the resolution of the virtual Haken's conjecture, CAT(0) cube complexes have since been a central object of study in geometric group theory and their study has led to ground-breaking advances in 3–manifold theory. The class of groups admitting proper cocompact actions on CAT(0) cube complexes is very broad and it includes hyperbolic 3-manifolds, most non-geometric 3 manifold groups, small cancelation groups and many others.
A revolutionary work of Sageev shows that the entire structure of a CAT(0) cube complexes is encoded in its hyperplanes and the way they interact with one another. I will discuss Sageev's theorem which provides a recipe for constructing group actions on CAT(0) cube complexes using some very simple and purely set theoretical data.
I will give an introduction to Oliver Lorscheid’s theory of ordered blueprints – one of the more successful approaches to “the field of one element” – and sketch its relationship to Tits models for algebraic groups and moduli spaces of matroids. The basic idea for these applications is quite simple: given a scheme over Z defined by equations with coefficients in {0,1,-1}, there is a corresponding “blue model” whose K-points (where K is the Krasner hyperfield) sometimes correspond to interesting combinatorial structures. For example, taking closed K-points of a suitable blue model for a split reductive group scheme G over Z gives the Weyl group of G, and taking K-points of a suitable blue model for the Grassmannian G(r,n) gives the set of matroids of rank r on {1,...,n}.
Two of the most well-studied topics in geometric group theory are CAT(0) cube complexes and mapping class groups. This is in part because they both admit powerful combinatorial-like structures that encode their (coarse) geometry: hyperplanes for the former and curve graphs for the latter. In recent years, analogies between the two theories have become more apparent. For instance: there are counterparts of curve graphs for CAT(0) cube complexes and rigidity theorems for such counterparts that mirror the surface setting, and both can be studied using the machinery of hierarchical hyperbolicity. However, the considerably larger class of CAT(0) spaces is left out of this analogy, as the lack of a combinatorial-like structure presents a difficulty in importing techniques from those areas. In this talk, I will speak about recent work with Petyt and Spriano where we bring CAT(0) spaces into the picture by developing analogues of hyperplanes and curve graphs for them. The talk will be accessible to everyone, and all the aforementioned terms will be defined.
In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs.
As another application of our characterization of the second order adjoint state, we derive additional necessary optimality conditions in terms of the value function. These results generalize a classical relationship between the adjoint states and the derivatives of the value function to the case of viscosity differentials.
The last part of the talk is devoted to sufficient optimality conditions. We show how the necessary conditions lead us directly to a non-smooth version of the classical verification theorem in the framework of viscosity solutions.
This talk is based on joint work with Wilhelm Stannat: W. Stannat, L. Wessels, Peng's maximum principle for stochastic partial differential equations, SIAM J. Control Optim., 59 (2021), pp. 3552–3573 and W. Stannat, L. Wessels, Necessary and Sufficient Conditions for Optimal Control of Semilinear Stochastic Partial Differential Equations, https://arxiv.org/abs/2112.09639, 2022.
Given a graph G with edges labeled by a group, a construction of Zaslavsky gives a rank-1 lift of the graphic matroid M(G) that respects the group-labeling. For which finite groups can we construct a rank-t lift of M(G) with t > 1 that respects the group-labeling? We show that this is possible if and only if the group is the additive subgroup of a non-prime finite field. We assume no knowledge of matroid theory.
In this introductory talk, we describe an older result (with David Dumas) that relates hyperbolic affine spheres over polygons to polynomial Pick differentials in the plane. All the definitions will be developed. In the last few minutes, I will quickly introduce two analytic problems in other directions that I struggle with.
In the world of moderate, everyday turbulence of fluids flowing across planes and down pipes, a quiet revolution is taking place. Applied mathematicians can today compute 'exact coherent structures', i.e. numerically precise 3D, fully nonlinear Navier-Stokes solutions: unstable equilibria, traveling waves, and (relative) periodic orbits. Experiments carried out at Georgia Tech today yield measurements as detailed as the numerical simulations; our experimentalists measure 'exact coherent structures' and trace out their unstable manifolds. What emerges is a dynamical systems theory of low-Reynolds turbulence as a walk among sets of weakly unstable invariant solutions.
We take you on a tour of this newly breached, hitherto inaccessible territory. Mastery of fluid mechanics is no prerequisite, and perhaps a hindrance: the talk is aimed at anyone who had ever wondered why - if no cloud is ever seen twice - we know a cloud when we see one? And how do we turn that into mathematics?
Link to the online seminar: https://gatech.zoom.us/j/94538442915
We study nonparametric dependence detection with the proposed binary expansion approximation of uniformity (BEAUTY) approach, which generalizes the celebrated Euler's formula, and approximates the characteristic function of any copula with a linear combination of expectations of binary interactions from marginal binary expansions. This novel theory enables a unification of many important tests through approximations from some quadratic forms of symmetry statistics, where the deterministic weight matrix characterizes the power properties of each test. To achieve a robust power, we study test statistics with data-adaptive weights, referred to as the binary expansion adaptive symmetry test (BEAST). By utilizing the properties of the binary expansion filtration, we show that the Neyman-Pearson test of uniformity can be approximated by an oracle weighted sum of symmetry statistics. The BEAST with this oracle provides a benchmark of feasible power against any alternative by leading all existing tests with a substantial margin. To approach this oracle power, we develop the BEAST through a regularized resampling approximation of the oracle test. The BEAST improves the empirical power of many existing tests against a wide spectrum of common alternatives and provides clear interpretation of the form of dependency when significant. This is joint work with Zhigen Zhao and Wen Zhou.
I will describe my recent joint work with Jacob Bedrossian and Sam Punshon-Smith on the formation of small scales in passively-advected scalars being mixed by a fluid evolving by the Navier-Stokes equation. Our main result is a confirmation of Batchelor's law, a power-law for the spectral density of a passively advected scalar in the so-called Batchelor regime of infinite Schmidt number. Along the way I will describe how this small-scale formation is intimately connected with dynamical questions, such as the connection between shear-straining in the fluid and sensitive dependence on initial conditions (Lyapunov exponents). Time-permitting I will describe some work-in-progress as well as interesting open problems in the area.
We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Spectral methods are the gold standard for parameterizing manifolds of solutions for ODEs because of their high precision and amenability to computer assisted proofs. However, these methods suffer from several drawbacks. In particular, the parameterizations are costly to compute and time-stepping is far more complicated than other methods. In this talk we demonstrate how computing these parameterizations and accurately time-stepping can be reduced to a related manifold learning problem. The latter problem is solved by training a deep neural network to interpolate charts for a low dimensional manifold embedded in a high dimensional Euclidean space. This training is highly parallelizable and need only be performed once. Once the neural network is trained, it is capable of parameterizing invariant manifolds for the ODE and time-stepping with remarkable efficiency and precision.
Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics, convex geometry, fair allocations, combinatorics, spectral graph theory, network design, and random processes. In an instance of a determinant maximization problem, we are given a collection of vectors $U = {v_1, \ldots, v_n}$ in $d$ dimensions, and a goal is to pick a subset $S$ of given vectors to maximize the determinant of the matrix $\sum_{i \in S} v_i v_i^T$. Often, the set $S$ of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint ($|S| \leq k$) or matroid constraint ($S$ is a basis of a matroid defined on the vectors). In this talk, we give a polynomial-time deterministic algorithm that returns an $r^{O(r)}$-approximation for any matroid of rank $r \leq d$. Our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improves any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the determinant function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.
This talk is based on joint work with Adam Brown, Madhusudhan Pittu, Mohit Singh, and Prasad Tetali.
Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. In this talk, we will consider the background necessary for an approach to this problem. Specifically, we will survey some essential notions and terminology related to low-dimensional contact and symplectic topology. These will involve Dehn surgery, tightness, overtwistedness, concave and convex symplectic fillings, and open book decompositions. We will also look at some results about these and mention some research trends.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz... />
Abstract: In the field of structural dynamics, engineers heavily rely on high-fidelity models of the structure at hand to predict its dynamic response and identify potential threats to its integrity.
The structure under investigation, be it an aircraft wing or a MEMS device, is typically discretised with finite elements, giving rise to a very large system of nonlinear ODEs. Due to the high dimensionality, the solution of such systems is very expensive in terms of computational time. For this reason, a large amount of literature in this field is devoted to the development of reduced order models of much lower dimensionality, able to accurately reproduce the original system’s dynamics. Not only the lower dimensionality increases the computational speed, but also provides engineers with interpretable and manageable models of complex systems, which can be easily coupled with data and uncertainty quantification, and whose parameter space can be easily explored. Slow invariant manifolds prove to be the perfect candidate for dimensionality reduction, however their computation for large scale systems has only been proposed in recent years (see Gonzalez et al. (2019), Haller et al. (2020), AV et al. (2019)).
In this talk, the Direct Parametrisation of Invariant Manifolds method (DPIM) will be presented. The theoretical basis of the method is provided by the results of Cabré, Fontich and de la Llave and its algorithmic implementation relies on the parametrisation method for invariant manifolds proposed by Haro et al.. The idea is to parametrise the invariant manifold around a fixed point through a power series expansion which can be solved recursively for each monomial in the reduced coordinates. The main limitation of the original algorithm is the necessity to operate in diagonal representation, which is unfeasible for large finite element systems as it would require the computation of the whole eigenspectrum. The main novelty of the proposed method lies in the expression of the normal homological equation directly in physical coordinates, which is the key aspect that permits its application to large scale systems.
The talk will focus on problems in structural dynamics in both autonomous and nonautonomous settings. The accuracy of the reduction will be shown on several examples, covering phenomena like internal resonances and parametric resonances. Finally, the current limitations and future developments of the method will be discussed.
Polynomial systems can be effectively solved by exploiting structure present in their Galois group. Esterov determined two conditions for which the Galois group of a sparse polynomial system is imprimitive, and showed that the Galois group is the symmetric group otherwise. A system with an imprimitive Galois group can be decomposed into simpler systems, which themselves may be further decomposed. Esterov's conditions give a stopping criterion for decomposing these systems and leads to a recursive algorithm for efficient solving.
Given an immersed, Maslov-0, exact Lagrangian filling of a Legendrian knot, if the filling has a vanishing index and action double point, then through Lagrangian surgery it is possible to obtain a new immersed, Maslov-0, exact Lagrangian filling with one less double point and with genus increased by one. We show that it is not always possible to reverse the Lagrangian surgery: not every immersed, Maslov-0, exact Lagrangian filling with genus g ≥ 1 and p double points can be obtained from such a Lagrangian surgery on a filling of genus g − 1 with p+1 double points. To show this, we establish the connection between the existence of an immersed, Maslov-0, exact Lagrangian filling of a Legendrian Λ that has p double points with action 0 and the existence of an embedded, Maslov-0, exact Lagrangian cobordism from p copies of a Hopf link to Λ. We then prove that a count of augmentations provides an obstruction to the existence of embedded, Maslov-0, exact Lagrangian cobordisms between Legendrian links. Joint work with Noemie Legout, Maylis Limouzineau, Emmy Murphy, Yu Pan and Lisa Traynor.
After an introduction to how to think about the mapping class groupand its cohomology, I will discuss a recent theorem of mine saying
that passing to the level-l subgroup does not change the rational cohomology in a stable range.
It is well known that the wave operators cos(t (−∆)) and sin(t (−∆)) are not bounded on Lp(Rn), for n≥2 and 1≤p≤∞, unless p=2 or t=0. In fact, for 1 < p < ∞ these operators are bounded from W2s(p),p to Lp(Rn) for s(p) := (n−1)/2 | 1/p − 1/2 |, and this exponent cannot be improved. This phenomenon is symptomatic of the behavior of Fourier integral operators, a class of oscillatory operators which includes wave propagators, on Lp(Rn).
In this talk, I will introduce a class of Hardy spaces HFIOp (Rn), for p ∈ [1,∞],on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on Lp(Rn).
However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on Lp(Rn). In particular, we shall indicate how one can use this invariance to obtain the optimal fixed-time Lp regularity for wave equations with rough coefficients. We shall also mention the connection of these spaces to the phenomenon of local smoothing.
This talk is based on joint work with Andrew Hassell and Pierre Portal (Aus- tralian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat- Sen University).
Sleep and wake states are driven by interactions of neuronal populations in many areas of the human brain, such as the brainstem, midbrain, hypothalamus, and basal forebrain. The timing of human sleep is strongly modulated by the 24 h circadian rhythm and the homeostatic sleep drive, the need for sleep that depends on the history of prior awakening. The parameters dictating the evolution of the homeostatic sleep drive may vary with development or interindividual characteristics and have been identified as important parameters for generating the transition from multiple sleeps to a single sleep episode per day. Features of the mean firing rate of the neurons in the suprachiasmatic nucleus (SCN), the central pacemaker in humans, may differ with seasonality. In this talk, I will present our analysis of changes in sleep patterning under variation of homeostatic and circadian parameters using a mathematical model for human sleep-wake regulation. I will also talk about the fundamental tools we employ to understand the dynamics of the model, such as the construction of a circle map that captures the timing of sleep onsets on successive days. Analysis of the structure and bifurcations in the map reveals changes in the average number of sleep episodes per circadian day in a period-adding-like structure caused by the separate or combined effects of circadian and homeostatic variation. Time permitting, I will talk about some of our current work on modeling sleep patterns in early childhood using experimental data.
I will consider the long-time influence of deterministic and stochastic perturbations of dynamical systems and diffusion processes with a first integral . A diffusion process on the Reeb graph of the first integral determines the long-time behavior of the perturbed system. In particular, I will consider stochasticity of long time behavior of deterministic systems close to a system with a conservation law. Which of the invariant measures of the non-perturbed system will be limiting for a given class of perturbations also will be discussed.
Note the different time!
I present some results concerning the size of magnetic fields that support zero modes for the three dimensional Dirac equation and related problems for spinor equations. Critical quantities measuring this size are the 3/2 norm of the magnetic field B and the 3 norm of the vector potential A. The point is that the spinor structure enters the analysis in a crucial way. This is joint work with Rupert Frank at LMU Munich.
I will review several parallel GPU-based approaches to better understand multistable dynamics of simple neural networks and global bifurcation unfolding of systems with deterministic chaos.
Quadratic forms and their hidden convexity have been studied for decades, dating back to famous theorems by Toeplitz-Hausdorff, Dines and Brickman. It has very rich connection to optimization via Yakubovich's S-lemma. I will introduce these results, as well as an ongoing work of obtaining convex hull via aggregations, where we introduced the closely related notion of hidden hyperplane convexity.
Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. We will discuss a construction of small symplectic caps, using ideas first laid out by Gay in 2002, for rational homology balls bounded by lens spaces. This allows us to explicitly understand embeddings of these rational balls in CP2 that were earlier understood only through almost toric fibrations. This is joint work with John Etnyre, Hyunki Min, and Lisa Piccirillo.
This is an ongoing project. We make use of two dual Lawrence polytopes $P$ and $P*$ of a graph $G$, to study invariants of the graph. The $h$-vector of the graphic (resp. cographic) matroid complex associated to $G$ coincides with the $h^*$-vector of the Lawrence polytope $P$ (resp. $P^*$). In general, the $h$-vector is an invariant defined for an abstract simplicial complex, which encodes the number of faces of different dimensions. The $h^*$-vector, a.k.a. the $\delta$-polynomial, is an invariant defined for a rational polytope, which is obtained by dilating the polytope. By dissecting the Lawrence polytopes, we may study the $h$-vectors associated to the graph $G$ at a finer level. In particular, we understand activities and reduced divisors of the graph $G$ in a more geometric way. I will try to make the talk self-contained.
The study of Poisson approximations of the process of recurrences to small subsets in the phase spaces of chaotic dynamical systems, started in 1991, have developed by now into a large active area of the dynamical systems theory. In this talk, I will present some new results. This is a joint work with Prof. Leonid Bunimovich.
Networks of coupled oscillators are studied in biology, chemistry, physics, and engineering. The Kuramoto model is a simple dynamical system that models the nonlinear interaction among coupled oscillators. It has received widespread attention since it is simple enough to be analyzed rigorously yet complex enough to exhibit interesting emergent behaviors.
One such emergent behavior is the spontaneous synchronization of oscillators into special configurations. In the past decades, rigorous analysis of such synchronization configurations has been the focus of intensive studies.
In this talk, we explore the new insight to this problem provided by an algebraic and tropical approach.
An outstanding problem for surface bundles over surfaces, closely related to the symplectic geography problem in dimension four, is to determine for which fiber and base genera there are examples with non-zero signatures. I will report on our recent progress (joint with M. Korkmaz), which resolves the problem for all fiber and base genera except for 18 pairs at the time of writing.
For steady two-dimensional incompressible flows with a single eddy (i.e. nested closed streamlines), Prandtl (1905) and Batchelor (1956) proposed that in the limit of vanishing viscosity the vorticity is constant in an inner region separated from the boundary layer. By constructing higher order approximate solutions of the Navier-Stokes equations and establishing the validity of Prandtl boundary layer expansion, we give a rigorous proof of the existence of Prandtl-Batchelor flows on a disk with the wall velocity slightly different from the rigid-rotation. The leading order term of the flow is the constant vorticity solution (i.e. rigid rotation) satisfying the Batchelor-Wood formula. For an annulus with wall velocities slightly different from the rigid-rotation, we also constructed Prandtl-Batchelor flows, whose leading order terms are rotating shear flows. This is a joint work with Chen Gao, Mingwen Fei and Tao Tao.
The list-$k$-coloring problem is to decide, given a graph $G$ and a list assignment $L$ of $G$ from $V(G)$ to subsets of $\{1,...,k\}$, whether $G$ has a coloring $f$ such that $f(v)$ in $L(v)$ for all $v$ in $V(G)$. The list-$k$-coloring problem is a generalization of the $k$-coloring problem. Thus for $k\geq 3$, both the $k$-coloring problem and the list-$k$-coloring problem are NP-Hard. This motivates studying the complexity of these problems restricted to graphs with a fixed forbidden induced subgraph $H$, which are called $H$-free graphs.
In this talk, I will present a polynomial-time algorithm to solve the list-5-coloring $H$-free graphs with $H$ being the union of $r$ copies of mutually disjoint 3-vertex paths. Together with known results, it gives a complete complexity dichotomy of the list-5-coloring problem restricted to $H$-free graphs. This is joint work with Sepehr Hajebi and Sophie Spirkl.
An order-n Latin square is an n by n array of n symbols such that each row and column contains each symbol exactly once. Latin squares were famously studied by Euler in the 1700s, and at present they are still a central object of study in modern extremal and probabilistic combinatorics. In this talk, I will give some history about Latin squares, share some simple-to-state yet notoriously difficult open problems, and present some of my own research on Latin squares.
A three-manifold is a space that locally looks like the Euclidean three-dimensional space. The study of three-manifolds has been at the heart of many beautiful constructions in low dimensional topology. This talk will provide a quick tour through some fundamental results about three-manifolds that were discovered between the late nineteenth century and the Fifties.
Recording: https://us02web.zoom.us/rec/share/cIdTfvS0tjar04MWv9ltWrVxAcmsUSFvDznprS...
We revisit the notion of noise stability in the hypercube and show how one can replace the usual heat semigroup with more general stochastic processes. We will then introduce a re-normalized Brownian motion, embedding the discrete hypercube into the Wiener space, and analyze the noise stability along its paths. Our approach leads to a new quantitative form of the 'Majority is Stablest' theorem from Boolean analysis and to progress on the 'most informative bit' conjecture of Kumar and Courtade.
The mathematical theory of the recovery of quantum states stored in a quantum memory, is intimately related to the subadditivity property of the entropy function, and the class of states known as quantum Markov chains. In this talk we will introduce some of the basic ideas of this area of quantum information theory. We discuss a theorem regarding recovery of a widely studied class of quantum states, the matrix product states, and its implication for the mutual information stored over separated regions of a one dimensional quantum memory. This is joint work with Pavel Svetlichnyy and Shivan Mittal.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
The three-body problem, on the dynamics of three masses under mutual gravity, serves as a model for the motion of a spacecraft relative to the Earth-Moon or Sun-Earth system. We describe the equations of motion for the three-body problem and the geometric objects that organize the dynamics: equilibriums points, periodic and quasi-periodic orbits, and their stable and unstable manifolds. As it turns out, trajectories that follow these manifolds require zero energy cost. We describe several methods to design low energy spacecraft trajectories from Earth to Moon, as well as maneuvers to change the inclination of the orbit of a satellite relative to the ecliptic. This is based on joint works with E. Belbruno, F. Topputo, A. Delshams, and P. Roldan.
Dense kernel matrices resulting from pairwise evaluations of a kernel function arise naturally in machine learning and statistics. Previous work in constructing sparse transport maps or sparse approximate inverse Cholesky factors of such matrices by minimizing Kullback-Leibler divergence recovers the Vecchia approximation for Gaussian processes. However, these methods often rely only on geometry to construct the sparsity pattern, ignoring the conditional effect of adding an entry. In this work, we construct the sparsity pattern by leveraging a greedy selection algorithm that maximizes mutual information with target points, conditional on all points previously selected. For selecting k points out of N, the naive time complexity is O(N k^4), but by maintaining a partial Cholesky factor we reduce this to O(N k^2). Furthermore, for multiple (m) targets we achieve a time complexity of O(N k^2 + N m^2 + m^3) which is maintained in the setting of aggregated Cholesky factorization where a selected point need not condition every target. We directly apply the selection algorithm to image classification and recovery of sparse Cholesky factors. By minimizing Kullback-Leibler divergence, we apply the algorithm to Cholesky factorization, Gaussian process regression, and preconditioning with the conjugate gradient, improving over k-nearest neighbors particularly in high dimensional, unusual, or otherwise messy geometries with non-isotropic kernel functions.
One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.
The Lovasz Local Lemma is a powerful tool to prove existence of combinatorial structures satisfying certain properties. In a constructive proof of the LLL, Moser and Tardos introduced a proof technique that is now referred to as the entropy compression method. In this talk I will describe the main idea of the method and apply it to a problem easily solved using the LLL. I will also describe recent applications of the idea to various graph coloring problems.
We study a particular distinguished component (the 'Hitchin component') of the space of surface group representations to SL(3,\R). In this setting, both Hitchin (via Higgs bundles) and the more ancient subject of affine spheres associate a bundle of holomorphic differentials over Teichmuller space to this component of the character variety. We focus on a ray of holomorphic differentials and provide a formula, tropical in appearance, for the asymptotic holonomy of the representations in terms of the local geometry of the differential. Alternatively, we show how the associated equivariant harmonic maps to a symmetric space converge to a harmonic map to a building, with geometry determined by the differential. All of this is joint work with John Loftin and Andrea Tamburelli, and all the constructions and definitions will be (likely briskly) explained.
Many fluid flows display at a wide range of space and time scales. Turbulent and multiphase flows can include small eddies or particles, but likewise large advected features. This challenge makes some degree of multi-scale modeling or homogenization necessary. Such models are restricted, though: they should be numerically accurate, physically consistent, computationally expedient, and more. I present two tools crafted for this purpose. First, the fast macroscopic forcing method (Fast MFM), which is based on an elliptic pruning procedure that localizes solution operators and sparse matrix-vector sampling. We recover eddy-diffusivity operators with a convergence that beats the best spectral approximation (from the SVD), attenuating the cost of, for example, targeted RANS closures. I also present a moment-based method for closing multiphase flow equations. Buttressed by a recurrent neural network, it is numerically stable and achieves state-of-the-art accuracy. I close with a discussion of conducting these simulations near exascale. Our simulations scale ideally on the entirety of ORNL Summit's GPUs, though the HPC landscape continues to shift.
The burning number $b(G)$ of a graph $G$ is the smallest integer $k$ such that $G$ can be covered by $k$ balls of radii respectively $0,\dots,k-1$, and was introduced independently by Brandenburg and Scott at Intel as a transmission problem on processors \cite{alon} and Bonato, Janssen and Roshanbin as a model for the spread of information in social networks.
The Burning Number Conjecture \cite{bonato} claims that $b(G)\leq \left\lceil\sqrt{n}\right\rceil$, where $n$ is the number of vertices of $G$. This bound tight for paths. The previous best bound for this problem, by Bastide et al. \cite{bastide}, was $b(G)\leq \sqrt{\frac{4n}{3}}+1$.
We prove that the Burning Number Conjecture holds asymptotically, that is $b(G)\leq (1+o(1))\sqrt{n}$.
Following a brief introduction to graph burning, this talk will focus on the general ideas behind the proof.
Random and irregular growth is all around us. We see it in the form of cancer growth, bacterial infection, fluid flow through porous rock, and propagating flame fronts. Simple models for these processes originated in the '50s with percolation theory and have since given rise to many new models and interesting mathematics. I will introduce a few models (percolation, invasion percolation, first-passage percolation, diffusion-limited aggregation, ...), along with some of their basic properties.
TBA
We estimate the Riesz basis (RB) bounds obtained in Hruschev, Nikolskii and Pavlov' s classical characterization of exponential RB. As an application, we improve previously known estimates of the RB bounds in some classical cases, such as RB obtained by an Avdonin type perturbation, or RB which are the zero-set of sine-type functions. This talk is based on joint work with S. Nitzan
The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.
Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.
(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others. The talk is suitable for both professors and students.)
The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.
Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.
(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others.
The talk is suitable for both professors and students.)
Elastic beam Hamiltonians on single-layer graphs are constructed out of Euler-Bernoulli beams, each governed by a scalar valued fourth-order Schrödinger operator equipped with a real symmetric potential. Unlike the second-order Schrödinger operator commonly applied in quantum graph literature, here the self-adjoint vertex conditions encode geometry of the graph by their dependence on angles at which edges are met. In this talk, I will first consider spectral properties of this Hamiltonian with periodic potentials on a special equal-angle lattice, known as graphene or honeycomb lattice. I will also discuss spectral properties for the same operator on lattices in the geometric neighborhood of graphene. This talk is based on a joint work with Mahmood Ettehad (University of Minnesota),https://arxiv.org/pdf/2110.05466.pdf.
A Coxeter group is a (not necessarily finite) group given by certain types of generators and relations. Examples of finite Coxeter groups include dihedral groups, symmetric groups, and reflection groups. They play an important role in various areas. In this talk, I will discuss why I am interested in Coxeter groups from a combinatorial perspective - the geometric concepts associated with the finite Coxeter groups form the language of Coxeter matroids, which are generalizations of ordinary matroids. In particular, finite Coxeter groups are related to Coxeter matroids in the same way as symmetric groups are related to ordinary matroids. The main reference for this talk is Chapter 5 of Borovik-Gelfand-White's book Coxeter Matroids. I will only assume basic group theory, but not familiarity with matroids.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
We will begin with a brief overview of several parallel and hybrid computing approaches including CUDA, OpenAcc, OpenMP, and OpenMPI, followed by a demonstration of how we can leverage these technologies to study complex dynamics arising from diverse nonlinear systems. First, we discuss multistable rhythms in oscillatory 4-cell central pattern generators (CPGs) of inhibitory coupled neurons. We show how network topology and intrinsic properties of the cells affect dynamics, and how even simple circuits can exhibit a variety of mono/multi-stable rhythms including pacemakers, half-center oscillators, multiple traveling-waves, fully synchronous states, as well as various chimeras. We then discuss symbolic methods and parametric sweeps to analyze isolated neuron dynamics such as bursting, tonic spiking and chaotic mixed-mode oscillations, the bifurcations that underlie transitions between activity types, as well as emergent network phenomena through synergistic interactions seen in realistic neural circuits and animal CPGs. We also demonstrate how such symbolic methods can help identify the universal principles governing both simple and complex dynamics, and chaotic structure in various Lorenz-like systems, their key self-similar organizing structures in 2D parameter space, as well as detailed computational reconstructions of 3D bifurcation surfaces.
In the problem of online portfolio selection as formulated by Cover (1991), the trader repeatedly distributes her capital over $ d $ assets in each of $ T > 1 $ rounds, with the goal of maximizing the total return. Cover proposed an algorithm called Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with
an $ O(d\log(T)) $ regret in terms of the logarithmic return. Without imposing any restrictions on the market, this guarantee is known to be worst-case optimal, and no other algorithm attaining it has been discovered so far. Unfortunately, Cover's algorithm crucially relies on computing the expectation over certain log-concave density in R^d, so in a practical implementation this expectation has to be approximated via sampling, which is computationally challenging. In particular, the fastest known implementation, proposed by Kalai and Vempala in 2002, runs in $ O( d^4 (T+d)^{14} ) $ per round, which rules out any practical application scenario. Proposing a practical algorithm with a near-optimal regret is a long-standing open problem. We propose an algorithm for online portfolio selection with a near-optimal regret guarantee of $ O( d \log(T+d) ) $ and the runtime of only $ O( d^2 (T+d) ) $ per round. In a nutshell, our algorithm is a variant of the follow-the-regularized-leader scheme, with a time-dependent regularizer given by the volumetric barrier for the sum of observed losses. Thus, our result gives a fresh perspective on the concept of volumetric barrier, initially proposed in the context of cutting-plane methods and interior-point methods, correspondingly by Vaidya (1989) and Nesterov and Nemirovski (1994). Our side contribution, of independent interest, is deriving the volumetrically regularized portfolio as a variational approximation of the universal portfolio: namely, we show that it minimizes Gibbs's free energy functional, with accuracy of order $ O( d \log(T+d) ) $. This is a joint work with Remi Jezequel and Pierre Gaillard.
Dirac proved that every $n$-vertex graph with minimum degree at least $n/2$ contains a hamiltonian cycle. Moreover, every graph with minimum degree $k \geq 2$ contains a cycle of length at least $k+1$, and this can be further improved if the graph is 2-connected. In this talk, we prove analogs of these theorems for hypergraphs. That is, we give sharp minimum degree conditions that imply the existence of long Berge cycles in uniform hypergraphs. This is joint work with Alexandr Kostochka and Grace McCourt.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz... />
In this talk I will generalize a simple trick to produce splitting for the separatrices of (the time-one map of) a simple pendulum, to hyperbolic tori of any dimension $m\geq 2$. The examples will be constructed in the Gevrey class, and the splitting is bounded from below by a term of the form $\exp (-c(1/\eps)^a)$, where $a=\frac{1}{2(\alpha-1)(m-2)}$. This will be compared to usual upper bounds in the same setting.
The analysis of tensor data, i.e., arrays with multiple directions, has become an active research topic in the era of big data. Datasets in the form of tensors arise from a wide range of scientific applications. Tensor methods also provide unique perspectives to many high-dimensional problems, where the observations are not necessarily tensors. Problems in high-dimensional tensors generally possess distinct characteristics that pose great challenges to the data science community.
In this talk, we discuss several recent advances in statistical tensor learning and their applications in computational imaging, social network, and generative model. We also illustrate how we develop statistically optimal methods and computationally efficient algorithms that interact with the modern theories of computation, high-dimensional statistics, and non-convex optimization.
This presentation, which is based on the work Sun [2], is dedicated to describing the complete integrability of the Benjamin–Ono (BO) equation on the line when restricted to every N-soliton mani- fold, denoted by UN . We construct (generalized) action–angle coordinates which establish a real analytic symplectomorphism from UN onto some open convex subset of R2N and allow to solve the equation by quadrature for any such initial datum. As a consequence, UN is the universal covering of the manifold of N-gap potentials for the BO equation on the torus as described by G ́erard–Kappeler [1]. The global well-posedness of the BO equation on UN is given by a polynomial characterization and a spectral char- acterization of the manifold UN . Besides the spectral analysis of the Lax operator of the BO equation and the shift semigroup acting on some Hardy spaces, the construction of such coordinates also relies on the use of a generating functional, which encodes the entire BO hierarchy. The inverse spectral formula of an N-soliton provides a spectral connection between the Lax operator and the infinitesimal generator of the very shift semigroup. The construction of action–angle coordinates for each UN constitutes a first step towards the soliton resolution conjecture of the BO equation on the line.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Operational weather and ocean forecasting proceeds as a sequence of time intervals. During each interval numerical models produce a forecast, observations are collected and a comparison between the two is made. This comparison is used, in a process called data assimilation (DA), to construct observation-informed initial conditions for the forecast in the next time interval. Many DA algorithms are in use, but they all share the need to solve a high-dimensional (>1010) system of linear equations. Constructing and solving this system in the limited amount of time available between the reception of the observations and the start of the next time interval is highly non-trivial for three reasons. 1) As the numerical models are computationally demanding, it is generally impossible to construct the full linear system. 2) Its high dimensionality makes it impossible to store the system as a matrix in memory. Consequently, it is not possible to directly invert it. 3) The operational time-constraints strongly limit the number of iterations that can be used by iterative linear solvers. By adapting DA algorithms to use parallelization, it is possible to leverage the computational power of superclusters to construct a high-rank approximation to the linear system and solve it using less then ~20 iterations. In this talk, I will first introduce the two most popular families of DA algorithms: Kalman filters and variational DA. After this, I will discuss some of the adaptations that have been developed to enable parallelization. Among these are ensemble Kalman filters, domain localization, the EVIL (Ensemble Variational Integrated Localized) and saddle point algorithms.
Federated learning is a distributed learning paradigm where multiple agents, each only with access to local data, jointly learn a global model. There has recently been an explosion of research aiming not only to improve the accuracy rates of federated learning, but also provide certain guarantees around social good properties such as total error or fairness. In this talk, I describe two papers analyzing federated learning through the lens of cooperative game theory (both joint with Jon Kleinberg).
In the first paper, we discuss fairness in federated learning, which relates to how error rates differ between federating agents. In this work, we consider two notions of fairness: egalitarian fairness (which aims to bound how dissimilar error rates can be) and proportional fairness (which aims to reward players for contributing more data). For egalitarian fairness, we obtain a tight multiplicative bound on how widely error rates can diverge between agents federating together. For proportional fairness, we show that sub-proportional error (relative to the number of data points contributed) is guaranteed for any individually rational federating coalition. The second paper explores optimality in federated learning with respect to an objective of minimizing the average error rate among federating agents. In this work, we provide and prove the correctness of an efficient algorithm to calculate an optimal (error minimizing) arrangement of players. Building on this, we give the first constant-factor bound on the performance gap between stability and optimality, proving that the total error of the worst stable solution can be no higher than 9 times the total error of an optimal solution (Price of Anarchy bound of 9).
Relevant Links: https://arxiv.org/abs/2010.00753, https://arxiv.org/abs/2106.09580, https://arxiv.org/abs/2112.00818
Bio:
Kate Donahue is a fifth year computer science PhD candidate at Cornell advised by Jon Kleinberg. She works on algorithmic problems relating to the societal impact of AI such as fairness, human/AI collaboration and game-theoretic models of federated learning. Her work has been supported by an NSF fellowship and recognized by a FAccT Best Paper award. During her PhD, she has interned at Microsoft Research, Amazon, and Google.
One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz... />
In this talk, I will present results concerning the existence and the precise description of periodic solutions of the Navier-Stokes equations with a time- independent forcing, obtained in collaboration with Jan Bouwe van den Berg (VU Amsterdam), Jean-Philippe Lessard (McGill) and Lennaert van Veen (Ontario TU).
These results are obtained by combining numerical simulations, a posteriori error estimates, interval arithmetic, and a fixed point theorem applied to a quasi-Newton operator, which yields the existence of an exact solution in a small and explicit neighborhood of the numerical one.
I will first introduce the main ideas and techniques required for this type of approach on a simple example, and then discuss their usage in more complex settings like the Navier-Stokes equations.
This talk is part of the Atlanta Combinatorics Colloquium. Note the time (4pm) and location (Instructional Center 105).
It is easy to partition the vertices of any graph into two sets where each vertex has at least as many neighbours across as on its own side; take any maximal cut! Can we do the opposite? This is not possible in general, but Füredi conjectured in 1988 that it should nevertheless be possible on a random graph. I shall talk about our recent proof of Füredi's conjecture: with high probability, the random graph $G(n,1/2)$ on an even number of vertices admits a partition of its vertex set into two parts of equal size in which $n−o(n)$ vertices have more neighbours on their own side than across.
Joint Topology Seminar @ GaTech
Given n points on a disk, we will describe how to build an A-infinity category based on the instanton Floer complex of links, and explain why it is finitely generated. This is based on work in progress with Ko Honda.
Joint Topology Seminar @ GaTech
There exist many different diagrammatic descriptions of 4-manifolds, with the usual claim that "such and such a diagram uniquely determines a smooth 4-manifold up to diffeomorphism". This raises higher order questions: Up to what diffeomorphism? If the same diagram is used to produce two different 4-manifolds, is there a diffeomorphism between them uniquely determined up to isotopy? Are such isotopies uniquely determined up to isotopies of isotopies? Such questions become important if one hopes to use "diagrams" to study spaces of diffeomorphisms between manifolds. One way to achieve these higher order versions of uniqueness is to ask that a diagram uniquely determine a contractible space of 4-manifolds (i.e. a 4-manifold bundle over a contractible space). I will explain why some standard types of diagrams do not do this and give at least one type of diagram that does do this.
We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.
Every multi-soliton manifold of the Benjamin–Ono equation on the line is invariant under the Benjamin–Ono flow. Its generalized action–angle coordinates allow to solve this equation by quadrature and we have the explicit expression of every multi-solitary wave.
The Matroid Minors Project of Geelen, Gerards, and Whittle describes the structure of minor-closed classes of matroids representable by a matrix over a fixed finite field. To use these results to study specific classes, it is important to study the matroids in the class containing spanning cliques. A spanning clique of a matroid M is a complete-graphic restriction of M with the same rank as M.
In this talk, we will describe the structure of dyadic matroids with spanning cliques. The dyadic matroids are those matroids that can be represented by a real matrix each of whose nonzero subdeterminants is a power of 2, up to a sign. A subclass of the dyadic matroids is the signed-graphic matroids. In the class of signed-graphic matroids, the entries of the matrix are determined by a signed graph. Our result is that dyadic matroids with spanning cliques are signed-graphic matroids and a few exceptional cases.
The main results in this talk will come from joint work with Ben Clark, James Oxley, and Stefan van Zwam. This talk will include a brief introduction to matroids.
Abstract: How big is a group? One possible notion of the size of the group is the cohomological dimension, which is the largest n for which a group G can have non—trivial cohomology in degree n, possibly with twisted coefficients. Following the work of Bestvina, Bux and Margalit, we compute the cohomological dimension of the terms Johnson filtration of a closed surface. No background is required for this talk.
Let $H$ be a separable Hilbert space and let $\{x_n\}$ be a Bessel sequence or a frame for $H$ which does not contain any zero elements. We say that $\{x_n\}$ is a normalizable Bessel sequence or normalizable frame if the normalized sequence $\{x_n/||x_n||\}$ remains a Bessel sequence or frame. In this talk, we will present characterizations of normalizable and non-normalizable frames . In particular, we prove that normalizable frames can only have two formulations. Perturbation theorems tailored for normalizable frames will be also presented. Finally, we will talk about some open questions related to the normalizable frames.
I will talk about some recent work on the stability problem of shear flows and vortices as solutions of the Euler equations in 2D. Our results include nonlinear stability theorems for monotonic shear flows and point vortices, as well as linear stability theorems for more general flows. This is joint work with Hao Jia.
In the late 20th century, statistical physicists introduced a chemical reaction model called ballistic annihilation. In it, particles are placed randomly throughout the real line and then proceed to move at independently sampled velocities. Collisions result in mutual annihilation. Many results were inferred by physicists, but it wasn’t until recently that mathematicians joined in. I will describe my trajectory through this model. Expect tantalizing open questions.
With very minor assumptions, I show that periodic orbits in
an ODE can persist under (singular) perturbations of including a delay
term. These perturbations change the phase space from finite to
infinite dimensions. The results apply to electrodynamics and give new
approaches to handle state-dependent, small, nested, and distributed
delays.
I will discuss and explain some motivations, the new methods, sketches
of the proofs, and possible applications. I will end the talk giving
some ideas of work in progress and possible future works.
We will introduce the foundations of model theory, by defining languages, models, and theories. Then we will look at a couple proofs of the compactness theorem, state Gödel's completeness theorem, and prove that any planar graph is four colorable. Expect a lot of examples, and I hope everyone comes away understanding the foundations of this wonderful theory.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
In this talk we will discuss a shooting method designed for solving two point boundary value problems in a setting where a system has integrals of motion. We will show how it can be applied to obtain certain families of orbits in the circular restricted three body problem. These include transverse ejection/collisions from one primary body to the other, families of periodic orbits, orbits passing through collision, and orbits connecting fixed points to ejections or collisions.
This is joint work with Shane Kepley and Jason Mireles James.
In this talk we will go over the Hardly-Littlewood circle method, and the major and minor arc decomposition. We shall then see a toy-example of the High-Low decomposition, and proceed with defining sparse families and sparse domination. We will conclude by explaining why sparse domination is of interest to us when studying $L^p$ bounds. This talk aims to be accessible to people without a strong background in the area. Some basic concepts of real and harmonic analysis will be useful (e.g. $L^p$ spaces, Fourier transform, Holder inequality, the Hardy-Littlewood Maximal function, etc)
For graphs with maximum degree $\Delta$, a greedy algorithm shows $\chi(G) \leq \Delta + 1$. Brooks improved this to $\chi(G) \leq \Delta$ when $G$ has no cliques of size $\Delta + 1$, provided $\Delta \geq 3$. If is conjectured that if one forbids other graphs, the bound can be pushed further: for instance, Alon, Krivelevich, and Sudakov conjecture that, for any graph $F$, there is a constant $c(F) > 0$ such that $\chi(G) \leq (c(F) + o(1)) \Delta / \log\Delta$ for all $F$-free graphs $G$ of maximum degree $\Delta$. The only graphs $F$ for which this conjecture has been verified so far---by Alon, Krivelevich, and Sudakov themselves---are the so-called almost bipartite graphs, i.e., graphs that can be made bipartite by removing at most one vertex. Equivalently, a graph is almost bipartite if it is a subgraph of the complete tripartite graph $K_{1,t,t}$ for some $t \in \N$. The best heretofore known upper bound on $c(F)$ for almost bipartite $F$ is due to Davies, Kang, Pirot, and Sereni, who showed that $c(K_{1,t,t}) \leq t$. We prove that in fact $c(F) \leq 4$ for any almost bipartite graph $F$, thus making the bound independent of $F$ in all the known cases of the conjecture. We also establish a more general version of this result in the setting of DP-coloring (also known as correspondence coloring), which we give a gentle introduction to. Finally, we consider consequences of this result in the context of sublinear algorithms.
This is joint work with Anton Bernshteyn and Abhishek Dhawan.
One of the most interesting and surprising features of manifold topology is the existence of topological 4-manifold that admit infinitely many smooth structures. In these talks I will discuss what is known about these “exotic” smooth structures on open manifolds, starting with R^4 and then moving on to other open 4-manifolds. We will also go over various constructions and open questions about these manifolds.
The Potts model is a distribution on q-colorings of a graph, used to represent spin configurations of a system of particles. Intuitively we expect most configurations to be "solid-like" at low temperatures and "gas-like" at high temperatures. We prove a precise version of this statement for d-regular expander graphs. We also consider the question of whether or not there are efficient algorithms for approximate counting and sampling from the model, and show that such algorithms exist at almost all temperatures. In this talk, I will introduce the different tools we use in our proofs, which come from both statistical physics (polymer models, cluster expansion) and combinatorics (a new container-like result, Karger's randomized min-cut algorithm). This is joint work with Charlie Carlson, Ewan Davies, Nicolas Fraiman, Alexandra Kolla, and Aditya Potukuchi.
The phylogenetic birth-death process is a probabilistic model of evolution that
is widely used to analyze genetic data. In a striking result, Louca & Pennell
(Nature, 2020) recently showed that this model is statistically unidentifiable,
meaning that an arbitrary number of different evolutionary hypotheses are
consistent with any given data set. This grave finding has called into question
the conclusions of a large number of evolutionary studies which relied on this
model.
In this talk, I will give an introduction to the phylogenetic birth-death
process, and explain Louca and Pennell's unidentifiability result. Then, I will
describe recent positive results that we have obtained, which establish that, by
restricting the evolutionary hypothesis space in certain biologically plausible
ways, statistical identifiability is restored. Finally, I will discuss some
complementary hardness-of-estimation results which show that, even in identifiable
model classes, obtaining reliable inferences from finite amounts of data may be
extremely challenging.
No background in this area is assumed, and the talk will be accessible to a
mathematically mature audience. This is joint work with Brandon Legried.
Zoom link: https://gatech.zoom.us/j/99936668317
I will highlight recent interplay between problems in extremal combinatorics and real algebraic geometry. This sheds a new light on undecidability of graph homomorphism density inequalities in extremal combinatorics, trace inequalities in linear algebra, and symmetric polynomial inequalities in real algebraic geometry. All of the necessary notions will be introduced in the talk. Joint work with Jose Acevedo, Sebastian Debus and Cordian Riener.
I'll talk about some 2D billiards, the most visual class of dynamical systems, where orbits (rays) move along straight lines within a billiard table with elastic reflections off the boundary. Elliptic flowers are built “around" convex polygons, and the boundary of corresponding billiard tables consists of the arcs of ellipses. It will be explained why some classes of such elliptic flowers demonstrate a never expected before dynamics, and why it raises a variety of (seemingly new) questions in geometry (particularly in 3D), in bifurcation theory (particularly about singularities of wave fronts and creation of wave trains), in statistical mechanics, quantum chaos, and perhaps some more. The talk will be concluded by showing a free movie. Everything (including various definitions of ellipses) will be explained/reminded.
We study the local and global dynamics of mean curvature flow with cylindrical singularities. We find the most generic dynamic behavior of such singularities, and show that the singularities with the most generic dynamic behavior are robust. We also show that the most generic singularities are isolated and type-I. Among applications, we prove that the singular set structure of the generic mean convex mean curvature flow has certain patterns, and the level set flow starting from a generic mean convex hypersurface has low regularity. This is joint work with Jinxin Xue (Tsinghua University)
Through the pioneering numerical computations of Fermi-Pasta-Ulam (mid 50s) and Kruskal-Zabusky (mid 60s) it was observed that nonlinear equations modeling wave propagation asymptotically decompose as a superposition of “traveling waves” and “radiation”. Since then, it has been a widely believed (and supported by extensive numerics) that “coherent structures” together with radiations describe the long-time asymptotic behavior of generic solutions to nonlinear dispersive equations. This belief has come to be known as the “soliton resolution conjecture”. Roughly speaking it tells that, asymptotically in time, the evolution of generic solutions decouples as a sum of modulated solitary waves and a radiation term that disperses. This remarkable claim establishes a drastic “simplification” to the complex, long-time dynamics of general solutions. It remains an open problem to rigorously show such a description for most dispersive equations. After an informal introduction to dispersive equations, I will illustrate how to understand the long-time behavior solutions to dispersive waves via various results I obtained over the years.
We consider a random walk on the $d\ge 3$ dimensional discrete torus starting from vertices chosen independently and uniformly at random. In this talk, we discuss the fluctuation behavior of the size of the range of the random walk trajectories at a time proportional to the size of the torus. The proof relies on a refined analysis of tail estimates for hitting time. We also discuss related results and open problems. This is based on joint work with Partha Dey.
The mathematically rigorous derivation of nonlinear Boltzmann equations from first principles in interacting physical systems is an extremely active research area in Analysis, Mathematical Physics, and Applied Mathematics. In classical physical systems, rigorous results of this type have been obtained for some models. In the quantum case on the other hand, the problem has essentially remained open. In this talk, I will explain how a cubic quantum Boltzmann equation arises within the fluctuation dynamics around a Bose-Einstein condensate, within the quantum field theoretic description of an interacting Boson gas. This is based on joint work with Michael Hott.
Join Zoom Meeting at https://gatech.zoom.us/j/92873362365
We will introduce some basic notions needed to talk about the question of decidability for roots of polynomials with coefficients in a specified ring R in the sense of Hilbert's tenth problem with an emphasis on rings of number theoretic interest. We will also attempt to give an overview of the literature on the topic and recent lines of work.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Oscillations are ubiquitous in the brain, but their role is not completely understood. In this talk we will focus on the study of oscillations in neuronal networks. I will introduce some neuronal models and I will show how tools from dynamical systems theory, such as the parameterization method for invariant manifolds or the separatrix map, can be used to provide a thorough analysis of the oscillatory dynamics. I will show how the conclusions obtained may contribute to unveiling the role of oscillations in certain cognitive tasks.
In recent years, there has been increased interest in using quantum computing for the purposes of solving problems in combinatorial optimization. No prior knowledge of quantum computing is necessary for this talk; in particular, the talk will be divided into three parts: (1) a gentle high-level introduction to the basics of quantum computing, (2) a general framework for solving combinatorial optimization problems with quantum computing (the Quantum Approximate Optimization Algorithm introduced by Farhi et al.), (3) and some recent results that my colleagues and I have found. Our group has looked at the Max-Cut problem and have developed a new quantum algorithm that utilizes classically-obtained warm-starts in order to improve upon already-existing quantum algorithms; this talk will discuss both theoretical and experimental results associated with our approach with our main results being that we obtain a 0.658-approximation for Max-Cut, our approach provably converges to the Max-Cut as a parameter (called the quantum circuit depth) increases, and (on small graphs) are approach is able to empirically beat the (classical) Goemans-Williamson algorithm at a relatively low quantum circuit-depth (i.e. using a small amount of quantum resources). This work is joint with Jai Moondra, Bryan Gard, Greg Mohler, and Swati Gupta.
Let $A$ be drawn uniformly at random from the set of all $n \times n$ symmetric matrices with entries in $\{-1,1\}$. What is the probability that $A$ is singular? This is a classical problem at the intersection of probability and combinatorics. I will give an introduction to this type of question and sketch a proof that the singularity probability of $A$ is exponentially small in $n$. This is joint work with Marcelo Campos, Marcus Michelen and Julian Sahasrabudhe.
One of the most beautiful aspects of math is the interplay between its different fields. We will discuss one such interaction by studying topology using tools from combinatorics and group theory. In particular, given a surface (two-dimensional manifold) S, we construct the curve complex of S, which is a graph that encodes topological data about the surface. We will then state a seminal result of Ivanov: the symmetries of a surface S are in a natural bijection with the symmetries of its curve complex. In the direction of the proof of Ivanov's result, we will touch on some tools we have when working with infinite graphs.
Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
We show that for any natural number n, the set of domains containing absolutely periodic orbits of order n are dense in the set of bounded strictly convex domains with smooth boundary. The proof that such an orbit exists is an extension to billiard maps of the results of a paper by Gonchenko, Shilnikov, and Turaev, where it is proved that such maps are dense in Newhouse domains in regions of real-analytic area-preserving two-dimensional maps. Our result is a step toward disproving a conjecture that no absolutely periodic billiard orbits of infinite order exist in Euclidean billiards and is also an indication that Ivrii's Conjecture about the measure set of periodic orbits may not be true.
The ring of multivariate polynomials carries a natural action of the symmetric group. Quotienting by the ideal generated by the polynomials which are invariant under this action yields the "coinvariant algebra," an object with many beautiful algebraic and combinatorial properties. We will survey these properties and then discuss recent generalizations where the multivariate polynomials may contain anti-commuting ("superspace") variables. This talk is based on joint work with Brendon Rhoades.
Morse theory establishes a celebrated link between classical gradient dynamics and the topology of the
underlying phase space. It provided the motivation for two independent developments. On the one hand, Conley's
theory of isolated invariant sets and Morse decompositions, which is a generalization of Morse theory, is able
to encode the global dynamics of general dynamical systems using topological information. On the other hand,
Forman's discrete Morse theory on simplicial complexes, which is a combinatorial version of the classical
theory, and has found numerous applications in mathematics, computer science, and applied sciences.
In this talk, we introduce recent work on combinatorial topological dynamics, which combines both of the
above theories and leads as a special case to a dynamical Conley theory for Forman vector fields, and more
general, for multivectors. This theory has been developed using the general framework of finite topological
spaces, which contain simplicial complexes as a special case.
How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.
In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.
I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.
https://gatech.zoom.us/my/margalit?pwd=b3RhY3pVZUdlRUR3S1FLZzhFR1RVUT09
In recent years tremendous progress was made in understanding the ``inviscid damping" phenomenon near shear flows and vortices, which are steady states for the 2d incompressible Euler equation, especially at the linearized level. However, in real fluids viscosity plays an important role. It is natural to ask if incorporating the small but crucial viscosity term (and thus considering the Navier Stokes equation in a high Reynolds number regime instead of Euler equations) could change the dynamics in any dramatic way. It turns out that for the perturbative regime near a spectrally stable monotonic shear flows in an infinite periodic channel (to avoid boundary layers and long wave instabilities), we can prove uniform-in-viscosity inviscid damping. The proof introduces techniques that provide a unified treatment of the classical Orr-Sommerfeld equation in a way analogous to Rayleigh equations.
Suppose n runners are running on a circular track of circumference 1, with all runners starting at the same time and place. Each runner proceeds at their own constant speed. We say that a runner is lonely at some point in time if the distance around the track to the nearest other runner is at least 1/n. For example, if there two runners then there will come a moment when they are at anitpodal points on the track, and at this moment both runners are lonely. The lonely runner conjecture asserts that for every runner there is a point in time when that runner is lonely. This conjecture is over 50 years old and remains widely open.
A coprime matching of two sets of integers is a matching that pairs every element of one set with a coprime element of the other set. We present a recent partial result on the lonely runner conjecture. Coprime matchings of intervals of integers play an central role in the proof of this result.
Joint work with Fei Peng
Zoom link to the seminar: https://gatech.zoom.us/j/91330848866
I will show how to construct a numerical scheme for solutions to linear Dirichlet-Poisson boundary problems which does not suffer of the curse of dimensionality. In fact we show that as the dimension increases, the complexity of this scheme increases only (low degree) polynomially with the dimension. The key is a subtle use of walk on spheres combined with a concentration inequality. As a byproduct we show that this result has a simple consequence in terms of neural networks for the approximation of the solution. This is joint work with Iulian Cimpean, Arghir Zarnescu, Lucian Beznea and Oana Lupascu.
The classical Keller-Lieb-Thirring inequality bounds the ground state energy of a Schrödinger operator by a Lebesgue norm of the potential. This problem can be rewritten as a minimization problem for the Rayleigh quotient over both the eigenfunction and the potential. It is then straightforward to see that the best potential is a power of the eigenfunction, and the optimal eigenfunction satisfies a nonlinear Schrödinger equation.
This talk concerns the analogous question for the smallest eigenvalue in the gap of a massive Dirac operator. This eigenvalue is not characterized by a minimization problem. By using a suitable Birman-Schwinger operator, we show that for sufficiently small potentials in Lebesgue spaces, an optimal potential and eigenfunction exists. Moreover, the corresponding eigenfunction solves a nonlinear Dirac equation.
This is joint work with Jean Dolbeaults, David Gontier and Fabio Pizzichillo
Join Zoom Meeting: https://gatech.zoom.us/j/91396672718
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Weather modeling in conjunction with Data Assimilation (DA) has proven to provide effective weather forecasts that can both help you plan your day to save your life. We often refer to the combination of weather models and DA as Numerical Weather Prediction (NWP). One of the most widely employed DA methods in NWP is a variational method called 4d-Var. In this method, a cost function involving the model background error and a series of observations over time is minimized to find the best initial condition from which to run your model so that model forecast is consistent with observations. 4d-Var has been shown to provide the most reliable weather forecasts to date, but is not without its pitfalls. In particular, 4d-Var depends heavily on a tangent linear model (TLM) and an adjoint to the tangent linear model. While conceptually simple, coding these two elements is extremely time intensive and difficult. A small change in the larger weather model can induce months of work on its TLM and adjoint delaying the benefits of improvements on the model side. In this talk I will introduce the 4d-var method in general and present work on a Hybrid Tangent Linear Model (HTLM) developed in [Payne 2021] which is aimed at improving TLMs as well as allowing the use of incomplete TLMs when model physics changes. I will also touch on the Joint Effort for Data Integration (JEDI) project which now includes an HTLM and how you can use JEDI for DA.
Robert Tarjan is the James S. McDonnell Distinguished University Professor of Computer Science at Princeton University. He has held academic positions at Cornell, Berkeley, Stanford, and NYU, and industrial research positions at Bell Labs, NEC, HP, Microsoft, and Intertrust Technologies. He has invented or co-invented many of the most efficient known data structures and graph algorithms. He was awarded the first Nevanlinna Prize from the International Mathematical Union in 1982 for “for outstanding contributions to mathematical aspects of information science,” the Turing Award in 1986 with John Hopcroft for “fundamental achievements in the design and analysis of algorithms and data structures,” and the Paris Kanellakis Award in Theory and Practice in 1999 with Daniel Sleator for the invention of splay trees. He is a member of the U.S. National Academy of Sciences, the U. S. National Academy of Engineering, the American Academy of Arts and Sciences, and the American Philosophical Society. <br />
Data structures are everywhere in computer software. Classical data structures are specially designed to make each individual operation fast. A more flexible approach is to design the structure so that it adapts to its use. This idea has produced data structures that perform well in practice and have surprisingly good performance guarantees. In this talk I’ll review some recent work on such data structures, specifically on self-adjusting search trees and self-adjusting heaps.
In-person. Streaming available via zoom: Zoom link: https://us06web.zoom.us/j/83392531099?pwd=UHh2MDFMcGErbzFtMHBZTmNZQXM0dz09
We introduce a large family of one-dimensional full branch maps which generalize the classical “intermittency maps” by admitting two neutral fixed points and possibly also critical points and/or singularities. We study the statistical properties of these maps for various parameter values, including the existence (and non-existence) of physical measures, and their properties such as decay of correlations and limit theorems. In particular we describe a new mechanism for relatively persistent non-statistical chaotic dynamics. This is joint work with Douglas Coates and Muhammad Mubarak.
We will talk a little about realizing automorphisms of a free group as graph maps and how to use Stallings folds to study them.
Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra. Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear. In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions. These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.
Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.
Cohomology groups of moduli spaces of curves are fruitfully studied from several mathematical perspectives, including geometric group theory, stably homotopy theory, and quantum algebra. Algebraic geometry endows these cohomology groups with additional structures (Hodge structures and Galois representations), and the Langlands program makes striking predictions about which such structures can appear. In this talk, I will present recent results inspired by, and in some cases surpassing, such predictions. These include the vanishing of odd cohomology on moduli spaces of stable curves in degrees less than 11, generators and relations for H^11, and new constructions of unstable cohomology on M_g.
Based on joint work with Jonas Bergström and Carel Faber; with Sam Canning and Hannah Larson; with Melody Chan and Søren Galatius; and with Thomas Willwacher.
We will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina which hopes to answer the question: What is “Big Out(Fn)”? This group will consist of proper homotopy classes of proper homotopy equivalences of locally finite, infinite graphs. We will then discuss some classification theorems related to the coarse geometry of these groups. This is joint work with Hannah Hoganson and Sanghoon Kwak.
Gaussian processes (GPs) are widely employed as versatile modeling and predictive tools in spatial statistics, functional data analysis, computer modeling and diverse applications of machine learning. They have been widely studied over Euclidean spaces, where they are specified using covariance functions or covariograms for modelling complex dependencies. There is a growing literature on GPs over Riemannian manifolds in order to develop richer and more flexible inferential frameworks. While GPs have been extensively studied for asymptotic inference on Euclidean spaces using positive definite covariograms, such results are relatively sparse on Riemannian manifolds. We undertake analogous developments for GPs constructed over compact Riemannian manifolds. Building upon the recently introduced Matérn covariograms on a compact Riemannian manifold, we employ formal notions and conditions for the equivalence of two Matérn Gaussian random measures on compact manifolds to derive the microergodic parameters and formally establish the consistency of their maximum likelihood estimates as well as asymptotic optimality of the best linear unbiased predictor.
Each point x in Gr(r, n) corresponds to an r × n matrix Ax which gives rise to a matroid Mx on its columns. Gel’fand, Goresky, MacPherson, and Serganova showed that the sets {y ∈ Gr(r, n)|My = Mx} form a stratification of Gr(r, n) with many beautiful properties. However, results of Mnëv and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals Ix of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann-Cayley algebra may be used to derive non-trivial elements of Ix geometrically when the combinatorics of the matroid is sufficiently rich.
A distinctive feature of nonlinear evolution equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and, as a consequence, predicting blowup and understanding its nature is a central problem of the modern analysis of nonlinear PDEs.
In this talk we concentrate on wave maps – a geometric nonlinear wave equation – and we discuss the existence and stability of self-similar solutions, as in all higher dimensions they appear to drive the generic blowup behavior. We outline a novel framework for studying global-in-space stability of such solutions; we then men-tion some long-awaited results that we thereby obtained, and, finally, we discuss the new mathematical challenges that our approach generates.
The chromatic index $\chi'(G)$ of a graph $G$ is the least number of colors assigned to the edges of $G$ such that no two adjacent edges receive the same color. The total chromatic number $\chi''(G)$ of a graph $G$ is the least number of colors assigned to the edges and vertices of $G$ such that no two adjacent edges receive the same color, no two adjacent vertices receive the same color and no edge has the same color as its two endpoints. The chromatic index and the total chromatic number are two of few fundamental graph parameters, and their correlation has always been a subject of intensive study in graph theory.
By definition, $\chi'(G) \le \chi''(G)$ for every graph $G$. In 1984, Goldberg conjectured that for any multigraph $G$, if $\chi'(G) \ge \Delta(G) +3$ then $\chi''(G) = \chi'(G)$. We show that Goldberg's conjecture is asymptotically true. More specifically, we prove that for a multigraph $G$ with maximum degree $\Delta$ sufficiently large, $\chi''(G) = \chi'(G)$ provided $\chi'(G) \ge \Delta + 10\Delta^{35/36}$. When $\chi'(G) \ge \Delta(G) +2$, the chromatic index $\chi'(G)$ is completely determined by the fractional chromatic index. Consequently, the total chromatic number $\chi''(G)$ can be computed in polynomial-time in this case.
We study reducible surgeries on knots in S^3, developing thickness bounds for L-space knots that admit reducible surgeries and lower bounds on the slice genus of general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots. Our techniques involve the d-invariants and mapping cone formula from Heegaard Floer homology. This is joint work with Holt Bodish.
Let M be the 3-manifold obtained by r-surgery on the right handed trefoil knot. Classification of contact structures on such manifolds have been mostly understood for r \geq 1 and r=0. Etnyre-Min-Tosun has an upcoming work on the classification of the tight contact structures for all r. The fillability of contact structures on M is mostly understood if r is not between 0 and 1/2. In this talk, we will discuss the fillability of the contact structures M for 0
We will discuss two types of structured multi-objective optimization programs. In the first, the goal is to minimize a sum of functions described by a graph: each function is associated with a vertex, and there is an edge between vertices if two functions share a subset of their variables. Problems of this type arise in state estimation problems, including simultaneous localization and mapping (SLAM) in robotics, tracking, and streaming reconstruction problems in signal processing. We will show that under mild smoothness conditions, these types of problems exhibit a type of locality: if a node is added to the graph (changing the optimization problem), the optimal solution changes only for variables that are ``close’’ to the added node, immediately giving us a quick way to update the solution as the graph grows.
In the second part of the talk, we will consider a multi-task learning problem where the solutions are expected to lie in a low-dimensional subspace. This corresponds to a low-rank matrix recover problem where the columns of the matrix have been ``sketched’’ independently. We show that a novel convex relaxation of this problem results in optimal sample complexity bounds. These bounds demonstrate the statistical leverage we gain by solving the problem jointly over solving each individually.
Independent component analysis is a useful and general data analysis tool. It has found great successes in many applications. But in recent years, it has been observed that many popular approaches to ICA do not scale well with the number of components. This debacle has inspired a growing number of new proposals. But it remains unclear what the exact role of the number of components is on the information theoretical limits and computational complexity for ICA. Here I will describe our recent work to specifically address these questions and introduce a refined method of moments that is both computationally tractable and statistically optimal.
In this talk, using a technique introduced by P.~T.~Nam in 2012 and the Coulomb Uncertainty Principle, I will present the proof of new bounds on the excess charge for non relativistic atomic systems, independent of the particle statistics. These new bounds are the best bounds to date for bosonic systems. This is joint work with Juan Manel González and Trinidad Tubino.
Join Zoom Meeting: https://gatech.zoom.us/j/94786316294
I will describe the arithmetic of polynomials over idylls and various division algorithms and rules. For instance, that arithmetic might capture a total order/sign or an absolute value. These division algorithms will relate, for instance, the number of positive roots of a polynomial to the signs of the coefficients (Descartes's Rule of Signs).
Recent years have seen tremendous progress in high-accuracy solvers for Maximum Flow, Minimum-Cost Flow and general Linear Programs (LP). Progress on strongly polynomial solvers for combinatorial LP on the other hand has stalled. The computational gap between high-accuracy solvers and strongly polynomial solvers is linear in the number of variables only for combinatorial LP on directed graphs. For combinatorial LP beyond directed graphs this gap is currently quadratic and is known since the 1980s due to a seminal result by Tardos.
We finally break the quadratic gap and design a strongly polynomial interior-point-method for combinatorial LP, which reduces the gap to only a linear factor. Thus, we match the known linear gap for LP on directed graphs. Furthermore, we close the linear gap between feasibility and optimization of combinatorial LP.
Part 1 of a multi-part discussion.
Morse theory and Morse homology together give a method for understanding how the topology of a smooth manifold changes with respect to a filtration of the manifold given by sub-level sets. The Morse homology of a smooth manifold can be expressed using an algebraic object called a persistence module. A persistence module is a module graded by real numbers, and in this setup the grading on the module corresponds to the aforementioned filtration on the smooth manifold.
This is the first of a series of talks that aims to explain the relationship between Morse homology and persistence modules. In the first talk, I will give a rapid review of Morse theory and a review of Morse homology. An understanding of singular homology will be assumed.
With applications in the analysis of political districtings, Markov chains have become and essential tool for studying contiguous partitions of geometric regions. Nevertheless, there remains a dearth of rigorous results on the mixing times of the chains employed for this purpose. In this talk we'll discuss a sub-exponential bound on the mixing time of the Glauber dynamics chain for the case of bounded-size contiguous partition classes on certain grid-like classes of graphs.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
The complex connectivity structure unique to the brain network is believed to underlie its robust and efficient coding capability. One of many unique features of the mammalian brain network is its spatial embedding and hierarchical organization. I will discuss effects of these structural characteristics on network dynamics as well as their computational implications with a focus on the flexibility between modular and global computations and predictive coding.
Loosely speaking, the maximum likelihood threshold of a statistical model is the fewest number of data points needed to fit the model using maximum likelihood estimation. In this talk, I will discuss combinatorial and algebraic-geometric approaches to studying this poorly understood quantity for a certain class of Gaussian models. This is based on joint work with Sean Dewar, Steven Gortler, Tony Nixon, Meera Sitharam, and Louis Theran
Speaker will give the talk in person
Estimating the fixed-point of a contractive operator from empirical data is a fundamental computational and statistical task. In many practical applications including dynamic programming, the relevant norm is not induced by an inner product structure, which hinders existing techniques for analysis. In this talk, I will present recent advances in stochastic approximation methods for fixed-point equations in Banach spaces. Among other results, we discuss a novel variance-reduced stochastic approximation scheme, and establish its non-asymptotic error bounds. In contrast to worst-case guarantees, our bounds are instance-dependent, and achieve the optimal covariance structure in central limit theorems non-asymptotically.
Joint works with Koulik Khamaru, Martin Wainwright, Peter Bartlett, and Michael Jordan.
Given a Legendrian knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We proved that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, try to get the ideal of the proof and talk about the application which is about distinguishing Legendrian and Transverse knot.
The shuffle lattice is a partial order on words determined by two common types of genetic mutation: insertion and deletion. Curtis Greene discovered many remarkable enumerative properties of this lattice that are inexplicably connected to Jacobi polynomials. In this talk, I will introduce an alternate poset called the bubble lattice. This poset is obtained from the shuffle lattice by including transpositions. Using the structural relationship between bubbling and shuffling, we provide insight into Greene’s enumerative results. This talk is based on joint work with Henri Mülle.
In this talk, I will talk about the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I will discuss the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.
In this talk, we consider the problem of minimizing multi-modal loss functions with many local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss's landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea is to conducts 1D long-range exploration with a large smoothing radius along orthogonal directions, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. We also provide theoretical analysis on the convergence of the method on nonconvex landscape. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by a highly oscillating, deterministic noise. We provide a convergence theory under which the iterates converge to a tightened neighborhood of the solution, whose size is characterized by the noise frequency. We complement our theoretical analysis with numerical experiments to illustrate the performance of this approach.
An n-vertex graph is called C-Ramsey if it has no clique or independent set of size Clog n (i.e., if it has near-optimal Ramsey behavior). In this paper, we study edge-statistics in Ramsey graphs, in particular obtaining very precise control of the distribution of the number of edges in a random vertex subset of a C-Ramsey graph. This brings together two ongoing lines of research: the study of "random-like’’ properties of Ramsey graphs and the study of small-ball probability for low-degree polynomials of independent random variables.
The proof proceeds via an "additive structure’’ dichotomy on the degree sequence, and involves a wide range of different tools from Fourier analysis, random matrix theory, the theory of Boolean functions, probabilistic combinatorics, and low-rank approximation. One of the consequences of our result is the resolution of an old conjecture of Erdos and McKay, for which he offered one of his notorious monetary prizes.
(Joint work with Matthew Kwan, Ashwin Sah and Lisa Sauermann)
Disks are nice for many reasons. In this casual talk, I will try to convince you that it's even nicer than you think by presenting the Alexander's lemma. Just like in algebraic topology, we are going to rely on disks heavily to understand mapping class groups of surfaces. The particular method is called the Alexander's method. Twice the Alexander, twice the fun! No background in mapping class group is required.
Quantum mechanics and diffusion on a network, in the sense of a metric graph, are locally one-dimensional, but the way the graph is connected can add multidimensional features and some strange phenomena. Quantum graphs have been an active area of research since the 1990s. I’ll review the subject and share some ideas about analyzing Schrödinger and heat equations on metric graphs, through the associated eigenvalue problem and the heat kernel.
This talk is based on a 2022 article with David Borthwick and Kenny Jones, and on work in progress with David Borthwick, Anna Maltsev, and Haozhe Yu.
https://gatech.zoom.us/j/95197085752?pwd=WmtJUVdvM1l6aUJBbHNJWTVKcVdmdz09
Abstract: ODE eigenvalue problems often arise in the study of stability of traveling waves, in showing the second variation of a functional is positive definite, and in many other applications. For many eigenvalue problems, it is not possible to obtain an explicit eigen pair. Thus, one uses numerical methods to approximate the solution. By rigorously bounding all errors in the computation, including computer rounding errors via use of an interval arithmetic package, one may obtain a computer assisted proof that the true solution lies in a small neighborhood of an approximation. This allows one to prove stability of traveling waves, for example. In this talk, we discuss recent work regarding computer assisted proof of stability of waves, and discuss other areas of application, such as in identifying most probable paths of escape in stochastic systems.
We'll talk about problems of optimizing a quadratic function subject to quadratic constraints, in addition to a sparsity constraint that requires that solutions have only a few nonzero entries. Such problems include sparse versions of linear regression and principal components analysis. We'll see that this problem can be formulated as a convex conical optimization problem over a sparse version of the positive semidefinite cone, and then see how we can approximate such problems using ideas arising from the study of hyperbolic polynomials. We'll also describe a fast algorithm for such problems, which performs well in practical situations.
Morse theory and Morse homology together give a method for understanding how the topology of a smooth manifold changes with respect to a filtration of the manifold given by sub-level sets. The Morse homology of a smooth manifold can be expressed using an algebraic object called a persistence module. A persistence module is a module graded by real numbers, and in this setup the grading on the module corresponds to the aforementioned filtration on the smooth manifold.
This is the second of a series of talks that aims to explain the relationship between Morse homology and persistence modules. In this second talk, I will define persistence modules, explain how to compute Morse homology using persistence modules, and explain how the Künneth theorem and the cup product work with persistence modules. The material from the first part of this series will be assumed.
The classroom version of this event will be held in Skiles 005. Everyone on campus at Georgia Tech is highly encouraged to attend this version. The virtual version will be administered through Zoom (https://gatech.zoom.us/j/99514218896).
This talk will discuss an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exact topological invariants. To demonstrate the utility of the proposed method, the application is considered for the predictions of binding free energy (BFE) changes of protein-protein interactions (PPIs) induced by mutations with machine learning modeling. It has a great application in studying the SARS-CoV-2 virus' infectivity, antibody resistance, and vaccine breakthrough, which will be presented in this talk.
We prove a 1973 conjecture due to Erdős on the existence of Steiner triple systems with arbitrarily high girth. Our proof builds on the method of iterative absorption for the existence of designs by Glock, Kühn, Lo, and Osthus while incorporating a "high girth triangle removal process". In particular, we develop techniques to handle triangle-decompositions of polynomially sparse graphs, construct efficient high girth absorbers for Steiner triple systems, and introduce a moments technique to understand the probability our random process includes certain configurations of triples.
(Joint with Matthew Kwan, Mehtaab Sawhney, and Michael Simkin)
We show how to obtain a decomposition of an arbitrary closed, smooth, orientable 4-manifold from a loop of Morse functions on a surface or as a loop in the pants complex. A nice feature of all of these decompositions is that they can be encoded on a surface so that, in principle, 4-manifold topology can be reduced to surface topology. There is a good amount to be learned from translating between the world of Morse functions and the world of pants decompositions. We will allude to some of the applications of this translation and point the interested researcher to where they can learn more. No prior knowledge of these fields is assumed and there will be plenty of time for questions.
Around 1997, Shub and Smale proved that sufficiently good upper bounds
on the number of integer roots of polynomials in one variable --- as a function
of the input complexity --- imply a variant of P not equal to NP. Since then,
later work has tried to go half-way: Trying to prove that easier root counts
(over fields instead) still imply interesting separations of complexity
classes. Koiran, Portier, and Tavenas have found such statements over the real
numbers.
We present an analogous implication involving p-adic valuations:
If the integer roots of SPS polynomials (i.e., sums of products of sparse polynomials) of size s never yield more than s^{O(1)} distinct p-adic
valuations, then the permanents of n by n matrices cannot be computed by constant-free, division-free arithmetic circuits of size n^{O(1)}. (The
implication would be a new step toward separating VP from VNP.) We also show that this conjecture is often true, in a tropical geometric sense (paralleling a similar result over the real numbers by Briquel and Burgisser). Finally, we prove a special case of our conjectured valuation bound, providing a p-adic analogue of an earlier real root count for polynomial systems supported on circuits. This is joint work with Joshua Goldstein, Pascal Koiran, and Natacha Portier.
In many applied fields of research, like Geophysics, Medicine, Engineering, Economy, and Finance, to name a few, classical problems are the extraction of hidden information and features, like quasi-periodicities and frequency patterns, as well as the separation of different components contained in a given signal, like, for instance, its trend.
Standard methods based on Fourier and Wavelet Transform, historically used in Signal Processing, proved to be limited when nonlinear and non-stationary phenomena are present. For this reason in the last two decades, several new nonlinear methods have been developed by many research groups around the world, and they have been used extensively in many applied fields of research.
In this talk, we will briefly review the Hilbert-Huang Transform (a.k.a. Empirical Mode Decomposition method) and discuss its known limitations. Then, we will review the Iterative Filtering technique and we will introduce newly developed generalizations to handle multidimensional, multivariate, or highly non-stationary signals, as well as their time-frequency representation, via the so-called IMFogram. We will discuss the theoretical and numerical properties of these methods and show their applications to real-life data.
We will conclude the talk by reviewing the main problems which are still open in this research field.
We introduce a decomposition of a 4-manifold called a multisection, which is a mild generalization of a trisection. We show that these correspond to loops in the pants complex and provide an equivalence between closed smooth 4-manifolds and loops in the pants complex up to certain moves. In another direction, we will consider multisections with boundary and show that these can be made compatible with a Weinstein structure, so that any Weinstein 4-manifold can be presented as a collection of curves on a surface.
Structural graph theory has traditionally focused on graph classes that are closed under both vertex- and edge-deletion (such as, for each surface Σ, the class of all graphs which embed in Σ). A more recent trend, however, is to require only closure under vertex-deletion. This is typically the right approach for graphs with geometric, rather than topological, representations. More generally, it is usually the right approach for graphs that are dense, rather than sparse. I will discuss this paradigm, taking a closer look at classes with a forbidden vertex-minor.
Recent years have witnessed tremendous progress in developing and analyzing quantum computing algorithms for quantum dynamics simulation of bounded operators (Hamiltonian simulation). However, many scientific and engineering problems require the efficient treatment of unbounded operators, which frequently arise due to the discretization of differential operators. Such applications include molecular dynamics, electronic structure theory, quantum control and quantum machine learning. We will introduce some recent advances in quantum algorithms for efficient unbounded Hamiltonian simulation, including Trotter type splitting and the quantum highly oscillatory protocol (qHOP) in the interaction picture. The latter yields a surprising superconvergence result for regular potentials. In the end, I will discuss briefly how Hamiltonian simulation techniques can be applied to a quantum learning task achieving optimal scaling. (The talk does not assume a priori knowledge on quantum computing.)
The principal minor map takes an n by n square matrix to the length 2^n-vector of its principal minors. A basic question is to give necessary and sufficient conditions that characterize the image of various spaces of matrices under this map. In this talk, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. For complex symmetric matrices, this recovers a result of Oeding from 2011. If time permits, I will also give examples to prove that for general matrices no such finite characterization is possible. This is based on joint work with Cynthia Vinzant.
Braid groups belong to a broad class of groups known as Artin groups, which are defined by presentations of a particular form and have played a major role in geometric group theory and low-dimensional topology in recent years. These groups fall into two classes, finite-type and infinte-type Artin groups. The former come equipped with a powerful combinatorial structure, known as a Garside structure, while the latter are much less understood and present many challenges. However, if one restricts to the Artin monoid, then much of the combinatorial structure still applies in the infinite-type case. In a joint project with Rachael Boyd, Rose Morris-Wright, and Sarah Rees, we use geometric techniques to study the relation between the Artin monoid and the Artin group.
The flow of compressible fluids is governed by the Euler equations, and understanding the dynamics for large times is an outstanding open problem whose full resolution is unlikely to happen in our lifetimes. The main source of difficulty is that any global-in-time theory must incorporate singularities in the PDEs, a fact we have known even in one spatial dimension since Riemann’s 1860 work. In this 1D setting, mathematicians have successfully spent the past 160 years painting a nearly-full picture of fluid dynamics that incorporates singularities.
There is a monumental gap in our understanding of compressible fluids in the physical 3D setting compared to the 1D case. This is due in large to the (provable) inaccessibility of the technical PDE tools used in 1D when quantifying the dynamics in 3D. Nevertheless, Christodoulou’s 2007 celebrated breakthrough on shock singularities for the Euler equation has sparked a dramatic wave of results and ideas in multiple space dimensions that have the potential to make the first meaningful dent in the global-in-time theory of compressible fluids. Roughly, shocks are a form of singularity where the fluid solution remains regular but certain first derivatives blow up.
In this talk I will discuss the recent culmination of the wave of results initiated by Christodoulou: my work on the maximal classical development (MCD) for compressible fluids, joint with J. Speck. Roughly speaking, the MCD describes the largest region of spacetime where the Euler equations admit a classical solution. For an open set of smooth data, my work reveals the intimate relationship between shock singularity formation and the full structure of the MCD. This fully solves the 162 year old open problem of extending Riemann’s historic 1D result to 3D without symmetry assumptions. In addition to the mathematical contribution, the geo-analytic information of the MCD is precisely the correct “initial data” needed to physically describe the fluid “past” the initial shock singularity in a weak sense. I will also briefly discuss the countless open problems in the field, all of which can be viewed as “building blocks” which will shine the first lights onto the outstanding global-in-time open problem of fluids.
We study the long-time regime of the prediction with expert advice problem in both full information and adversarial bandit feedback setting. We show that with full information, the problem leads to second order parabolic partial differential equations in the Euclidean space. We exhibit solvable cases for this equation and discuss the optimal behavior of both agents. In the adversarial bandit feedback setting, we show that the problem leads to second order parabolic equations in the Wasserstein space which allows us to obtain novel regret bounds. Based on joint works with Erhan Bayraktar and Xin Zhang.
Fruitful interactions between arithmetic geometry and dynamical systems have emerged in recent years. In this talk I will illustrate how insights from complex dynamics can be employed to study problems from arithmetic geometry. And conversely how arithmetic geometry can be used in the study of dynamical systems. The motivating questions are inspired by a recurring phenomenon in arithmetic geometry known as `unlikely intersections' and conjectures of Pink and Zilber therein. More specifically, I will discuss work toward understanding the distribution of preperiodic points in subvarieties of families of rational maps.
In the 20th century, Thurston proved two classification theorems, one for surface homeomorphisms and one for branched covers of surfaces. While the theorems have long been understood to be analogous, we will present new work with Belk and Winarski showing that the two theorems are in fact special cases of one Ubertheorem. We will also discuss joint work with Belk, Lanier, Strenner, Taylor, Winarski, and Yurttas on algorithmic aspects of Thurston’s theorem. This talk is meant to be accessible to a wide audience.
Live streamed but not recorded:<br />
https://gatech.zoom.us/j/93724280805
The last decade has witnessed great interest in the study of divisors of graphs and a fascinating combinatorially-rich picture has emerged. The class of break divisors has attracted particular attention, for reasons both geometric and combinatorial. I will present several representation-theoretic results in this context.
I will demonstrate how certain quotients of polynomial rings by power ideals, already studied by Ardila-Postnikov, Sturmfels-Xu, Postnikov-Shapiro amongst others, arise by applying the method of orbit harmonics to break divisors. These quotients then naturally afford symmetric group representations which are not entirely understood yet. By describing the invariant spaces of these representations in terms of break divisors, I will answer a combinatorial question from the setting of cohomological Hall algebras.
In this talk we report on recent works (with A. Cosso, I. Kharroubi, H. Pham, M. Rosestolato) on the optimal control of (possibly path dependent) McKean-Vlasov equations valued in Hilbert spaces. On the other side we present the first ideas of a work with S. Federico, D. Ghilli and M. Rosestolato, on Mean Field Games in infinite dimension.
We will begin by presenting some examples for both topics and their relations. Then most of the time will be devoted to the first topic and the main results (the dynamic programming principle, the law invariance property of the value function, the Ito formula and the fact that the value function is a viscosity solution of the HJB equation, a first comparison result).
We conclude, if time allows, with the first ideas on the solution of the HJB-FKP system arising in mean Field Games in infinite dimension.
The purpose of this talk is to discuss our recent work on multi-frequency quasi-periodic cocycles, establishing continuity (both in cocycle and jointly in cocycle and frequency) of the Lyapunov exponent for non-identically singular cocycles. Analogous results for one-frequency cocycles have been known for over a decade, but the multi-frequency results have been limited to either Diophantine frequencies (continuity in cocycle) or SL(2,C) cocycles (joint continuity). We will discuss the main points of our argument, which extends earlier work of Bourgain.
A pervading question in the study of stochastic PDE is how small-scale random forcing in an equation combines to create nontrivial statistical behavior on large spatial and temporal scales. I will discuss recent progress on this topic for several related stochastic PDEs - stochastic heat, KPZ, and Burgers equations - and some of their generalizations. These equations are (conjecturally) universal models of physical processes such as a polymer in a random environment, the growth of a random interface, branching Brownian motion, and the voter model. The large-scale behavior of solutions on large scales is complex, and in particular depends qualitatively on the dimension of the space. I will describe the phenomenology, and then describe several results and challenging problems on invariant measures, growth exponents, and limiting distributions.
Memory bounds for continual learning
Abstract: Continual learning, or lifelong learning, is a formidable current challenge to machine learning. It requires the learner to solve a sequence of k different learning tasks, one after the other, while with each new task learned it retains its aptitude for earlier tasks; the continual learner should scale better than the obvious solution of developing and maintaining a separate learner for each of the k tasks. We embark on a complexity-theoretic study of continual learning in the PAC framework. We make novel uses of communication complexity to establish that any continual learner, even an improper one, needs memory that grows linearly with k, strongly suggesting that the problem is intractable. When logarithmically many passes over the learning tasks are allowed, we provide an algorithm based on multiplicative weights update whose memory requirement scales well; we also establish that improper learning is necessary for such performance. We conjecture that these results may lead to new promising approaches to continual learning.
Based on the joint work with Xi Chen and Christos Papadimitriou.
Two of the most influential theorems in discrete mathematics state, respectively, that diagonal Ramsey numbers grow exponentially and that error-correcting codes for noisy channels exist up to the information limit. The former, proved by Erdős in 1947 using random graphs, led to the development of the probabilistic method in combinatorics. The latter, proved by Shannon in 1948 using random codes, is one of the founding results of coding theory. Since then, the probabilistic method has been a cornerstone in the development of both Ramsey theory and coding theory. In this talk, we highlight a few important applications of the probabilistic method in these two parallel but interconnected worlds. We then present new results on Ramsey numbers of graphs and hypergraphs and codes correcting deletion errors, all based on probabilistic ideas.
This talk is concerned with α-Hölder-continuous weak solutions of the incompressible Euler equations. A great deal of recent effort has led to the conclusion that the space of Euler flows is flexible when α is below 1/3, the famous Onsager regularity. We show how convex integration techniques can be extended above the Onsager regularity to all α<1/2 in the case of the forced Euler equations. This leads to a new non-uniqueness theorem for any initial data. This work is joint with Aynur Bulut and Manh Khang Huynh.
Large sieve inequalities are a fundamental tool used to investigate prime numbers and exponential sums. I will explain my work that resolves a 1978 conjecture of S. Patterson (conditional on the Generalized Riemann hypothesis) concerning the bias of cubic Gauss sums. This explains a well-known numerical bias first observed by Kummer in 1846. One important byproduct of my work is a proof that
Heath-Brown's famous cubic large sieve is sharp, contrary to popular belief. This sheds light on some of the mysteries surrounding large sieve inequalities for certain families of arithmetic harmonics and gives strong clues on where to look next for further progress. This is based on joint work with Maksym Radziwill.
Weighted inequalities for singular integral operators are central in the study of non-homogeneous harmonic analysis. Two weight inequalities for singular integral operators, in-particular attracted attention as they can be essential in the perturbation theory of unitary matrices, spectral theory of Jacobi matrices and PDE's. In this talk, I will discuss several results concerning the two weight inequalities for various Calder\'on-Zygmund operators in both Euclidean setting and in the more generic setting of spaces of homogeneous type in the sense of Coifman and Weiss.
The two-weight conjecture for singular integral operators T was first raised by Nazarov, Treil and Volberg on finding the real variable characterization of the two weights u and v so that T is bounded on the weighted $L^2$ spaces. This conjecture was only solved completely for the Hilbert transform on R until recently. In this talk, I will describe our result that resolves a part of this conjecture for any Calder\'on-Zygmund operator on the spaces of homogeneous type by providing a complete set of sufficient conditions on the pair of weights. We will also discuss the existence of similar analogues for multilinear Calder\'on-Zygmund operators.
Refreshments available from 10:30 in Skiles Atrium. Talk will be streamed via https://gatech.zoom.us/j/94839708119?pwd=bmE1WXFTTzdFVDBtYzlvWUc3clFlZz09 but not recorded.
Algebraic Combinatorics originated in Algebra and Representation Theory, yet its objects and methods turned out applicable to other fields from Probability to Computer Science. Its flagship hook-length formula for the number of Standard Young Tableaux, or the beautiful Littlewood-Richardson rule have inspired large areas of study and development. We will see what lies beyond the reach of such nice product formulas and combinatorial interpretations and enter the realm of Computational Complexity and Asymptotics. We will also show how an 80 year old open problem on Kronecker coefficients of the Symmetric group lead to the disprove of the wishful approach towards the resolution of the algebraic P vs NP Millennium problem.
In the first talk of this series we introduced the definition of Chebyshev polynomials on compact subsets of the complex plane and discussed some properties. This week, after a short review of the first talk, we will start to discuss asymptotic properties of Chebyshev polynomials and how they are related with logarithmic potential theory. Our main focus will be the necessary concepts from potential theory needed in the study of asymptotic properties of Chebyshev polynomials.
Link:https://gatech.zoom.us/j/91232113113?pwd=MDhteEdtcENuME9kdXJmcUY0eWlSUT09
Large-scale computing systems are massively important, using over 1% of the world's electricity. It is vital that these systems be both fast and resource-efficient, and scheduling theory is a key tool towards that goal. However, prior scheduling theory is not equipped to handle large multiserver systems, with little extant theoretical analysis, and no optimality results.
I focus on two important multiserver scheduling systems: The one-server-per-job (M/G/k) model, and the multiple-servers-per-job (MSJ) model. In the M/G/k, we prove the first optimality result, demonstrating that the Shortest Remaining Processing Time (SRPT) policy yields asymptotically optimal mean response time in the heavy traffic limit. In the MSJ model, we prove the first mean response analysis for any scheduling policy, for a novel policy called ServerFilling. Moreover, we introduce the ServerFilling-SRPT policy, for which we present the first asymptotic optimality result for the MSJ model. Each result progresses by proving novel bounds on relevant work, and using novel methods to convert those bounds to bounds on mean response time. These results push the state of the art of scheduling theory ever closer to applicability to today's large-scale computing systems.
Algebraic matroids record the algebraic dependencies among elements in a field extension, similar to the linear dependencies of vectors in a vector space. Realizing a given matroid by elements in a field extension can depend on the characteristic of the field. I will talk about the possible characteristic sets of algebraic matroids. An essential tool is the one-dimensional group construction of an algebraic matroid, which turns the realization problem for algebraic matroids into a linear problem over the endomorphism ring of a one-dimensional algebraic group.
Understanding the behavior of large physical systems is a problem of fundamental importance in mathematical physics. Analysis of systems of many interacting particles is key for understanding various phenomena from physical sciences (e.g. gases in non-equilibrium, galactic dynamics) to social sciences (e.g. modeling social networks). Similarly, the description of systems of weakly nonlinear interacting waves, referred to as wave turbulence theory, finds a wide range of applications from solid state physics and water waves to plasma theory and oceanography. However, with the size of the system of interest being extremely large, deterministic prediction of its behavior is practically impossible, and one resorts to an averaging description. In this talk, we will discuss about kinetic theory, which is a mesoscopic framework to study the qualitative properties of large systems. As we will see, the main idea behind kinetic theory is that, in order to identify averaging quantities of large systems, one studies their asymptotic behavior as the size of the system tends to infinity, with the hope that the limiting effective equation will reveal properties observed in a system of large, but finite size. We will focus on the Boltzmann equation, which is the effective equation for systems of interacting particles, and its higher order extensions, as well as the kinetic wave equation which describes systems of many nonlinearly interacting waves.
The Torelli group of a surface is a natural yet mysterious subgroup of the mapping class group. We will discuss a few recent results about finiteness properties of the Torelli group, as well as a result about the cohomological dimension of the Johnson filtration.
Smooth ergodic theory provides a framework for studying systems exhibiting dynamical chaos, features of which include sensitive dependence with respect to initial conditions, correlation decay (even for deterministic systems!) and complicated fractal-like attractor geometry. This talk will be an overview of some of these ideas as they apply to evolutionary PDE, with an emphasis on dissipative semilinear parabolic problems, and a discussion of some of my work in this direction, joint with: Lai-Sang Young and Sam Punshon-Smith.
Mobile sampling concerns finding surfaces upon which any function with Fourier transform supported in a symmetric convex set must have some large values. We shall describe a sharp sufficient for mobile sampling in terms of the surface density introduced by Unnikrishnan and Vetterli. Joint work with Mishko Mitkovski and Manasa Vempati.
A central question in the field of optimal transport studies optimization problems involving two measures on a common metric space, a source and a target. The goal is to find a mapping from the source to the target, in a way that minimizes distances. A remarkable fact discovered by Caffarelli is that, in some specific cases of interest, the optimal transport maps on a Euclidean metric space are Lipschitz. Lipschitz regularity is a desirable property because it allows for the transfer of analytic properties between measures. This perspective has proven to be widely influential, with applications extending beyond the field of optimal transport.
In this talk, we will further explore the Lipschitz properties of transport maps. Our main observation is that, when one seeks Lipschitz mappings, the optimality conditions mentioned above do not play a major role. Instead of minimizing distances, we will consider a general construction of transport maps based on interpolation of measures, and introduce a set of techniques to analyze the Lipschitz constant of this construction. In particular, we will go beyond the Euclidean setting and consider Riemannian manifolds as well as infinite-dimensional spaces.
Some applications, such as functional inequalities, normal approximations, and generative diffusion models will also be discussed.
Gutzwiller semi-classical quantization, Boven-Sinai-Ruelle dynamical zeta functions for chaotic dynamical systems, statistical mechanics partition functions, and path integrals of quantum field theory are often presented in ways that make them appear as disjoint, unrelated theories. However, recent advances in describing fluid turbulence by its dynamical, deterministic Navier-Stokes underpinning, without any statistical assumptions, have led to a common field-theoretic description of both (low-dimension) chaotic dynamical systems, and (infinite-dimension) spatiotemporally turbulent flows.
I have described the remarkable experimental progress connecting turbulence to deterministic dynamics in the Sept 24, 2023 colloquium (the recoding is available on the website below). In this seminar I will use a lattice discretized field theory in 1 and 1+1 dimension to explain how temporal `chaos', `spatiotemporal chaos' and `quantum chaos' are profitably cast into the same field-theoretic framework.
https://ChaosBook.org/overheads/spatiotemporal/
The talk will also be on Zoom: GaTech.zoom.us/j/95338851370
Chip-firing asks a simple question: Given a group of people and an initial integer distribution of dollars among the people including people in debt, can we redistribute the money so that no one ends up in debt? This simple question with its origins in combinatorics can be reformulated using concepts from graph theory, linear algebra, graph orientation algorithms, and even divisors in Riemann surfaces. This presentation will go over a summary of Part 1 of Divisors and Sandpiles by Scott Corry and David Perkinson. Moreover, we will cover three various approaches to solve this problem: a linear algebra approach with the Laplacian, an algorithmic approach with Dhar's algorithm, and an algebraic geometry approach with a graph-theoretic version of the Riemann-Roch theorem by Baker and Norine. If we have time, we will investigate additional topics from Part 2 and Part 3. As true to the title, there will be a non-alcoholic drinking game involved with this presentation and participation will be completely voluntary. Limited refreshments (leftover Coca-Cola I found in the grad student lounge) and plastic cups will be served.
After finishing the proof of equivalence of the Chebyshev constant of a set and its logarithmic capacity, we will start to discuss classical and recent results on estimates and asymptotics of Chebyshev numbers.
We consider a service system where incoming tasks are instantaneously assigned to one out of many heterogeneous server pools. All the tasks sharing a server pool are executed in parallel and the execution times do not depend on the class of the server pool or the number of tasks currently contending for service. However, associated with each server pool is a utility function that does depend on the class of the server pool and the number of tasks currently sharing it. These features are characteristic of streaming and online gaming services, where the duration of tasks is mainly determined by the application but congestion can have a strong impact on the quality-of-service (e.g., video resolution and smoothness). We derive an upper bound for the mean overall utility in steady-state and introduce two load balancing policies that achieve this upper bound in a large-scale regime. Also, the transient and stationary behavior of these asymptotically optimal load balancing policies is characterized in the same large-scale regime.
The classroom version of this event will be held in Skiles 005. Everyone on campus at Georgia Tech is highly encouraged to attend this version. The virtual version will be administered through Zoom. (Link: https://gatech.zoom.us/j/91063740629 )
Reaction networks are commonly used to model a variety of physical systems ranging from the microscopic world like cell biology and chemistry, to the macroscopic world like epidemiology and evolution biology. A biologically relevant property that reaction networks can have is absolute concentration robustness (ACR), which refers to when a steady-state species concentration is maintained even when initial conditions are changed. Networks with ACR have been observed experimentally, for example, in E. coli EnvZ-OmpR and IDHKP-IDH systems. Another reaction network property that might be desirable is multistationarity-the capacity for two or more steady states, since it is often associated with the capability for cellular signaling and decision-making.
While the two properties seem to be opposite, having both properties might be favorable as a biochemical network may require robustness in its internal operation while maintaining flexibility as a signal-response mechanism. As such, our driving motivation is to explore what network structures can produce ACR and multistationarity. We show that it is highly atypical for both properties to coexist in very small and very large reaction networks without special structures. However, it is possible for them to coexist in certain classes of reaction networks. I will discuss in detail one such class of networks, which consists of multisite phosphorylation-dephosphorylation cycles with a ``paradoxical enzyme".
Most of us have been taught geometry from the perspective of equations and how those equations act on a given space. But in the 1870’s, Felix Klein’s Erlangen program was more concerned about the maps that preserved the geometric structures of a space rather than the equations themselves. In this talk, I will present some modern results from this perspective and show details of how to reconstruct the equations that preserve geometric structures.
https://gatech.zoom.us/j/98358157136
Mammalian cortical networks are known to process sensory information utilizing feedforward and feedback connections along the cortical hierarchy as well as intra-areal connections between different cortical layers. While feedback and feedforward signals have distinct layer-specific connectivity motifs preserved across species, the computational relevance of these connections is not known. Motivated by predictive coding theory, we study how expected and unexpected visual information is encoded along the cortical hierarchy, and how layer-specific feedforward and feedback connectivity contributes to differential, context-dependent representations of information across cortical layers, by analyzing experimental recordings of neural populations and also by building a recurrent neural network (RNN) model of the cortical microcircuitry. Experimental evidence shows that information about identity of the visual inputs and expectations are encoded in different areas of the mouse visual cortex, and simulations with our RNNs which incorporate biologically plausible connectivity motifs suggest that layer-specific feedforward and feedback connections may be the key contributor to this differential representation of information.
Marcus (1972) and de Oliveira (1982) conjectured bounds on the determinantal range of the sum of a pair of normal matrices with prescribed eigenvalues. We show that this determinantal range is a flattened solid twisted permutahedron, which is, in turn, a finite union of flattened solid twisted hypercubes with prescribed vertices. This complete geometric description, in particular, proves the conjecture. Our techniques are based on classical Lie theory, geometry, and combinatorics. I will give a pre-seminar that will be accessible to 1st year graduate students who like matrices, and provides an easy introduction to the topic. This is joint work with Matt Speck.
Sampling is a fundamental and widespread algorithmic primitive that lies at the heart of Bayesian inference and scientific computing, among other disciplines. Recent years have seen a flood of works aimed at laying down the theoretical underpinnings of sampling, in analogy to the fruitful and widely used theory of convex optimization. In this talk, I will discuss some of my work in this area, focusing on new convergence guarantees obtained via a proximal algorithm for sampling, as well as a new framework for studying the complexity of non-log-concave sampling.
A source of richness in Teichmüller theory is that Teichmüller spaces have descriptions both in terms of group representations and in terms of hyperbolic structures and complex structures. A program in higher-rank Teichmüller theory is to understand to what extent there are analogous geometric interpretations of Hitchin components. In this talk, we will give a natural description of the SL(3,R) Hitchin component in terms of higher complex structures as first described by Fock and Thomas. Along the way, we will describe higher complex structures in terms of jets and discuss intrinsic structural features of Fock-Thomas spaces.
In this talk we will discuss an optimal control problem for stochastic differential delay equations. We will only consider the case with delays in the state. We will show how to rewrite the problem in a suitable infinite-dimensional Hilbert space. Then using the dynamic programming approach we will characterize the value function of the problem as the unique viscosity solution of an infinite dimensional Hamilton-Jacobi-Bellman equation. We will discuss partial C^{1,α}-regularity of the value function. This regularity result is particularly interesting since it permits to construct a candidate optimal feedback map which may allow to find an optimal feedback control. Finally we will discuss some ideas about the case in which delays also appear in the controls.
This is a joint work with S. Federico and A. Święch.
The theory of graph quasirandomness studies graphs that "look like" samples of the Erdős--Rényi
random graph $G_{n,p}$. The upshot of the theory is that several ways of comparing a sequence with
the random graph turn out to be equivalent. For example, two equivalent characterizations of
quasirandom graph sequences is as those that are uniquely colorable or uniquely orderable, that is,
all colorings (orderings, respectively) of the graphs "look approximately the same". Since then,
generalizations of the theory of quasirandomness have been obtained in an ad hoc way for several
different combinatorial objects, such as digraphs, tournaments, hypergraphs, permutations, etc.
The theory of graph quasirandomness was one of the main motivations for the development of the
theory of limits of graph sequences, graphons. Similarly to quasirandomness, generalizations of
graphons were obtained in an ad hoc way for several combinatorial objects. However, differently from
quasirandomness, for the theory of limits of combinatorial objects (continuous combinatorics), the
theories of flag algebras and theons developed limits of arbitrary combinatorial objects in a
uniform and general framework.
In this talk, I will present the theory of natural quasirandomness, which provides a uniform and
general treatment of quasirandomness in the same setting as continuous combinatorics. The talk will
focus on the first main result of natural quasirandomness: the equivalence of unique colorability
and unique orderability for arbitrary combinatorial objects. Although the theory heavily uses the
language and techniques of continuous combinatorics from both flag algebras and theons, no
familiarity with the topic is required as I will also briefly cover all definitions and theorems
necessary.
This talk is based on joint work with Alexander A. Razborov.
We shall discuss the asymptotics of singular values of the transfer matrices of ergodic Schroedinger and block-Schroedinger operators. At a fixed value of the spectral parameter, the logarithmic asymptotics is almost surely given by the Lyapunov exponents; however, this is not, in general, true simultaneously for all the values of the parameter. We shall try to explain the importance of these sets in various problems of spectral theory, and then review some of the earlier works on the subject and present some new results. Based on joint work with I. Goldsheid.
This talk will be online. Meeting ID: 919 5236 6315. Pleas note the unusual time!
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Let T be an invertible measure preserving transformation of a standard infinite measure space (X,m). Then a Poisson suspension (X*,m*,T*) of the dynamical system (X,m,T) is a well studied object in ergodic theory (especially for the last 20 years). It has physical applications as a model for the ideal gas consisting of countably many non-interacting particles. A natural problem is to develop a nonsingular counterpart of the theory of Poisson suspensions. The following will be enlightened in the talk:
--- description of the m-nonsingular (i.e. preserving the equivalence class of m) transformations T such that T* is m*-nonsingular
---algebraic and topological properties of the group of all m*-nonsingular Poisson suspensions
--- an interplay between dynamical properties of T and T*
--- an example of a "phase transition" in the ergodic properties of T* depending on the scaling of m
--- applications to Kazhdan property (T), stationary (nonsingular) group actions and the Furstenberg entropy.
(joint work with Z. Kosloff and E. Roy)
We will continue to discuss lower and upper estimates of Widom factors. We will also introduce Cantor-type sets, constructed as the intersection of the level domains for simple sequences of polynomials. Using these Cantor-type sets we will prove some results on growth of Widom factors.
We demonstrate for the first time that ill-conditioned, non-smooth, constrained distributions in very high dimensions, upwards of 100,000, can be sampled efficiently in practice. Our algorithm incorporates constraints into the Riemannian version of Hamiltonian Monte Carlo and maintains sparsity. This allows us to achieve a mixing rate independent of condition numbers. On benchmark data sets from systems biology and linear programming, our algorithm outperforms existing packages by orders of magnitude. In particular, we achieve a 1,000-fold speed-up for sampling from the largest published human metabolic network (RECON3D). Our package has been incorporated into the COBRA toolbox. This is joint work with Yin Tat Lee, Ruoqi Shen, and Santosh Vempala.
The Zariski closure of the central path (which interior point algorithms track in convex optimization problems such as linear and semidefinite programs) is an algebraic curve, called the central curve. Its degree has been studied in relation to the complexity of these interior point algorithms. We show that the degree of the central curve for generic semidefinite programs is equal to the maximum likelihood degree of linear concentration models. This is joint work with Serkan Hoşten and Angélica Torres.
The finite index subgroups of a finitely presented group generate a topology on the group. We will discuss using examples how this relates to the organization of a group's finite quotients, and introduce the ideas of profinite rigidity and flexibility.
Speaker will be in person, but also livestreamed but not recorded at https://gatech.zoom.us/j/98355006347
Modern neural networks are usually over-parameterized—the number of parameters exceeds the number of training data. In this case the loss function tends to have many (or even infinite) global minima, which imposes a challenge of minima selection on optimization algorithms besides the convergence. Specifically, when training a neural network, the algorithm not only has to find a global minimum, but also needs to select minima with good generalization among many others. We study the mechanisms that facilitate global minima selection of optimization algorithms, as well as its connection with good generalization performance. First, with a linear stability theory, we show that stochastic gradient descent (SGD) favors global minima with flat and uniform landscape. Then, we build a theoretical connection of flatness and generalization performance based on a special multiplicative structure of neural networks. Connecting the two results, we develop generalization bounds for neural networks trained by SGD. Our bounds take the optimization process into consideration. Furthermore, we study the behavior of optimization algorithms around manifold of minima and reveal the exploration of algorithms from one minimum to another.
The fundamental groups of knot complements have lots of finite quotients. We give a criterion for a hyperbolic knot in the three-sphere to be distinguished (up to isotopy and mirroring) from every other knot in the three-sphere by the set of finite quotients of its fundamental group, and we use this criterion as well as recent work of Baldwin-Sivek to show that there are infinitely many hyperbolic knots distinguished (up to isotopy and mirroring) by finite quotients.
We consider the 1+1 dimensional vector valued Principal Chiral Field model (PCF) obtained as a simplification of the Vacuum Einstein Field equations under the Belinski-Zakharov symmetry. PCF is an integrable model, but a rigorous description of its evolution is far from complete. Here we provide the existence of local solutions in a suitable chosen energy space, as well as small global smooth solutions under a certain non degeneracy condition. We also construct virial functionals which provide a clear description of decay of smooth global solutions inside the light cone. Finally, some applications are presented in the case of PCF solitons, a first step towards the study of its nonlinear stability.
The braid group has many applications throughout the world of math due to its simple yet rich structure. In this talk we will focus on the Burau representation of the braid group, which has important implications in knot theory. Most notably, the open problem of faithfulness of the Burau representation of the braid group on 4 strands is equivalent to whether or not the Jones polynomial can detect the unknot. The Burau representation has a topological interpretation that uses the mapping class definition of the braid group. We'll introduce the braid group first and then discuss the Burau representation. We will go through examples for small n and discuss the proof of nonfaithfulness for n > 4. Time permitting, we may introduce the Gassner representation.
We study a family of dynamical systems obtained by coupling a chaotic (Anosov) map on the two-dimensional torus -- the chaotic variable -- with the identity map on the one-dimensional torus -- the neutral variable -- through a dissipative interaction. We show that the two systems synchronize, in the sense that the trajectories evolve toward an attracting invariant manifold, and that the full dynamics is conjugated to its linearization around the invariant manifold. When the interaction is small, the evolution of the neutral variable is very close to the identity and hence the neutral variable appears as a slow variable with respect to the fast chaotic variable: we show that, seen on a suitably long time scale, the slow variable effectively follows the solution of a deterministic differential equation obtained by averaging over the fast variable.
The seminar can also be accessed online via zoom link: Meeting ID: 961 2577 3408
Group extensions are a natural way of building complicated groups out of simpler ones. We will develop techniques used to study group extensions. Through these techniques, we will motivate and discuss connections to the cohomology of groups.
In the past few decades the problem of reconstructing high-dimensional functions taking values in abstract spaces from limited samples has received increasing attention, largely due to its relevance to uncertainty quantification (UQ) for computational science and engineering. These UQ problems are often posed in terms of parameterized partial differential equations whose solutions take values in Hilbert or Banach spaces. Impressive results have been achieved on such problems with deep learning (DL), i.e. machine learning with deep neural networks (DNN). This work focuses on approximating high-dimensional smooth functions taking values in reflexive and typically infinite-dimensional Banach spaces. Our novel approach to this problem is fully algorithmic, combining DL, compressed sensing, orthogonal polynomials, and finite element discretization. We present a full theoretical analysis for DNN approximation with explicit guarantees on the error and sample complexity, and a clear accounting of all sources of error. We also provide numerical experiments demonstrating the efficiency of DL at approximating such high-dimensional functions from limited data in UQ applications.
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
The talk is about controllability for uncertain linear systems. Our approach is
based on the Controllability Function (CF) method proposed by V.I. Korobov in
1979. The CF method is a development of the Lyapunov function method and the
dynamic programming method. The CF includes both approaches at a certain values
of parameters. The main advance of the CF method is finiteness of the time of motion
(settling-time function).
In the talk the feedback synthesis problem for a chain of integrators system
with continuous bounded unknown perturbations is considered. This problem consist
in constructing a control in explicit form which depends on phase coordinates and
steers an arbitrary initial point from a neighborhood of the origin to the origin in a
finite time (settling-time function). Besides the control is satisfies some preassigned
constrains. The range of the unknown perturbations such that the control solving the
synthesis problem for the system without the perturbations also solves the synthesis
problem for the perturbed system are found. This study shows the relations between
the range of perturbations and the bounds of the settling-time function.
In particular the feedback synthesis problem for the motion of a material
point with allowance for friction is solved.
Keywords: chain of integrators, finite-time stability, robust control, settling
time estimation, uncertain systems, unknown bounded perturbation
We will continue to focus on Cantor-type sets we introduced last week. Using them we will consider maximal growth rate and irregular behavior of Widom factors (nth Chebyshev number divided by nth power of logarithmic capacity). We will also discuss a recent result of Jacob Christiansen, Barry Simon and Maxim Zinchenko, which shows that Widom factors of Parreau-Widom sets are uniformly bounded.
Assemblies are subsets of neurons whose coordinated excitation is hypothesized to represent the subject's thinking of an object, idea, episode, or word. Consequently, they provide a promising basis for a theory of how neurons and synapses give rise to higher-level cognitive phenomena. The existence (and pivotal role) of assemblies was first proposed by Hebb, and has since been experimentally confirmed, as well as rigorously proven to emerge in the model of computation in the brain recently developed by Papadimitriou & Vempala. In light of contemporary studies which have documented the creation and activation of sequences of assemblies of neurons following training on tasks with sequential decisions, we study here the brain's mechanisms for working with sequences in the assemblies model of Papadimitriou & Vempala. We show that (1) repeated presentation of a sequence of stimuli leads to the creation of a sequence of corresponding assemblies -- upon future presentation of any contiguous sub-sequence of stimuli, the corresponding assemblies are activated and continue until the end of the sequence; (2) when the stimulus sequence is projected to two brain areas in a "scaffold", both memorization and recall are more efficient, giving rigorous backing to the cognitive phenomenon that memorization and recall are easier with scaffolded memories; and (3) existing assemblies can be quite easily linked to simulate an arbitrary finite state machine (FSM), thereby capturing the brain's ability to memorize algorithms. This also makes the assemblies model capable of arbitrary computation simply in response to presentation of a suitable stimulus sequence, without explicit control commands. These findings provide a rigorous, theoretical explanation at the neuronal level of complex phenomena such as sequence memorization in rats and algorithm learning in humans, as well as a concrete hypothesis as to how the brain's remarkable computing and learning abilities could be realized.
Joint work with Christos Papadimitriou and Santosh Vempala.
The extremal function of a class of matroids maps each positive integer n to the maximum number of elements of a simple matroid in the class with rank at most n. We will present a result concerning the role of finite groups in minor-closed classes of matroids, and then use it to determine the extremal function for several natural classes of representable matroids. We will assume no knowledge of matroid theory. This is joint work with Jim Geelen and Peter Nelson.
We expect to have an online option available: https://gatech.zoom.us/j/98355006347
The ocean and atmosphere are density stratified fluids. A wide variety of gravity waves propagate in their interior, redistributing energy and mixing the fluid, affecting global climate balances. Stratified fluids with narrow regions of rapid density variation with respect to depth (pycnoclines) are often modelled as layered flows. We shall adopt this model and examine horizontally propagating internal waves within a three-layer fluid, with a focus on mode-2 waves which have oscillatory vertical structure and whose observations and modelling have only recently started. Mode-2 waves (typically) occur within the linear spectrum of mode-1 waves (i.e. they travel at lower speeds than mode-1 waves), and thus mode-2 solitary waves are generically associated with an unphysical resonant mode-1 infinite oscillatory tail. We will show that these tail oscillations can be found to have zero amplitude, thus resulting in families of localised solutions (so called embedded solitary waves) in the Euler equations. This is the first example we know of embedded solitary waves in the Euler equations, and would imply that these waves are longer lived that previously thought.
In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2. Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4. Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces. This is joint work in progress with Fabiola Manjarrez-Gutierrez.
We will describe an elegant construction of potential counterexamples to the Smooth 4-Dimensional Poincaré Conjecture whose input is a fibered, homotopy-ribbon knot in the 3-sphere. The construction also produces links that are potential counterexamples to the Generalized Property R Conjecture, as well as balanced presentations of the trivial group that are potential counterexamples to the Andrews-Curtis Conjecture. We will then turn our attention to generalized square knots (connected sums of torus knots with their mirrors), which provide a setting where the potential counterexamples mentioned above can be explicitly understood. Here, we show that the constructed 4-manifolds are diffeomorphic to the 4-sphere; but the potential counterexamples to the other conjectures persist. In particular, we present a new, large family of geometrically motivated balanced presentations of the trivial group. Along the way, we give a classification of fibered, homotopy-ribbon disks bounded by generalized square knots up to isotopy and isotopy rel-boundary. This talk is based on joint work with Alex Zupan.
When can we find perfect matchings in hypergraphs whose vertices represent group elements and whose edges represent solutions to systems of linear equations? A prototypical problem of this type is the Hall-Paige conjecture, which asks for a characterisation of the groups whose multiplication table (viewed as a Latin square) contains a transversal. Other problems expressible in this language include the toroidal n-queens problem, Graham-Sloane harmonious tree-labelling conjecture, Ringel's sequenceability conjecture, Snevily's subsquare conjecture, Tannenbaum's zero-sum conjecture, and many others. All of these problems have a similar flavour, yet until recently they have been approached in completely different ways, using algebraic tools ranging from the combinatorial Nullstellensatz to Fourier analysis. In this talk we discuss a unified approach to attack these problems, using tools from probabilistic combinatorics. In particular, we will see that a suitably randomised version of the Hall-Paige conjecture can be used as a black-box to settle many old problems in the area for sufficiently large groups. Joint work with Alexey Pokrosvkiy
The classical Hele-Shaw flow describes the motion of incompressible viscous fluid, which occupies part of the space between two parallel, nearby plates. With source and drift, the equation is used in models of tumor growth where cells evolve with contact inhibition, and congested population dynamics. We consider the flow with Hölder continuous source and Lipschitz continuous drift. We show that if the free boundary of the solution is locally close to a Lipschitz graph, then it is indeed Lipschitz, given that the Lipschitz constant is small. This is joint work with Inwon Kim.
Have you ever wanted to marry topology, hyperbolic geometry, and geometric group theory, all at once?* Bowden-Hensel-Webb do this and more when they embark on their study of Diff0(S). In this talk, we will discuss the main theorems of Bowden-Hensel-Webb's paper, the most notable of which is (arguably) the lack of uniform perfection of Diff0(S). We will then summarize the main tools they use to prove these results. (Note: the perspectives on Diff0(S) in this talk will DIFFer greatly from those used in the diffeomorphism groups class.)
*If you answered "yes" for your personal life as opposed to your academic life: that's ok, I won't judge (if you don't tell me).
I discuss a sharp quantitative stability result for the Sobolev inequality with explicit constants. Moreover, the constants have the optimal behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant.
The problem of estimation of smooth functionals of unknown parameters of statistical models will be discussed in the cases of high-dimensional log-concave location models (joint work with Martin Wahl) and infinite dimensional Gaussian models with unknown covariance operator. In both cases, the minimax optimal error rates have been obtained in the classes of H\”older smooth functionals with precise dependence on the sample size, the complexity of the parameter (its dimension in the case of log-concave location models or the effective rank of the covariance in the case of Gaussian models) and on the degree of smoothness of the functionals. These rates are attained for different types of estimators based on two different methods of bias reduction in functional estimation.
We will discuss work of Michael Aizenman, Francois Germinet, Abel Klein, and Simone Warzel from 2007 on optimal Bernoulli decompositions of random variables and applications thereof. We will briefly discuss the basic properties of such decompositions, and demonstrate the existence of decompositions for which the contribution of the Bernoulli disorder is optimized in various ways.
We will then go through a proof of almost sure spectral localization (at the bottom of the spectrum) for continuous random Schroedinger operators with arbitrary bounded disorder. This proof relies on a Bernoulli decomposition of the disorder combined with a slightly stronger variant of the 2005 result from Jean Bourgain and Carlos Kenig showing such localization when the disorder is Bernoulli.
In the minimum eigenvalue problem we are given a collection of rank-1 symmetric matrices, and the goal is to find a subset whose sum has large minimum eigenvalue, subject to some combinatorial constraints. The constraints on which subsets we can select, could be cardinality, partition, or more general matroid base constraints. Using pipage rounding and a matrix concentration inequality, we will show a randomised algorithm which achieves a (1- epsilon) approximation for the minimum eigenvalue problem when the matrices have constant size, subject to any matroid constraint.
The bulk of the talk will be background on “pipage rounding, pessimistic estimators and matrix concentration” adapted from the paper with that title by Nicholas J. A. Harvey and Neil Olver. The application to the minimum eigenvalue problem is joint work with Aditi Laddha and Mohit Singh.
Zoom link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT0... />
In 1955, Mark Kac introduced a simple model to study the evolution of a gas of particles undergoing pairwise collisions. Although extremely simplified in such a way to be rigorously treatable, the model maintains interesting aspects of gas dynamics. In recent years, we worked with M. Loss and others to extend the analysis to more "realistic" versions of the original model.
I will introduce the Kac model and present some standard and more recent results. These results refer to a system with a fixed number of particles and at fixed kinetic energy (micro canonical ensemble) or temperature (canonical ensemble). I will introduce a "Grand Canonical" version of the Kac system and discuss new results on it.
Legendrian knots are smooth knots which are compatible with an ambient contact structure. They are an essential object of study in contact and symplectic geometry, and many easily posed questions about these knots remain unanswered. In this talk I will introduce Legendrian knots, their properties, some of their invariants. Expect lots of pictures.
Two knots are said to be concordant if they jointly form the boundary of a cylinder in four-dimensional Euclidean space. In the symplectic setting, we say they are Lagrangian concordant if the knots are Legendrian and the cylinder is Lagrangian. Interestingly, Lagrangian concordance is, unlike smooth concordance, not a symmetric relation. In this talk, I'll discuss various strategies that can be used to obstruct Lagrangian concordance, from basic invariants of Legendrian knots, to the Chekanov-Eliashberg DGA, to building new obstructions from Weinstein cobordisms.
Bayesian optimal experimental design (OED) is a principled framework for maximizing information gained from limited data in Bayesian inverse problems. Unfortunately, conventional methods for OED are prohibitive when applied to expensive models with high-dimensional parameters. In this talk, I will present fast and scalable computational methods for large-scale Bayesian OED with infinite-dimensional parameters, including data-informed low-rank approximation, efficient offline-online decomposition, projected neural network approximation, and a new swapping greedy algorithm for combinatorial optimization.
We will discuss the work of Ding-Smart (2019) which showed Anderson localization at the bottom of the spectrum for random discrete Schroedinger operators with arbitrary bounded noise, i.e. without any supposition of regularity of the distribution. In this talk, we will discuss at a high level the basic idea behind a multi-scale analysis, as well as the usual ingredients involved in one: resolvent decay at large scales and the Wegner-type estimate.
We will then discuss the obstacles posed by singular distributions, and the various methods used to overcome these obstacles in various regimes, discussing briefly the transfer matrix method used for d=1 by Carmona-Klein-Martinelli (1987) before examining the unique continuation principles used by Bourgain-Kenig (2005) and the Ding-Smart work which are used in d=2 in the continuum and discrete cases respectively, highlighting the unique challenges arising in the discrete case.
Solitons are particle-like solutions to dispersive evolution equations
whose shapes persist as time goes by. In some situations, these solitons
appear due to the balance between nonlinear effects and dispersion, in
other situations their existence is related to topological properties of
the model. Broadly speaking, they form the building blocks for the
long-time dynamics of dispersive equations.
In this talk I will present joint work with W. Schlag on long-time decay
estimates for co-dimension one type perturbations of the soliton for the
1D focusing cubic Klein-Gordon equation (up to exponential time scales),
and I will discuss our previous work on the asymptotic stability of the
sine-Gordon kink under odd perturbations. While these two problems are
quite similar at first sight, we will see that they differ by a subtle
cancellation property, which has significant consequences for the
long-time dynamics of the perturbations of the respective solitons.
The edge-coloring problem (ECP) for a multigraph $G=(V, E)$ is to color its edges with minimum number of colors such that no two adjacent vertices receive the same color. ECP can be naturally formulated as an integer program, and its linear programming relaxation is referred to as the fractional edge-coloring problem (FECP). The optimal value of ECP (resp. FECP) is called the chromatic index (resp. fractional chromatic index) of $G$, denoted by $\chi^{\prime}(G)$ (resp. $\chi^*(G)$). Let $\Delta(G)$ be the maximum degree of $G$ and let $ \mathcal{W}^*(G) $ be the fractional density of $G$, defined by $$ \mathcal{W}^*(G) = \max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor|U|/2\rfloor}. $$ Seymour showed that $\chi^*(G)=\max \{\Delta(G), \mathcal{W}^*(G)\}$. Moreover, the Goldberg-Seymour Conjecture is confirmed Chen, Jing, and Zang states that $\chi^{\prime}(G) \leq \max \{\Delta(G)+1,\lceil\mathcal{W}^*(G)\rceil\}$. Chen, Zang and Zhao developed an algorithm that calculates $ \mathcal{W}^*(G) $ in strongly polynomial time. Inspired by their results, we consider the fractional $ f $-edge-coloring problem ($ f $-FECP) for a given function $ f:V\to \mathbb N_+ $, which is a generalization of FECP: each spanning subgraph induced by a color class has degree at most $ f(v) $ at each vertex $ v\in V $. We give a strongly polynomial-time algorithm for calculating the corresponding fractional $ f $-density $$ \mathcal{W}^*_{f}(G)=\max _{U \subseteq V,|U| \geq 2}\frac{|E(U)|}{\lfloor f(U) / 2\rfloor}. $$
This talk will be about connections between spectral problems for canonical systems and non-linear Fourier transforms (NLFTs). Non-linear Fourier transform is closely connected to Dirac systems, which form a subclass of canonical systems of differential equations. This connection allows one to find analogs of results on inverse spectral problems for canonical systems in the area of NLFT. In particular, NLFTs of discrete sequences, discussed in the lecture notes by Tao and Thiele, are related to spectral problems for periodic measures and the theory of orthogonal polynomials.
I will start the talk with the basics of non-linear Fourier transforms, then connect NLFTs to canonical systems. Then I will present an explicit algorithm for inverse spectral problems developed by Makarov and Poltoratski for locally-finite periodic spectral measures, as well as an extension of their work to certain classes of non-periodic spectral measures. Finally I will return to NLFT and translate the results for inverse spectral problems to results for NLFT.
The Anderson tight binding model describes an electron moving in a disordered material. Such models are, depending on various parameters of the system, either expected to or known to display a phenomenon known as Anderson localization, in which this disorder can "trap" electrons. Different versions of this phenomenon can be characterized spectrally or locally. We will review both the dominant methods and some of the foundational results in the study of these systems in arbitrary dimension, before shifting our focus to aspects of the one-dimensional theory.
Specifically, we will examine the transfer matrix method, which allows us to leverage the Furstenberg theory of random matrix products to understand the asymptotics of generalized eigenfunctions. From this, we will briefly sketch a proof of localization given originally in Jitomirskaya-Zhu (2019). Finally, we will discuss recent work of the speaker combining the argument in Jitomirskaya-Zhu with certain probabilistic results to prove localization for a broader class of models.
In this continuing joint work with Benjamin Arras, we explore connections between covariance representations and Stein's method. In particular, via Stein's kernels we obtain quantitative high-dimensional CLTs in 1-Wasserstein distance when the limiting Gaussian probability measure is anisotropic. The dependency on the parameters is completely explicit and the rates of convergence are sharp.
We'll begin with a primer on hyperbolic and stable polynomials, which have been popular in recent years due to their many surprising appearances in combinatorics and algebra. We will cover a sketch of the famous Branden Borcea characterization of univariate stability preservers in the first part of the talk. We will then discuss more our recent work on multivariate hyperbolic polynomials which are invariant under permutations of their variables and connections to this Branden Borcea characterization.
Zoom Link: https://gatech.zoom.us/j/99596774152
We give a polynomial-time algorithm for online covering IPs with a competitive ratio of O(\log mn) when the constraints are revealed in random order, essentially matching the best possible offline bound of \Omega(\log n) and circumventing the \Omega(\log m \log n) lower bound known in adversarial order. We then leverage this O(\log mn)-competitive algorithm to solve this problem in the prophet setting, where constraints are sampled from a sequence of known distributions. Our reduction in fact relies only on samples from these distributions, in a manner evocative of prior work on single-sample prophet inequalities for online packing problems. We present sample guarantees in the prophet setting, as well as in the setting where random samples from an adversarial instance are revealed at the outset.
This talk is based on joint work with Anupam Gupta and Roie Levin, part of which appeared at FOCS 2021.
Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.
The classroom version of this event will be held in Skiles 005. Everyone on campus at Georgia Tech is highly encouraged to attend this version. The virtual version will be administered through Zoom. (Link: https://gatech.zoom.us/j/95527383236)
In embryonic development, formation of blood vessels in the retina of the eye is critically dependent on prior establishment of a mesh of astrocytes. Astrocytes emerge from the optic nerve head and then migrate over the retinal surface in a radially symmetric manner and mature through differentiation. We develop a PDE model describing the migration and differentiation of astrocytes and study the appropriateness of the model equation components that combines approximate Bayesian computation (ABC) and sensitivity analysis (SA). Comparing numerical simulations to experimental data, we identify model components that can be removed via model reduction using ABC+SA.
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Conservation laws and Lyapunov functions are powerful tools for proving the global existence or stability of solutions to PDEs, but for most complex systems these tools are insufficient to completely understand non-perturbative dynamics. In this talk I will discuss a complex-scalar PDE which may be seen as a toy model for vortex stretching in fluid flow, and cannot be neatly categorized as conservative nor dissipative.
In a recent series of papers, we have shown (using computer-assisted-proofs) that this equation exhibits rich dynamical behavior existing globally in time: non-trivial equilibria, homoclinic orbits, heteroclinic orbits, and integrable subsystems foliated by periodic orbits. On the other side of the coin, we show several mechanisms by which solutions can blowup.
A translation surface is obtained by identifying edges of polygons in the plane to create a compact Riemann surface equipped with a nonzero holomorphic one-form. Every Riemann surface can be given as an algebraic curve via its Jacobian variety. We aim to construct explicitly the underlying algebraic curves from their translation surfaces, given as polygons in the plane. The key tools in our approach are discrete Riemann surfaces, which allow us to approximate the Riemann matrices, and then, via theta functions, the equations of the curves. In this talk, I will present our algorithm and numerical experiments. From the newly found Riemann matrices and equations of curves, we can then make several conjectures about the curves underlying the Jenkins-Strebel representatives, a family of examples that until now, lived squarely on the analytic side of the transcendental divide between Riemann surfaces and algebraic curves.
Speaker will present in person
In this talk, we obtain new computational insights into two classical areas of statistics: generalization and sampling. In the first part, we study generalization: the performance of a learning algorithm on unseen data. We define a notion of generalization for non-converging training with local descent approaches via the stability of loss statistics. This notion yields generalization bounds in a similar manner to classical algorithmic stability. Then, we show that more information from the training dynamics provides clues to generalization performance.
In the second part, we discuss a new method for constructing transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.
The classical braid groups can be viewed from many different angles and admit generalizations in just as many directions. Surface braid groups are a topological generalization of the braid groups that have close connections with mapping class groups of surfaces. In this talk we review a recent result on minimal nonabelian finite quotients of braid groups. In considering the analogous problem for surface braid groups, we construct nilpotent nonabelian quotients by generalizing the Heisenberg group. These Heisenberg quotients do not arise as quotients of the braid group.
We will talk about our results on the elasticity and stability of the
collision of two kinks with low speed v>0 for the nonlinear wave
equation of dimension 1+1 known as the phi^6 model. We will show that
the collision of the two solitons is "almost" elastic and that, after
the collision, the size of the energy norm of the remainder and the size
of the defect of the speed of each soliton can be, for any k>0, of the
order of any monomial v^{k} if v is small enough.
References:
This talk is based on our current works:
On the collision problem of two kinks for the phi^6 model with low speed
[https://arxiv.org/abs/2211.09749]
Approximate kink-kink solutions for the phi^6 model in the low-speed
limit [https://arxiv.org/abs/2211.09714]
We show that Erdős-Renyi random graph with constant density has correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger’s conjecture for correspondence colouring. The proof follows from a sufficient condition for correspondence colourability in terms of the numbers of independent sets, following Bernshteyn's method. We conjecture the truth to be of order $O(n/\log n)$ as suggested by the random correspondence assignment. This is joint work with Zdenek Dvorak.
When do commuting homeomorphisms of S^2 have a common fixed point? Christian Bonatti gave the first sufficient condition: Commuting diffeomorphisms sufficiently close to the identity in Diff^+(S^2) always admit a common fixed point. In this talk we present a result of Michael Handel that extends Bonatti's condition to a much larger class of commuting homeomorphisms. If time permits, we survey results for higher genus surfaces due to Michael Handel and Morris Hirsch, and connections to certain compact foliated 4-manifolds.
We study $L^p$ bounds on Nikodym maximal functions associated to spheres. In contrast to the spherical maximal functions studied by Stein and Bourgain, our maximal functions are uncentered: for each point in $\mathbb R^n$, we take the supremum over a family of spheres containing that point. This is joint work with Georgios Dosidis and Jongchon Kim.
Live-stream link: https://gatech.zoom.us/j/93100501365?pwd=bWFEeURxek5pWG1BRjN4MHcvYllYQT0... />
Passcode provided in talk announcement
We describe a geometric framework to study Newton's
equations on infinite-dimensional configuration spaces of
diffeomorphisms and smooth probability densities. It turns out that
several important PDEs of hydrodynamical origin can be described in
this framework in a natural way. In particular, the so-called Madelung
transform between the Schrödinger-type equations on wave functions and
Newton's equations on densities turns out to be a Kähler map between
the corresponding phase spaces, equipped with the Fubini-Study and
Fisher-Rao information metrics. This is a joint work with G.Misiolek
and K.Modin.
I will discuss the soliton resolution and asymptotic stability problems for the sine-Gordon equation. It is known that the obstruction to the asymptotic stability for the sine-Gordon equation in the energy space is the existence of small breathers which is also closely related to the emergence of wobbling kinks. Our stability analysis gives a criterion for the weight which is sharp up to the endpoint so that the asymptotic stability holds. This is joint work with Jiaqi Liu and Bingying Lu.
Zoom link to the talk: https://gatech.zoom.us/j/91558578481
In this talk, we will consider stochastic processes on (random) graphs. They arise naturally in epidemiology, statistical physics, computer science and engineering disciplines. In this set-up, the vertices are endowed with a local state (e.g., immunological status in case of an epidemic process, opinion about a social situation). The local state changes dynamically as the vertex interacts with its neighbours. The interaction rules and the graph structure depend on the application-specific context. We will discuss (non-equilibrium) approximation methods for those systems as the number of vertices grow large. In particular, we will discuss three different approximations in this talk: i) approximate lumpability of Markov processes based on local symmetries (local automorphisms) of the graph, ii) functional laws of large numbers in the form of ordinary and partial differential equations, and iii) functional central limit theorems in the form of Gaussian semi-martingales. We will also briefly discuss how those approximations could be used for practical purposes, such as parameter inference from real epidemic data (e.g., COVID-19 in Ohio), designing efficient simulation algorithms etc.
We will continue our discussion of the key ingredients of a multi-scale analysis, namely resolvent decay and the Wegner type estimate. After briefly discussing how the Wegner estimate is obtained in the regime of regular noise, we will discuss the strategy used in Bourgain-Kenig (2005) and Ding-Smart (2018) to obtain analogues thereof using some form of unique continuation principle.
From here, we'll examine the quantitative unique continuation principle used by Bourgain-Kenig, and the lack of any even qualitative analogue on the two-dimensional lattice. From here, we'll discuss the quantitative probabilistic unique continuation result used in Ding-Smart.
A line arrangement is a collection of lines in the projective plane. The intersection lattice of the line arrangement is the set of all lines and their intersections, ordered with respect to reverse inclusion. A line arrangement is called free if the Jacobian ideal of the line arrangement is saturated. The underlying motivation for this talk is a conjecture of Terao which says that whether a line arrangement is free can be detected from its intersection lattice. This raises a question - in what ways does the saturation of the Jacobian ideal depend on the geometry of the lines and not just the intersection lattice? A main objective of the talk is to introduce planar rigidity theory and show that 'infinitesimal rigidity' is a property of line arrangements which is not detected by the intersection lattice, but contributes in a very precise way to the saturation of the Jacobian ideal. This connection builds a theory around a well-known example of Ziegler. This is joint work with Jessica Sidman (Mt. Holyoke College) and Will Traves (Naval Academy).
Every knot in S^3 bounds a PL (piecewise-linear) disk in the four ball. But this is no longer true for knots in other three manifolds, as demonstrated first by Akbulut, who constructed a knot which does not bound any PL disk in a specific contractible four manifold. Then Levine showed that there exist knots that do not bound a PL disk in any homology four ball. What happens if we relax the condition of bounding PL disk to bounding a PL surface with some given genus? I will discuss the joint work with Hom and Stoffregen, where we proved that for each n, there exists a knot K_n in an integer homology sphere that does not bound a PL surface of genus n in any homology four ball. This talk is meant to be accessible to a broad audience.
We study the problem of designing a robust parameter-independent feedback control input that steers, with minimum energy, the average of a linear system submitted to parameter perturbations where the states are initialized and finalized according to a given initial and final distribution. We formulate this problem as an optimal transport problem, where the transport cost of an initial and final state is the minimum energy of the ensemble of linear systems that have started from the initial state and the average of the ensemble of states at the final time is the final state. The by-product of this formulation is that using tools from optimal transport, we are able to design a robust parameter-independent feedback control with minimum energy for the ensemble of uncertain linear systems. This relies on the existence of a transport map which we characterize as the gradient of a convex function.
I will discuss the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. The proof uses tools from microlocal analysis and a detailed description of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator at low frequencies.
A \emph{list assignment} $L$ gives to each vertex $v$ in a graph $G$ a
list $L(v)$ of
allowable colors. An \emph{$L$-coloring} is a proper coloring $\varphi$ such that
$\varphi(v)\in L(v)$ for all $v\in V(G)$. An \emph{$L$-recoloring move} transforms
one $L$-coloring to another by changing the color of a single vertex. An
\emph{$L$-recoloring sequence} is a sequence of $L$-recoloring moves. We study
the problem of which hypotheses on $G$ and $L$ imply that for that every pair
$\varphi_1$ and $\varphi_2$ of $L$-colorings of $G$ there exists an $L$-recoloring
sequence that transforms $\varphi_1$ into $\varphi_2$. Further, we study bounds on
the length of a shortest such $L$-recoloring sequence.
We will begin with a survey of recoloring and list recoloring problems (no prior
background is assumed) and end with some recent results and compelling
conjectures. This is joint work with Stijn Cambie and Wouter Cames van
Batenburg.
A meromorphic inner function is a bounded analytic function on the upper half plane with unit modulus almost everywhere on the real line and a meromorphic continuation to the complex plane. Meromorphic inner functions and equivalently meromorphic Herglotz functions play a central role in inverse spectral theory of differential operators. In this talk, I will discuss some uniqueness problems for meromorphic inner functions and their applications to inverse spectral theory of canonical Hamiltonian systems as Borg-Marchenko type results.
The pants complex of a surface has as its 0-cells the pants decompositions of a surface and as its 1-cells some elementary moves relating two pants decompositions; the 2-cells are disks glued along certain cycles in the 1-skeleton of the complex. In "Pants Decompositions of Surfaces," Hatcher proves that this complex is contractible.
During this interactive talk, we will aim to understand the structure of the pants complex and some of the important tools that Hatcher uses in his proof of contractibility.
We study the locations of complex zeroes of independence polynomials of bounded degree hypergraphs. For graphs, this is a long-studied subject with applications to statistical physics, algorithms, and combinatorics. Results on zero-free regions for bounded-degree graphs include Shearer's result on the optimal zero-free disk, along with several recent results on other zero-free regions. Much less is known for hypergraphs. We make some steps towards an understanding of zero-free regions for bounded-degree hypergaphs by proving that all hypergraphs of maximum degree $\Delta$ have a zero-free disk almost as large as the optimal disk for graphs of maximum degree $\Delta$ established by Shearer (of radius $\sim1/(e\Delta)$). Up to logarithmic factors in $\Delta$ this is optimal, even for hypergraphs with all edge-sizes strictly greater than $2$. We conjecture that for $k\geq 3$, there exist families of $k$-uniform linear hypergraphs that have a much larger zero-free disk of radius $\Omega(\Delta^{-1/(k-1)})$. We establish this in the case of linear hypertrees. Joint work with David Galvin, Gwen McKinley, Will Perkins and Prasad Tetali.
I will give a hopefully accessible introduction to some work on
tropical moduli spaces of curves and abelian varieties. I will report
on joint work with Madeline Brandt, Juliette Bruce, Margarida Melo,
Gwyneth Moreland, and Corey Wolfe, in which we find new rational
cohomology classes in the moduli space A_g of abelian varieties using
tropical techniques. And I will try to touch on a new point of view on
this topic, namely that of differential forms on tropical moduli
spaces, following the work of Francis Brown.
We shall discuss the quantum dynamics associated with ergodic
Schroedinger operators with singular continuous spectrum. Upper bounds
on the transport moments have been obtained for several classes of
one-dimensional operators, particularly, by Damanik--Tcheremchantsev,
Jitomirskaya--Liu, Jitomirskaya--Powell. We shall present a new method
which allows to recover most of the previous results and also to
obtain new results in one and higher dimensions. The input required to
apply the method is a large-deviation estimate on the Green function
at a single energy. Based on joint work with S. Sodin.
The talk will be online at https://gatech.zoom.us/j/96285037913
In his seminal 1991 paper, Thomas M. Cover introduced a simple and elegant mathematical model for trading on the stock market. This model, which later on came to be known as online portfolio selection (OPS), is specified with only two integer parameters: the number of assets $d$ and time horizon $T$. In each round $t \in \{1, ..., T\}$, the trader selects a portfolio--distribution $p_t \in R^d_+$ of the current capital over the set of $d$ assets; after this, the adversary generates a nonnegative vector $r_t \in R^d_+$ of returns (relative prices of assets), and the trader's capital is multiplied by the "aggregated return'' $\langle p_{t}, r_{t} \rangle$. Despite its apparent simplicity, this model captures the two key properties of the stock market: (i) it "plays against'' the trader; (ii) money accumulates multiplicatively. In the 30 years that followed, the OPS model has received a great deal of attention from the learning theory, information theory, and quantitative finance communities.
In the same paper, Cover also proposed an algorithm, termed Universal Portfolios, that admitted a strong performance guarantee: the regret of $O(d \log (T))$ against the best portfolio in hindsight, and without any restrictions of returns or portfolios. This guarantee was later on shown to be worst-case optimal, and no other algorithm attaining it has been found to date. Unfortunately, exact computation of a universal portfolio amounts to averaging over a log-concave distribution, which is a challenging task. Addressing this, Kalai and Vempala (2002) achieved the running time of $O(d^4 T^{14})$ per round via log-concave sampling techniques. However, with such a running time essentially prohibiting all but "toy'' problems--yet remaining state-of-the-art--the problem of finding an optimal and practical OPS algorithm was left open.
In this talk, after discussing some of the arising challenges, I shall present a fast and optimal OPS algorithm proposed in a recent work with R. Jezequel and P. Gaillard (arXiv:2209.13932). Our algorithm combines regret optimality with the runtime of $O(d^2 T)$, thus dramatically improving state of the art. As we shall see, the motivation and analysis of the proposed algorithm are closely related to establishing a sharp bound on the accuracy of the Laplace approximation for a log-concave distribution with a polyhedral support, which is a result of independent interest.
Zoom link to the talk: https://gatech.zoom.us/j/98280978183
Continuing from where we left off, we will go through the proof of the probabilistic unique continuation result in Ding-Smart (2018) for solutions of the eigenequation on large finite boxes in the two-dimensional lattice. We'll briefly discuss the free sites formalism necessary to carry out the multiscale analysis as well, before going through technical lemmas concerning bounds on solutions to our eigenequation on large finite rectangles in the lattice as they propagate from a boundary.
Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.
https://gatech.zoom.us/j/98358157136
Analyzing when noisy trajectories, in the two dimensional plane, of a stochastic dynamical system exit the basin of attraction of a fixed point is specifically challenging when a periodic orbit forms the boundary of the basin of attraction. Our contention is that there is a distinguished Most Probable Escape Path (MPEP) crossing the periodic orbit which acts as a guide for noisy escaping paths in the case of small noise slightly away from the limit of vanishing noise. It is well known that, before exiting, noisy trajectories will tend to cycle around the periodic orbit as the noise vanishes, but we observe that the escaping paths are stubbornly resistant to cycling as soon as the noise becomes at all significant. Using a geometric dynamical systems approach, we isolate a subset of the unstable manifold of the fixed point in the Euler-Lagrange system, which we call the River. Using the Maslov index we identify a subset of the River which is comprised of local minimizers. The Onsager-Machlup (OM) functional, which is treated as a perturbation of the Friedlin-Wentzell functional, provides a selection mechanism to pick out a specific MPEP. Much of the talk is focused on the system obtained by reversing the van der Pol Equations in time (so-called IVDP). Through Monte-Carlo simulations, we show that the prediction provided by OM-selected MPEP matches closely the escape hatch chosen by noisy trajectories at a certain level of small noise.
General audience lecture
Waves are ubiquitous in nature. Some wave phenomena are conspicuous, most notably in elastic objects, and in bodies of water. In electro-dynamics, quantum mechanics, and gravity, waves play a fundamental role but are much more difficult to find. Over the past centuries, major scientific breakthroughs have been associated with the discovery of hidden wave phenomena in nature. Engineering has enabled our modern information based society by developing sophisticated methods which allow us to harness wave propagation. Seismic exploration relies on wave scattering in the discovery of natural resources. Medicine depends heavily on wave-based imaging technology such as MRI and CAT scans.
Mathematics has played a major role in the understanding of wave propagation, and its many intricate phenomena including reflection, diffraction, and refraction. In its most basic form, the wave equation is a linear partial differential equation (PDE). However, modern science and engineering rely heavily on nonlinear PDEs which can exhibit many surprising and delicate properties. Mathematical analysis continues to evolve rapidly driven in part by the many open questions surrounding nonlinear PDEs and their solutions. This talk will survey some of the mathematics involved in our understanding of waves, both linear and nonlinear.
Macdonald polynomials are a family of symmetric functions that are known to have remarkable connections to a well-studied particle model called the asymmetric simple exclusion process (ASEP). The modified Macdonald polynomials are obtained from the classical Macdonald polynomials using an operation called plethysm. It is natural to ask whether the modified Macdonald polynomials specialize to the partition function of some other particle system.
We answer this question in the affirmative with a certain multispecies totally asymmetric zero-range process (TAZRP). This link motivated a new tableaux formula for modified Macdonald polynomials. We present a Markov process on those tableaux that projects to the TAZRP and derive formulas for stationary probabilities and certain correlations, proving a remarkable symmetry property. This talk is based on joint work with Arvind Ayyer and James Martin.
Recent advances in data-driven modeling approaches have proven highly successful in a wide range of fields in science and engineering. In this talk, I will briefly discuss several ubiquitous challenges with the conventional model development / discretization / parameter inference / model revision loop that our methodology attempts to address. I will present our weak form methodology which has proven to have surprising performance properties. In particular, I will describe our equation learning (WSINDy) and parameter estimation (WENDy) algorithms. Lastly, I will discuss applications to several benchmark problems illustrating how our approach addresses several of the above issues and offers advantages in terms of computational efficiency, noise robustness, and modest data needs (in an online learning context).
The study of representations of fundamental groups of surfaces into Lie groups is captured by the character variety. One main tool to study character varieties are Higgs bundles, a complex geometric tool. They fail to see the mapping class group symmetry. I will present an alternative approach which replaces Higgs bundles by so-called higher complex structures, given in terms of commuting nilpotent matrices. The resulting theory has many similarities to the non-abelian Hodge theory. Joint with Georgios Kydonakis and Charlie Reid.
Mathematics lecture
In the 1950s Phil Anderson made a prediction about the effect of random impurities on the conductivity properties of a crystal. Mathematically, these questions amount to the study of solutions of differential or difference equations and the associated spectral theory of self-adjoint operators obtained from an ergodic process. With the arrival of quasicrystals, in addition to random models, nonrandom lattice models such as those generated by irrational rotations or skew-rotations on tori have been studied over the past 30 years.
By now, an extensive mathematical theory has developed around Anderson’s predictions, with several questions remaining open. This talk will attempt to survey certain aspects of the field, with an emphasis on the theory of SL(2,R) cocycles with an irrational or Diophantine rotation on the circle as base dynamics. In this setting, Artur Avila discovered about a decade ago that the Lyapunov exponent is piecewise affine in the imaginary direction after complexification of the circle. In fact, the slopes of these affine functions are integer valued. This is easy to see in the uniformly hyperbolic case, which is equivalent to energies falling into the gaps of the spectrum, due to the winding number of the complexified Lyapunov exponent. Remarkably, this property persists also in the non-uniformly hyperbolic case, i.e., on the spectrum of the Schrödinger operator. This requires a delicate continuity property of the Lyapunov exponent in both energy and frequency. Avila built his global theory (Acta Math. 2015) on this quantization property. I will present some recent results with Rui HAN (Louisiana) connecting Avila’s notion of acceleration (the slope of the complexified Lyapunov exponent in the imaginary direction) to the number of zeros of the determinants of finite volume Hamiltonians relative to the complex toral variable. This connection allows one to answer questions arising in the supercritical case of Avila’s global theory concerning the measure of the second stratum, Anderson localization on this stratum, as well as settle a conjecture on the Hölder regularity of the integrated density of states.
The Intermediate Long Wave equation (ILW) describes long internal gravity waves in stratified fluids. As the depth parameter in the equation approaches zero or infinity, the ILW formally approaches the Kortweg-deVries equation (KdV) or the Benjamin-Ono equation (BO), respectively. Kodama, Ablowitz and Satsuma discovered the formal complete integrability of ILW and formulated inverse scattering transform solutions. If made rigorous, the inverse scattering method will provide powerful tools for asymptotic analysis of ILW. In this talk, I will present some recent results on the ILW direct scattering problem. In particular, a Lax pair formulation is clarified, and the spectral theory of the Lax operators can be studied. Existence and uniqueness of scattering states are established for small interaction potential. The scattering matrix can then be constructed from the scattering states. The solution is related to the theory of analytic functions on a strip. This is joint work with Peter Perry.
In this talk we discuss our proof of a recent conjecture of Guo and Poznanovi\'{c} concerning chains in certain 01-fillings of moon polyominoes. A key ingredient of our proof is a correspondence between words $w$ and pairs $(\mathcal{W}(w), \mathcal{M}(w))$ of increasing tableaux such that $\mathcal{M}(w)$ determines the lengths of the longest strictly increasing and strictly decreasing sequences in every subinterval of $w$. (It will be noted that similar and well-studied correspondences like RSK insertion and Hecke insertion fail in this regard.) To define our correspondence we make use of Thomas and Yong's K-infusion operator and then use it to obtain the bijections that prove the conjecture of Guo and Poznanovi\'{c}. (Joint work with D. Saracino.)
The braid group (which encodes the braiding of n strands) has a canonical projection to the symmetric group (recording where the ends of the strands go). We ask the question: what are the extensions of the symmetric group by abelian groups that arise as quotients of the braid group, by a refinement of this canonical projection? To answer this question, we study a particular twisted coefficient system for the symmetric group, called the integral pair module. In this module, we find the maximal submodule in each commensurability class. We find the cohomology classes characterizing each such extension, and for context, we describe the second cohomology group of the symmetric group with coefficients in the most interesting of these modules. This is joint work with Trevor Nakamura.
Given a discrete set $\Lambda\subseteq\mathbb{R}$ and an interval $I$, define the sequence of complex exponentials in $L^2(I)$, $\mathcal{E}(\Lambda)$, by $\{e^{2\pi i\lambda t}\colon \lambda\in\Lambda\}$. A fundamental result in harmonic analysis says that if $\mathcal{E}(\frac{1}{b}\mathbb{Z})$ is an orthogonal basis for $L^2(I)$ for any interval $I$ of length $b$. It is also well-known that there exist sets $\Lambda$, which may be irregular, such that sets $\mathcal{E}(\Lambda)$ form nonorthogonal bases (known as Riesz bases) for $L^2(S)$, for $S\subseteq\mathbb{R}$ not necessarily an interval.
Given $\mathcal{E}(\Lambda)$ that forms a Riesz basis for $L^2[0,1]$ and some 0 < a < 1, Avdonin showed that there exists $\Lambda'\subseteq \Lambda$ such that $\mathcal{E}(\Lambda')$ is a Riesz basis for $L^2[0,a]$ (called basis extraction). Lyubarskii and Seip showed that this can be done in such a way that $\mathcal{E}(\Lambda \setminus \Lambda')$ is also a Riesz basis for $L^2[a,1]$ (called basis splitting). The celebrated result of Kozma and Nitzan shows that one can extract a Riesz basis for $L^2(S)$ from $\mathcal{E}(\mathbb{Z})$ where $S$ is a union of disjoint subintervals of $[0,1]$.
In this talk we construct sets $\Lambda_I\subseteq\mathbb{Z}$ such that the $\mathcal{E}(\Lambda_I)$ form Riesz bases for $L^2(I)$ for corresponding intervals $I$, with the added compatibility property that unions of the sets $\Lambda_I$ generate Riesz bases for unions of the corresponding intervals. The proof of our result uses an interesting assortment of tools from analysis, probability, and number theory. We will give details of the proof in the talk, together with examples and a discussion of recent developments. The work discussed is joint with Shauna Revay (GMU and Accenture Federal Services (AFS)), and Goetz Pfander (Catholic University of Eichstaett-Ingolstadt).
A fundamental result in Asymptotic Geometric Analysis is Dvoretzky’s theorem, which asserts that almost euclidean structure is locally present in any high-dimensional normed space. V. MIlman promoted the random version of the “Dvoretzky Theorem” by introducing the “concentration of measure Phenomenon.” Quantifying this phenomenon is important in theory as well as in applications. In this talk I will explain how techniques from High-dimensional Probability can be exploited to obtain optimal bounds on the randomized Dvoretzky theorem. Based on joint work(s) with Petros Valettas.
We report on recent results regarding the limiting absorption principle for multi-dimensional, massless Dirac-type operators (implying absence of singularly continuous spectrum) and continuity properties of the associated spectral shift function.
We will motivate our interest in this circle of ideas by briefly describing the connection to the notion of the Witten index for a certain class of non-Fredholm operators.
This is based on various joint work with A. Carey, J. Kaad, G. Levitina, R. Nichols, D. Potapov, F. Sukochev, and D. Zanin.
The Fermi variety plays a crucial role in the study of periodic operators. In this talk, I will first discuss recent works on the irreducibility of the Fermi variety for discrete periodic Schr\"odinger operators. I will then discuss the applications to solve problems of embedded eigenvalues, isospectrality and quantum ergodicity.
The celebrated Hajnal-Szemerédi theorem gives best possible minimum degree conditions for clique-factors in graphs. There have been some recent variants of this result into several settings, each of which has some sort of randomness come into play. We will give a survey on these problems and the recent developments.
The estimation of distributions of complex objects from high-dimensional data with low-dimensional structures is an important topic in statistics and machine learning. Deep generative models achieve this by encoding and decoding data to generate synthetic realistic images and texts. A key aspect of these models is the extraction of low-dimensional latent features, assuming data lies on a low-dimensional manifold. We study this by developing a minimax framework for distribution estimation on unknown submanifolds with smoothness assumptions on the target distribution and the manifold. The framework highlights how problem characteristics, such as intrinsic dimensionality and smoothness, impact the limits of high-dimensional distribution estimation. Our estimator, which is a mixture of locally fitted generative models, is motivated by differential geometry techniques and covers cases where the data manifold lacks a global parametrization.
Note the unusual time!
In this talk we explore a class of $\lambda$-convex bodies, i.e., convex bodies with curvature at each point of their boundary bounded below by some $\lambda >0$. For such bodies, we solve two reverse isoperimetric problems.
In $\mathbb{R}^3$, we show that the intersection of two balls of radius $1/\lambda$ (a $\lambda$-convex lens) is the unique volume minimizer among all $\lambda$-convex bodies of given surface area. We also show a reverse inradius inequality in arbitrary dimension which says that the $\lambda$-convex lens has the smallest inscribed ball among all $\lambda$-convex bodies of given surface area.
This is a joint work with Kostiantyn Drach.
We will prove the key lemma underlying the probabilistic unique continuation result of Ding-Smart, namely that for "thin" tilted rectangles, boundedness on all of one of the long edges and on a 1-\varepsilon proportion of the opposite long edge implies a bound (in terms of the dimensions of the rectangle) on the whole rectangle (with high probability).
A well-known conjecture of Dennis Sullivan asserts that a hyperbolic n-manifold with n>2 cannot admit a complex structure. This conjecture is known to be true in dimension four but little is known in higher dimensions. In this talk, I will outline a new proof of the fact that a hyperbolic 4-manifold cannot support a complex structure. This new proof has some nice features, and it generalizes to show that all extended graph 4-manifolds with positive Euler number cannot support a complex structure. This is joint work with M. Albanese.
We study the minimum number of paths needed to express the edge set of a given graph as the symmetric difference of the edge sets of the paths. This can be seen as a weakening of Gallai’s path decomposition problem, and a variant of the “odd cover” problem of Babai and Frankl which asks for the minimum number of complete bipartite graphs whose symmetric difference gives the complete graph. We relate this “path odd-cover” number of a graph to other known graph parameters and prove some bounds. Joint work with Steffen Borgwardt, Calum Buchanan, Eric Culver, Bryce Frederickson, and Puck Rombach.
A link of an isolated complex surface singularity is the intersection of the surface with a small sphere centered at the singular point. The link is a smooth 3-manifold that carries a natural contact structure (given by complex tangencies); one might then want to study its symplectic or Stein fillings. A special family of Stein fillings, called Milnor fillings, can be obtained by smoothing the singular point of the original complex surface. We will discuss some properties and constructions of Milnor fillings and general Stein fillings, and ways to detect whether the link of singularity has Stein fillings that do not arise from smoothings.
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
G. W. Hill made major contributions to Celestial Mechanics. One of them is to develop his lunar theory as an alternative approach for the study of the motion of the Moon around the Earth, which is the classical Lunar Hill problem. The mathematical model we study is one of the extensions of the classical Hill approximation of the restricted three-body problem. Considering a restricted four body problem, with a hierarchy between the bodies: two larger bodies, a smaller one and a fourth infinitesimal body, we encounter the shapes of the three heavy bodies via oblateness. We first find that the triangular central configurations of the three heavy bodies is a scalene triangle. Through the application of the Hill approximation, we obtain the limiting Hamiltonian that describes the dynamics of the infinitesimal body in a neighborhood of the smaller body. As a motivating example, we identify the three heavy bodies with the Sun, Jupiter and the Jupiter’s Trojan asteroid Hektor.
The Deligne category of symmetric groups is the additive Karoubi closure of the partition category. The partition category may be interpreted, following Comes, via a particular linearization of the category of two-dimensional oriented cobordisms. In this talk we will use a generalization of this approach to the Deligne category coupled with the universal construction of two-dimensional topological theories to construct their multi-parameter monoidal generalizations, one for each rational function in one variable. This talk is based on joint work with M. Khovanov.
Mathapalooza! is the biggest math event of the Atlanta Science Festival
Mathapalooza! is back at this year's Atlanta Science Festival! Come join us on Saturday, March 18, for an afternoon of mathematical fun beginning at 1:00pm at the Paideia School. There will be interactive puzzles and games, artwork, music, stage acts, and mathematics in motion.
We will finish our proof of the key lemma for the probabilistic unique continuation principle used in Ding-Smart. We will also briefly recall enough of the theory of martingales to clarify a use of Azuma's inequality, and the basic definitions of \epsilon-nets and \epsilon-packings required to formulate the basic volumetric bound for these in e.g. the unit sphere, before using these to complete the proof.
A fundamental conjecture of tight closure theory is every weakly F-regular ring is strongly F -regular. There has been incremental progress on this conjecture since the inception of tight closure. Most notably, the conjecture has been resolved for rings graded over a field by Lyubeznik and Smith. Otherwise, known progress around the conjecture have required assumptions on the ring that are akin to being Gorenstein. We extend known cases by proving the equivalence of F -regularity classes for rings whose anti-canonical algebra is Noetherian on the punctured spectrum. The anti-canonical algebra being Noetherian for a strongly F -regular ring is conjectured to be a vacuous assumption. This talk is based on joint work with Ian Aberbach and Craig Huneke.
In this talk, we demonstrate the application of Neural Networks with Locally Converging Inputs (NNLCI) to simulate the scattering of electromagnetic waves around two-dimensional perfect electric conductors (PEC). The NNLCIs are designed to output high-fidelity numerical solutions from local patches of two coarse grid numerical solutions obtained by a convergent numerical scheme. Once trained, the NNLCIs can play the role of a computational cost-saving tool for repetitive computations with varying parameters. To generate the inputs to our NNLCI, we design on uniform rectangular grids a second-order accurate finite difference scheme that can handle curved PEC boundaries systematically. More specifically, our numerical scheme is based on the Back and Forth Error Compensation and Correction method together with the construction of ghost points via a level set framework, PDE-based extension technique, and what we term guest values. We illustrate the performance of our NNLCI subject to variations in incident waves as well as PEC boundary geometries.
In this talk, we will explore and make comparisons between various models that exist for spherical tensor categories associated to the category of representations of the quantum group U_q(SL_n). In particular, we will discuss the combinatorial model of Murakami-Ohtsuki-Yamada (MOY), the n-valent ribbon model of Sikora and the trivalent spider category of Cautis-Kamnitzer-Morrison (CKM). We conclude by showing that the full subcategory of the spider category from CKM, whose objects are monoidally generated by the standard representation and its dual, is equivalent as a spherical braided category to Sikora's quotient category. This proves a conjecture of Le and Sikora and also answers a question from Morrison's Ph.D. thesis.
Many results in extremal graph theory build on supersaturation of subgraphs. In other words, when a graph is dense enough, it contains many copies of a certain subgraph and these copies are then used as building blocks to force another subgraph of interest. Recently more success is found within this approach where one utilizes not only the large number of copies of a certain subgraph but a well-distributed collection of them to force the desired subgraph. We discuss some recent progress of this nature. The talk is built on joint work with Sean Longbrake, and with Sean Longbrake and Jie Ma.
The Fourier restriction conjecture and the Bochner-Riesz conjecture ask for Lebesgue space mapping properties of certain oscillatory integral operators. They both are central in harmonic analysis, are open in dimensions $\geq 3$, and notably have the same conjectured exponents. In the 1970s, H\"{o}rmander asked if a more general class of operators (known as H\"{o}rmander type operators) all satisfy the same $L^p$-boundedness as in the above two conjectures. A positive answer to H\"{o}rmander's question would resolve the above two conjectures and have more applications such as in the manifold setting. Unfortunately H\"{o}rmander's question is known to fail in all dimensions $\geq 3$ by the work of Bourgain and many others. It continues to fail in all dimensions $\geq 3$ even if one adds a ``positive curvature'' assumption which one does have in restriction and Bochner-Riesz settings. Bourgain showed that in dimension $3$ one always has the failure unless a derivative condition is satisfied everywhere. Joint with Shaoming Guo and Hong Wang, we generalize this condition to arbitrary dimension and call it ``Bourgain's condition''. We unify Fourier restriction and Bochner-Riesz by conjecturing that any H\"{o}rmander type operator satisfying Bourgain's condition should have the same $L^p$-boundedness as in those two conjectures. As evidence, we prove that the failure of Bourgain's condition immediately implies the failure of such an $L^p$-boundedness in every dimension. We also prove that current techniques on the two conjectures apply equally well in our conjecture and make some progress on our conjecture that consequently improves the two conjectures in higher dimensions. I will talk about some history and some interesting components in our proof.
This talk includes an interactive prop demonstration. There exist non-trivial loops in SO(3) (the familiar group of real life rotations) which can be visualized with Dirac's belt trick. Although the belt trick offers a vivid picture of a loop in SO(3), a belt is not a proof, so we will prove SO(n) is not simply connected (n>2), and find its universal covering group Spin(n) (n >2). Along the way we'll introduce the Clifford algebra and study its basic properties.
In the study of close to integrable Hamiltonian PDEs, a fundamental question is to understand the behavior of ''typical'' solutions. With this in mind it is natural to study the persistence of almost-periodic solutions and infinite dimensional invariant tori, which are indeed typical in the integrable case. Up to now almost all results in the literature deal with very regular solutions for model PDEs with external parameters giving a large modulation. In this talk I shall discuss a new result constructing Gevrey solutions for models with a weak parameter modulation.
This is a joint work with G.Gentile and M.Procesi.
Topological insulators are materials that exhibit unique physical properties due to their non-trivial topological order. One of the most notable consequences of this order is the presence of protected edge states as well as closure of bulk spectral gaps, which is known as the bulk-edge correspondence. In this talk, I will discuss the mathematical description of topological insulators and their related spectral properties. The presentation assumes only basic knowledge of spectral theory, and will begin with an overview of Floquet theory, Bloch bundles, and the Chern number. We will then examine the bulk-edge correspondence in topological insulators before delving into our research on closure of bulk spectral gaps for topological insulators with general edges. This talk is based on a joint work with Alexis Drouot.
Optimal transport has recently found applications in a variety of fields ranging from graphics to biology. Underlying these applications is a new statistical paradigm where the goal is to couple multiple data sources. It gives rise to interesting new questions ranging from the design of estimators to minimax rates of convergence. I will review several applications where the central problem consists in estimating transport maps. After studying optimal transport as a potential solution, I will argue that its entropic version is a good alternative model. In particular, it completely escapes the curse of dimensionality that plagues statistical optimal transport.
Special date and special room
We shall explain a simple remarkable stability phenomenon regarding the centers of the group algebras of the symmetric groups in n letters, as n goes to infinity. The same type of stability phenomenon extends to a wide class of finite groups including wreath products and finite general linear groups. Such stability has connections and applications to the cohomology rings of Hilbert schemes of n points on algebraic surfaces.
We will finish our proof of the key lemma for the probabilistic unique continuation principle used in Ding-Smart. We will also briefly recall enough of the theory of martingales to clarify a use of Azuma's inequality, and the basic definitions of \epsilon-nets and \epsilon-packings required to formulate the basic volumetric bound for these in e.g. the unit sphere, before using these to complete the proof.
The classic facility location problem seeks to open a set of facilities to minimize the cost of opening the chosen facilities and the total cost of connecting all the clients to their nearby open facilities. Such an objective may induce an unequal cost over certain socioeconomic groups of clients (i.e., total distance traveled by clients in such a group). This is important when planning the location of socially relevant facilities such as emergency rooms and grocery stores. In this work, we consider a fair version of the problem by minimizing the L_p-norm of the total distance traveled by clients across different socioeconomic groups and the cost of opening facilities, to penalize high access costs to open facilities across r groups of clients. This generalizes classic facility location (p = 1) and the minimization of the maximum total distance traveled by clients in any group (p = infinity). However, it is often unclear how to select a specific "p" to model the cost of unfairness. To get around this, we show the existence of a small portfolio of at most (log2r + 1) solutions for r (disjoint) client groups, where for any L_p-norm, at least one of the solutions is a constant-factor approximation with respect to any L_p-norm. We also show that such a dependence on r is necessary by showing the existence of instances where at least ~ sqrt(log2r) solutions are required in such a portfolio. Moreover, we give efficient algorithms to find such a portfolio of solutions. Additionally, We introduce the notion of refinement across the solutions in the portfolio. This property ensures that once a facility is closed in one of the solutions, all clients assigned to it are reassigned to a single facility and not split across open facilities. We give poly(exp(sqrt(r))-approximation for refinement in general metrics and O(log r)-approximation for the line and tree metrics. This is joint work with Swati Gupta and Mohit Singh.
Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.
There is a natural notion of `degree’ for functions from the symmetric group to the complex numbers, which translates roughly to saying the function counts certain weighted patterns. Low degree class functions have a classical interpretation in terms of the cycle structure of permutations. I will explain how to translate between pattern counts to cycle structure using a novel symmetric function identity analogous to the Murnaghan-Nakayama identity. This relationship allows one to lift many probabilistic properties of permutation statistics to certain non-uniform distributions, and I will present some results in this direction. This is joint work with Brendon Rhoades.
In this talk, we explore the fractional log-concavity property of generating polynomials of discrete distributions. This property is an analog to the Lorentzian [Branden-Huh’19]/log-concavity [Anari-Liu-OveisGharan-Vinzant’19] property of the generating polynomials of matroids. We show that multivariate generating polynomials without roots in a sector of the complex plane are fractionally log-concave. Furthermore, we prove that the generating polynomials of linear delta matroids and of the intersection between a linear matroid and a partition matroid have no roots in a sector, and thus are fractionally log-concave. Beyond root-freeness, we conjecture that for any subset F of {0,1}^n such that conv(F) has constantly bounded edge length, the generating polynomial for the uniform distribution over F is fractionally log-concave.
Based on joint works with Yeganeh Alimohammadi , Nima Anari and Kirankumar Shiragur.
In this talk, we present new gradient sliding results for constrained convex optimization with applications in image reconstruction and decentralized distributed optimization. Specifically, we will study classes of large-scale problems that minimizes a convex objective function over feasible set with linear constraints. We will show that by exploring the gradient sliding technique, the number of gradient evaluations of the objective function can be reduced by exploring the smoothness structure. Our results could lead to new decentralized algorithms for multi-agent optimization with graph topology invariant gradient/sampling complexity and new ADMM algorithms for solving total variation image reconstruction problems with accelerated gradient complexity.
In high dimensional contact and symplectic topology, finding interesting constructions for Legendrian submanifolds is an active area of research. Further, it is desirable that the constructions lend themselves nicely to computation of invariants. The doubling construction was defined by Ekholm, which uses Lagrangian fillings of a Legendrian knot in standard contact R^{2n-1} to produce a closed Legendrian submanifold in standard contact R^{2n+1}. Later Courte-Ekholm showed that symmetric doubles of embedded fillings are "uninteresting". In recent work the symmetric doubling construction was generalised to any contact manifold, giving two isotopic constructions related to open book decompositions of the ambient manifold. In a separate joint work with James Hughes, we explore the asymmetric doubling construction through Legendrian weaves.
Transport equations arise in the modelling of several complex systems, including mean field games. Such equations often involve nonlinear dependence of the solution in the drift. These nonlinear transport equations can be understood by developing a theory for transport equations with irregular drifts. In this talk, I will outline the well-posedness theory for certain transport equations in which the drift has a one-sided bound on the divergence, yielding contractive or expansive behavior, depending on the direction in which the equation is posed. The analysis requires studying the relationship between the transport and continuity equations and the associated ODE flow. The theory is then used to discuss certain nonlinear transport equations arising in the study of finite state-space mean field games. This is joint work with P.-L. Lions.
We show that if each edge of the complete bipartite graph $K_{n,n}$ is given a random list of $C(\log n)$ colors from $[n]$, then with high probability, there is a proper edge coloring where the color of each edge comes from the corresponding list. We also prove analogous results for Latin squares and Steiner triple systems. This resolves several related conjectures of Johansson, Luria-Simkin, Casselgren-Häggkvist, Simkin, and Kang-Kelly-Kühn-Methuku-Osthus. I will discuss some of the main ingredients which go into the proof: the Kahn-Kalai conjecture, absorption, and the Lovasz Local Lemma distribution. Based on joint work with Huy Tuan Pham.
We will look at a number of interesting examples — some proven, others merely conjectured — of Hamburger moment sequences in combinatorics. We will consider ways in which this positivity may be expected, for instance in different types of combinatorial statistics on perfect matchings that turn out to encode moments in noncommutative analogues of the classical Central Limit Theorem. We will also consider situations in which this positivity may be surprising, and where proving it would open up new approaches to a class of very hard open problems in combinatorics.
I'll talk about the structure of the collection of all n-ples of eigenvalues of elements of Zariski-dense subgroups D of SL(n,R). Subgroups like this appear, for instance, as the images of holonomy representations of geometric structures. Our focus is a deep and useful result of Benoist, which states that the natural cone one is led to consider here has strong convexity and non-degeneracy properties, and a succinct, qualitative characterization of the cones that so arise from Zariski-dense subgroups. The theorem comes out of a study of the dynamics of the actions of D on spaces of flags such as RP^n and the collection of open subsemigroups of SL(n,R). Everything in this talk is from Benoist’s paper Propriétés Asymptotiques des Groupes Linéaires (GAFA, 2002), and holds in far more generality than I'll state.
We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It also has several applications in optics
and medical imaging among others.
The problem can be recast as a geometric problem: Can one determine
the Riemannian metric of a Riemannian manifold with boundary by
measuring the distance function between boundary points? This is the
boundary rigidity problem.
We will also describe some recent results, joint with Plamen Stefanov
and Andras Vasy, on the partial data case, where you are making
measurements on a subset of the boundary.
The anomaly cancellation is a basic property of the Standard Model, crucial for its consistence. We consider a lattice chiral gauge theory of massless Wilson fermions interacting with a non-compact massiveU(1) field coupled with left- and right-handed fermions in four dimensions. We prove in the infinite volume limit, for weak coupling and inverse lattice step of the order of boson mass, that the anomaly vanishes up to subleading corrections and under the same condition as in the continuum. The proof is based on a combination of exact Renormalization Group, non-perturbative decay bounds of correlations and lattice symmetries.
The talk can be accessed via zoom: Meeting ID: 989 6686 9205
The general subject of the talk is spectral theory of discrete (tight-binding) Schrodinger operators on d-dimensional lattices. For operators with periodic potentials, it is known that the spectra of such operators are purely absolutely continuous. For random i.i.d. potentials, such as the Anderson model, it is expected and can be proved in many cases that the spectra are almost surely purely point with exponentially decaying eigenfunctions (Anderson local- ization). Quasiperiodic operators can be placed somewhere in between: depending on the potential sampling function and the Diophantine properties of the frequency and the phase, one can have a large variety of spectral types. We will consider quasiperiodic operators
(H(x)ψ)n =ε(∆ψ)n +f(x+n·ω)ψn, n∈Zd,
where ∆ is the discrete Laplacian, ω is a vector with rationally independent components, and f is a 1-periodic function on R, monotone on (0,1) with a positive lower bound on the derivative and some additional regularity properties. We will focus on two methods of proving Anderson localization for such operators: a perturbative method based on direct analysis of cancellations in the Rayleigh-Schr ̈odinger perturbation series for arbitrary d, and a non?perturbative method based on the analysis of Green?s functions for d = 1, originally developed by S. Jitomirskaya for the almost Mathieu operator.
The talk is based on joint works with S. Krymskii, L. Parnovski, and R. Shterenberg (per- turbative methods) and S. Jitomirskaya (non-perturbative methods).
Zoom link to the talk: https://gatech.zoom.us/j/94387417679
We will present the notion of Stein kernel, which provides generalizations of the integration by parts, a.k.a. Stein's formula, for the normal distribution (which has a constant Stein kernel, equal to its covariance). We will first focus on dimension one, where under good conditions the Stein kernel has an explicit formula. We will see that the Stein kernel appears naturally as a weighting of a Poincaré type inequality and that it enables precise concentration inequalities, of the Mills' ratio type. In a second part, we will work in higher dimensions, using in particular Max Fathi's construction of a Stein kernel through the so-called "moment maps" transportation. This will allow us to describe the performance of some shrinkage and thresholding estimators, beyond the classical assumption of Gaussian (or spherical) data. This presentation is mostly based on joint works with Max Fathi, Larry Goldstein, Gesine Reinert and Jon Wellner.
We will actually finish our proof of the key technical lemma for the quantitative unique continuation principle of Ding-Smart, reviewing briefly the volumetric bound from the theory of \varepsilon-coverings/nets/packings. From there, we will outline at a high level the strategy for the rest of the proof of the unique continuation principle using this key lemma, before starting the proof in earnest.
We consider the problem of estimating the factors of a rank-1 matrix with i.i.d. Gaussian, rank-1 measurements that are nonlinearly transformed and corrupted by noise. Considering two prototypical choices for the nonlinearity, we study the convergence properties of a natural alternating update rule for this nonconvex optimization problem starting from a random initialization. We show sharp convergence guarantees for a sample-split version of the algorithm by deriving a deterministic recursion that is accurate even in high-dimensional problems. Our sharp, non-asymptotic analysis also exposes several other fine-grained properties of this problem, including how the nonlinearity and noise level affect convergence behavior.
On a technical level, our results are enabled by showing that the empirical error recursion can be predicted by our deterministic sequence within fluctuations of the order n−1/2 when each iteration is run with n observations. Our technique leverages leave-one-out tools originating in the literature on high-dimensional M–estimation and provides an avenue for sharply analyzing higher-order iterative algorithms from a random initialization in other high-dimensional optimization problems with random data.
We study quantizations of SL_n-character varieties, which appears as moduli spaces for many geometric structures. Our main goal is to establish the existence of several quantum trace maps. In the classical limit, they reduce to the Fock-Goncharov trace maps, which are coordinate charts on moduli spaces of SL_n-local systems used in higher Teichmuller theory. In the quantized theory, the algebras are replaced with non-commutative deformations. The domains of the quantum trace maps are the SL_n-skein algebra and the reduced skein algebra, and the codomains are quantum tori, which are non-commutative analogs of Laurent polynomial algebras. In this talk, I will review the classical theory and sketch the definition of the quantum trace maps.
Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Abstract: In this talk I will explain a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for different equations in fluid dynamics. The new numerical framework is shown to be both robust and readily adaptable to several situations.
Joint work with Yongji Wang, Ching-Yao Lai and Tristan Buckmaster.
I will discuss some problems in geometric topology, and relate them to graph-theoretic properties. I will give some open problems, and answer questions of Kalai, Belolipetski, Gromov and others.
Intersection bodies are a popular construction in convex geometry. I will give an introduction on these objects, convex algebraic geometry, and starshaped sets in general. Then, we will analyze some features of intersection bodies and focus on the polyotopal case. Intersection bodies of polytopes are always semialgebraic sets and they are naturally related to hyperplane arrangements, which reveal their boundary structure. Finally, we will investigate their convexity, in the two-dimensional case. The exposition will be enriched by examples and computations. This is based on joint works with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.
The Kauffman bracket skein module S(M) of a 3-manifold M classifies polynomial invariants of links in M satisfying Kauffman bracket skein relations. Witten conjectured that the skein module (over a field, with generic A) is finite dimensional for any closed 3-manifold M. This conjecture was proved by Gunningham, Jordan, and Safronov, however their work does not lead to an explicit computation of S(M).
In fact, S(M) has been computed for a few specific families of closed 3-manifolds so far. We introduce a novel method of computing these skein modules for certain rational homology spheres. (This is joint work with R. Detcherry and E. Kalfagianni.)
Speaker will present in person
Semi-supervised learning refers to the problem of recovering an input-output map using many unlabeled examples and a few labeled ones. In this talk I will survey several mathematical questions arising from the Bayesian formulation of graph-based semi-supervised learning. These questions include the modeling of prior distributions for functions on graphs, the derivation of continuum limits for the posterior, the design of scalable posterior sampling algorithms, and the contraction of the posterior in the large data limit.
In this talk, I will present a method to construct nontrivial global solutions to some quasilinear wave equations in three space dimensions. Starting from a global solution to the geometric reduced system satisfying several pointwise estimates, we find a matching exact global solution to the original quasilinear wave equations. As an application of this method, we will construct nontrivial global solutions to Fritz John's counterexample $\Box u=u_tu_{tt}$ and the 3D compressible Euler equations without vorticity for $t\geq 0$.
Ramsey proved that for every positive integer r, every sufficiently large graph contains as an induced subgraph either a complete graph on r vertices or an independent set with r vertices. It is well known that every sufficiently large, connected graph contains an induced subgraph isomorphic to one of a large complete graph, a large star, and a long path. We prove an analogous result for 2-connected graphs. Similarly, for infinite graphs, every infinite connected graph contains an induced subgraph isomorphic to one of the following: an infinite complete graph, an infinite star, and a ray. We also prove an analogous result for infinite 2-connected graphs.
Are you tired of having to read a bunch of words during a seminar talk? Well, you’re in luck! This talk will be a (nearly) word-free exploration of a construction called unicorn paths. These paths are incredibly useful and can be used to show that both the curve graph and the arc graph of a surface are hyperbolic.
In first-passage percolation (FPP), we let $\tau_v$ be i.i.d. nonnegative weights on the vertices of a graph and study the weight of the minimal path between distant vertices. If $F$ is the distribution function of $\tau_v$, there are different regimes: if $F(0)$ is small, this weight typically grows like a linear function of the distance, and when $F(0)$ is large, the weight is typically of order one. In between these is the critical regime in which the weight can diverge but does so sublinearly. This talk will consider a dynamical version of critical FPP on the triangular lattice where vertices resample their weights according to independent rate-one Poisson processes. We will discuss results that show that if the sum of $F^{-1}(1/2+1/2^k)$ diverges, then a.s. there are exceptional times at which the weight grows atypically, but if the sum of $k^{7/8} F^{-1}(1/2+1/2^k)$ converges, then a.s. there are no such times. Furthermore, in the former case, we compute the Hausdorff and Minkowski dimensions of the exceptional set and show that they can be but need not be equal. Then we will consider what the model looks like when the weight of a long path is unusually small by considering an analogous construction to Kesten's incipient infinite cluster in the FPP setting. This is joint work with M. Damron, J. Hanson, W.-K. Lam.
Finally, we discuss a result related to work of Damron-Lam-Wang ('16) that the growth of the passage time to distance $n$ ($\mathbb{E}T(0,\partial B(n))$, where $B(n) = [-n,n]^2$) has the same order (up to a constant factor) as the sequence $\mathbb{E}T^{\text{inv}}(0,\partial B(n))$. This second passage time is the minimal total weight of any path from 0 to $\partial B(n)$ that resides in a certain embedded invasion percolation cluster. We discuss a result that claims this constant factor cannot be taken to be 1. This result implies that the time constant for the model is different than that for the related invasion model, and that geodesics in the two models have different structures. This was joint work with M. Damron.
This talk is centered around a symplectic approach to eigenvalue problems for systems of ordinary differential operators (e.g., Sturm-Liouville operators, canonical systems, and quantum graphs), multidimensional elliptic operators on bounded domains, and abstract self-adjoint extensions of symmetric operators in Hilbert spaces. The symplectic view naturally relates spectral counts for self-adjoint problems to the topological invariant called the Maslov index. In this talk, the notion of the Malsov index will be introduced in analytic terms and an overview of recent results on its role in spectral theory will be given.
Whitney showed that every planar triangulation without separating triangles is Hamiltonian. This result was extended to all $4$-connected planar graphs by Tutte. Hakimi, Schmeichel, and Thomassen showed the first lower bound $n/ \log _2 n$ for the number of Hamiltonian cycles in every $n$-vertex $4$-connected planar triangulation and in the same paper, they conjectured that this number is at least $2(n-2)(n-4)$, with equality if and only if $G$ is a double wheel. We show that every $4$-connected planar triangulation on $n$ vertices has $\Omega(n^2)$ Hamiltonian cycles. Moreover, we show that if $G$ is a $4$-connected planar triangulation on $n$ vertices and the distance between any two vertices of degree $4$ in $G$ is at least $3$, then $G$ has $2^{\Omega(n^{1/4})}$ Hamiltonian cycles.
The Laplacian of a graph is a real symmetric matrix given by $L=D-A$, where $D$ is the degree matrix of the graph and $A$ is the adjacency matrix. The Laplacian is a central object in spectral graph theory, and the spectrum of $L$ contains information on the graph. In the case of a random graph the Laplacian will be a random real symmetric matrix with dependent entries. These random Laplacian matrices can be generalized by taking $A$ to be a random real symmetric matrix and $D$ to be a diagonal matrix with entries equal to the row sums of $A$. We will consider the eigenvalues of general random Laplacian matrices, and the challenges raised by the dependence between $D$ and $A$. After discussing the bulk global eigenvalue behavior of general random Laplacian matrices, we will focus in detail on fluctuations of the largest eigenvalue of $L$ when $A$ is a matrix of independent Gaussian random variables. The asymptotic behavior of these Gaussian Laplacian matrices has a particularly nice free probabilistic interpretation, which can be exploited in the study of their eigenvalues. We will see how this interpretation can locate the largest eigenvalue of $L$ with respect to the largest entry of $D$. This talk is based on joint work with Kyle Luh and Sean O'Rourke.
We will start sketching the proof of the quantitative unique continuation principle used in Ding-Smart from their key lemma. We will discuss the proof of a growth lemma from our key lemma, which (roughly) says that with high probability, eigenfunctions which are small on a high proportion of sites do not grow too rapidly.
Lefschetz fibrations are very useful in the sense that they have one-one correspondence with the relations in the Mapping Class Groups and they can be used to construct exotic (homeomorphic but not diffeomorphic) 4-manifolds. In this series of talks, we will first introduce Lefschetz fibrations and Mapping Class Groups and give examples. Then, we will dive more into 4-manifold world. More specifically, we will talk about the history of exotic 4-manifolds and we will define the nice tools used to construct exotic 4-manifolds, like symplectic normal connect sum, Rational Blow-Down, Luttinger Surgery, Branch Covers, and Knot Surgery. Finally, we will provide various constructions of exotic 4-manifolds.
In this talk, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624].
We exploit the full symmetry of the problem by developing a block-diagonalization of the underlying matrix algebra and use it to improve bounds on several concrete instances. Our results imply that $\mathrm{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\mathrm{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\mathrm{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\mathrm{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all~$n$. The latter three bounds are computed using a new relaxation of the original semidefinite programming bound, by only requiring one small matrix block to be positive semidefinite. Our lower bound on $K_{13,n}$ implies that for each fixed $m \geq 13$, $\lim_{n \to \infty} \text{cr}(K_{m,n})/Z(m,n) \geq 0.8878 m/(m-1)$. Here $Z(m,n)$ is the Zarankiewicz number: the conjectured crossing number of $K_{m,n}$.
This talk is based on joint work with Sven Polak.
Zoom Link: Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Abstract: Linear response refers to the smooth change in the statistics of an observable in a dynamical system in response to a smooth parameter change in the dynamics. The computation of linear response has been a challenge, despite work pioneered by Ruelle giving a rigorous formula in Anosov systems. This is because typical linear perturbation-based methods are not applicable due to their instability in chaotic systems. Here, we give a new differentiable splitting of the parameter perturbation vector field, which leaves the resulting split Ruelle's formula amenable to efficient computation. A key ingredient of the overall algorithm, called space-split sensitivity, is a new recursive method to differentiate quantities along the unstable manifold.
In the second part, we discuss a new KAM method-inspired construction of transport maps. Transport maps are transformations between the sample space of a source (which is generally easy to sample) and a target (typically non-Gaussian) probability distribution. The new construction arises from an infinite-dimensional generalization of a Newton method to find the zero of a "score operator". We define such a score operator that gives the difference of the score -- gradient of logarithm of density -- of a transported distribution from the target score. The new construction is iterative, enjoys fast convergence under smoothness assumptions, and does not make a parametric ansatz on the transport map.
The Meeting on Applied Algebraic Geometry (MAAG 2023) is a regional gathering which attracts participants primarily from the South-East of the United States. Previous meetings took place at Georgia Tech in 2015, 2018, and 2019, and at Clemson in 2016.
For more information and to register, please visit https://sites.google.com/view/maag-2023. The registration is free until February 28th, 2023, and the registration fee will become $50 after that.
MAAG will be followed by a Macaulay2 Day on April 16.
Organizers: Abeer Al Ahmadieh, Greg Blekherman, Anton Leykin, and Josephine Yu.
Several experts from the M2internals group will give tutorials and lead the discussion. This is a part of Meeting on Applied Algebraic Geometry.
(Macaulay2 is a software system devoted to supporting research in algebraic geometry and commutative algebra.)
Zoom: https://gatech.zoom.us/j/98256586748?pwd=SkJLZ3ZKcjZsM0JkbGdyZ1Y3Tk9udz0... />
Meeting ID: 982 5658 6748<br />
Password: 929165
A class of graphs is said to be $\chi$-bounded with binding function $f$ if for every such graph $G$, it satisfies $\chi(G) \leq f(\omega(G)$, and polynomially $\chi$-bounded if $f$ is a polynomial. It was conjectured that chair-free graphs are perfectly divisible, and hence admit a quadratic $\chi$-binding function. In addition to confirming that chair-free graphs admit a quadratic $\chi$-binding function, we will extend the result by demonstrating that $t$-broom free graphs are polynomially $\chi$-bounded for any $t$ with binding function $f(\omega) = O(\omega^{t+1})$. A class of graphs is said to satisfy the Vizing bound if it admits the $\chi$-binding function $f(\omega) = \omega + 1$. It was conjectured that (fork, $K_3$)-free graphs would be 3-colorable, where fork is the graph obtained from $K_{1, 4}$ by subdividing two edges. This would also imply that (paw, fork)-free graphs satisfy the Vizing bound. We will prove this conjecture through a series of lemmas that constrain the structure of any minimal counterexample.
The group SL(n) x SL(n) acts on m-tuples of n x n matrices by simultaneous left-right multiplication. Visu Makam and the presenter showed the ring of invariants is generated by invariants of degree at most mn^4. We will also discuss geometric aspects of this action and connections to algebraic complexity and the notion of noncommutative rank.
The speaker will present in person.
This talk will have two parts. The first half will describe how to construct symplectic structures on trisected 4-manifolds. This construction is inspired by projective complex geometry and completely characterizes symplectic 4-manifolds among all smooth 4-manifolds. The second half will address a curious phenomenon: symplectic 4-manifolds appear to not admit any interesting connected sum decompositions. One potential explanation is that every embedded 3-sphere can be made contact-type. I will outline some strategies to prove this from a trisections perspective, describe some of the obstructions, and give evidence that these obstructions may be overcome.
We will discuss the one-phase Muskat problem concerning the free boundary of Darcy fluids in porous media. It is known that there exists a class of non-graph initial boundary leading to self-intersection at a single point in finite time (splash singularity). On the other hand, we prove that the problem has a unique global-in-time solution if the initial boundary is a periodic Lipschitz graph of arbitrary size. This is based on joint work with H. Dong and F. Gancedo.
We discuss flows (and group-connectivity) in signed graphs, and prove a new result about group-connectivity in 3-edge-connected signed graphs. This is joint work with Alejandra Brewer Castano and Kathryn Nurse.
This seminar has beeb cancelled and will be rescheduled next year. We discuss a kind of weak type inequality for the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operators that was first studied by Muckenhoupt and Wheeden and later by Sawyer. This formulation treats the weight for the image space as a multiplier, rather than a measure, leading to fundamentally different behavior. Such inequalities arise in the generalization of weak-type spaces to the matrix weighted setting and find applications in scalar two-weight norm inequalities via interpolation with change of measures. In this talk, I will discuss quantitative estimates obtained for $A_p$ weights, $p > 1$, that generalize those results obtained by Cruz-Uribe, Isralowitz, Moen, Pott and Rivera-Ríos for $p = 1$. I will also discuss an endpoint result for the Riesz potentials.
Understanding the route to thermalization of a physical system is a fundamental problem in statistical mechanics. When a system is initialized far from thermodynamical equilibrium, many interesting phenomena may arise. Among them, a lot of interest is attained by systems subjected to periodic driving (Floquet systems), which under certain circumstances can undergo a two-stage long dynamics referred to as "prethermalization", showing nontrivial physical features. In this talk, we present some prethermalization results for a class of lattice systems with quasi-periodic external driving in time. When the quasi-periodic driving frequency is large enough or the strength of the driving is small enough, we show that the system exhibits a prethermal state for exponentially long times in the perturbative parameter. Moreover, we focus on the case when the unperturbed Hamiltonian admits constants of motion and we prove the quasi-conservation of a dressed version of them. We discuss applications to perturbations of the Fermi-Hubbard model and the quantum Ising chain.
Join Zoom Meeting
https://gatech.zoom.us/j/96817326631
Several well-known problems in first-passage percolation relate to the behavior of infinite geodesics: whether they coalesce and how rapidly, and whether doubly infinite "bigeodesics'' exist. In the plane, a version of coalescence of "parallel'' geodesics has previously been shown; we will discuss new results that show infinite geodesics from the origin have zero density in the plane. We will describe related forthcoming work showing that geodesics coalesce in dimensions three and higher, under unproven assumptions believed to hold below the model's upper critical dimension. If time permits, we will also discuss results on the bigeodesic question in dimension three and higher.
We will finish the proof of the unique continuation theorem, starting with a brief discussion of the growth lemma discussed at our previous talk. After this, we will reduce unique continuation for untitled squares to unique continuation for tilted squares, and using the tilted square growth lemma prove such unique continuation result.
We present a polynomial-time algorithm for robustly learning an unknown affine transformation of the standard hypercube from samples, an important and well-studied setting for independent component analysis (ICA). Total variation distance is the information-theoretically strongest possible notion of distance in our setting and our recovery guarantees in this distance are optimal up to the absolute constant factor multiplying the fraction of corruption. Our key innovation is a new approach to ICA (even to outlier-free ICA) that circumvents the difficulties in the classical method of moments and instead relies on a new geometric certificate of correctness of an affine transformation. Our algorithm is based on a new method that iteratively improves an estimate of the unknown affine transformation whenever the requirements of the certificate are not met.
This week, we'll continue discussing the rational blowdown and use it to construct small exotic 4-manifolds. We will see how we can view the rational blowdown as a "monodromy substitution." Finally, if time allows, we will discuss knot surgery on 4-manifolds.
We show that in Euclidean 3-space any closed curve which contains the unit sphere in its convex hull has length at least $4\pi$, and characterize the case of equality, which settles a conjecture of Zalgaller. Furthermore, we establish an estimate for the higher dimensional version of this problem by Nazarov, which is sharp up to a multiplicative constant, and is based on Gaussian correlation inequality. Finally we discuss connections with sphere packing problems, and other optimization questions for convex hull of space curves. This is joint work with James Wenk.
Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Abstract: The timing of human sleep is strongly modulated by the 24 hour circadian rhythm, our internal biological clock, and the homeostatic sleep drive, one’s need for sleep which depends on prior awakening. The parameters dictating the evolution of the homeostatic sleep drive may vary with development and have been identified as important parameters for generating the transition from multiple sleeps to a single sleep episode per day. We employ piecewise-smooth ODE-based mathematical models to analyze developmentally-mediated transitions of sleep-wake patterns, including napping and non-napping behaviors. Our framework includes the construction of a circle map that captures the timing of sleep onsets on successive days. Analysis of the structure and bifurcations in the map reveals changes in the average number of sleep episodes per day in a period-adding-like structure. In two-state models of sleep-wake regulation, namely models that generate sleep and wake states, we observe saddle-node and border collision bifurcations in the maps. However, in our three-state model of sleep-wake regulation, which captures wake, rapid eye movement (REM) sleep, and non-REM sleep, these sequences are disrupted by period-doubling bifurcations and can exhibit bistability.
We show how the theory of Lorentzian polynomials extends to cones other than the positive orthant, and how this may be used to prove Hodge-Riemann relations of degree one for Chow rings. If time permits, we will show explicitly how the theory applies to volume polynomials of matroids and/or polytopes. Joint work with Petter Brändén.
There will be a pretalk 1-1:40pm in Skiles 006.
This talk is motivated by surprising connections between two very different approaches to 3-dimensional topology, and more precisely by the Kashaev-Murakami-Murakami Volume Conjecture, which relates the growth of colored Jones polynomials of a knot to the hyperbolic volume of its complement. I will discuss a closely related conjecture for diffeomorphisms of surfaces, based on the representation theory of the Kauffman bracket skein algebra of the surface, a quantum topology object closely related to the Jones polynomial of a knot. I will describe partial results obtained in joint work with Helen Wong and Tian Yang.
Hybrid version is available at: https://gatech.zoom.us/j/98003867540
Morphogenesis is the biological process that causes cells, tissues, or organisms to develop their shape. The theory of morphogenesis, proposed by Alan Turning, is a chemical model where biological cells differentiate and form patterns through intercellular reaction-diffusion mechanisms. Various reaction-diffusion models can produce a chemical pattern that mimics natural patterns. However, while they provide a plausible prepattern, they do not describe a mechanism in which the pattern is expressed. An alternative model is a mechanical model of the skin, initially described by Murray, Oster, and Harris. This model used mechanical interactions between cells without a chemical prepattern to produce structures like those observed in a Turing model. In this talk, we derive a modified version of the Murray, Oster, and Harris model incorporating nonlinear deformation effects. Since it is observed in some experiments that chemicals present in developing skin can cause or disrupt pattern formation, the mechanical model is coupled with a single diffusing chemical. Furthermore, it is observed that the interaction between tissue deformations with a diffusing chemical can cause a previously undescribed instability. This instability could describe both the pattern’s chemical patterning and mechanical expression without the need for a reaction-diffusion system.
I discuss some recent results, obtained jointly with David Wallauch, on the stability of self-similar wave maps under minimal regularity assumptions on the perturbation. More precisely, we prove the asymptotic stability of an explicitly known self-similar wave map in corotational symmetry. The key tool are Strichartz estimates for the linearized equation in similarity coordinates.
We will discuss proper q-colourings of sparse, bounded degree graphs when the maximum degree is near the so-called shattering threshold. This threshold, first identified in the statistical physics literature, coincides with the point at which all known efficient colouring algorithms fail and it has been hypothesized that the geometry of the solution space (the space of proper colourings) is responsible. This hypothesis is a cousin of the Overlap-Gap property of Gamarnik ‘21. Significant evidence for this picture was provided by Achlioptos and Coja-Oghlan ‘08, who drew inspiration from statistical physics, but their work only explains the performance of algorithms on random graphs (average-case complexity). We extend their work beyond the random setting by proving that the geometry of the solution space is well behaved for all graphs below the “shattering threshold”. This follows from an original result about the structure of uniformly random colourings of fixed graphs. Joint work with François Pirot.
The study of Egyptian fractions, representing rational numbers as the sum of distinct unit fractions, is one of the oldest areas of number theory. In this talk we will discuss some fascinating problems in the area, including both open problems and some recent progress, such as the solution to the Erdos-Graham conjecture: 1 can be written as the sum of unit fractions with denominators drawn from an arbitrary set of integers of positive density.
Dynamical Sampling is, in a sense, a hypernym classifying the set of inverse problems arising from considering samples of a signal and its future states under the action of a bounded linear operator. Recent works in this area consider questions such as when can a given frame for a separable Hilbert Space, $\{f_k\}_{k \in I} \subset H$, be represented by iterations of an operator on a single vector and what are necessary and sufficient conditions for a system, $\{T^n \varphi\}_{n=0}^{\infty} \subset H$, to be a frame? In this talk, we will discuss the connection between frames given by iterations of a bounded operator and the theory of model spaces in the Hardy-Hilbert Space as well as necessary and sufficient conditions for a system generated by the orbit of a pair of commuting bounded operators to be a frame. This is joint work with Carlos Cabrelli.
Join Zoom meeting: https://gatech.zoom.us/j/96113517745
It is often expected that the local statistical behavior of eigenvalues of some system depends only on its local properties; for instance, the local distribution of zeros of orthogonal polynomials should depend only on the local properties of the measure of orthogonality. This phenomenon is studied using an object called the Christoffel-Darboux kernel. The most commonly studied case is known as bulk universality, where the rescaled limit of Christoffel-Darboux kernels converges to the sine kernel. We will present a new approach which gives for the first time a completely local sufficient condition for bulk universality. This approach is based on a matrix version of the Christoffel-Darboux kernel and the de Branges theory of canonical systems, and it applies to other self-adjoint systems with 2x2 transfer matrices such as continuum Schrodinger and Dirac operators. The talk is based on joint work with Benjamin Eichinger (Technical University Wien) and Brian Simanek (Baylor University).
Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Abstract: In this talk, I will discuss problems and results in the rigorous statistical mechanics of particle systems in a one-dimensional lattice.
I will briefly describe the classical examples, such as the Ising model and its various generalizations concerning the
existence of the free energy, thermodynamic limit and the phase transition phenomenon.
Towards the end of the talk, I will talk about a recent work in collaboration with Jorge Littin, on a generalization of the
Khanin and Sinai model with random interactions for which one can prove that there exists a critical behavior in the free
energy for some parameters of the model and on the other side one can also have uniqueness of the equilibrium state.
In 1990, Mess gave a proof of Thurston's earthquake theorem using the Anti-de Sitter geometry. Since then, several of Mess's ideas have been used to investigate the correspondence between surfaces in 3-dimensional Anti de Sitter space and Teichmüller theory.
In this spirit, we investigate the problem of the existence of vector fields giving infinitesimal earthquakes on the hyperbolic plane, using the so-called Half-pipe geometry which is the dual of Minkowski geometry in a suitable sense. In particular, we recover Gardiner's theorem, which states that any Zygmund vector field on the circle can be represented as an infinitesimal earthquake. Our findings suggest a connection between vector fields on the hyperbolic plane and surfaces in 3-dimensional Half-pipe space, which may be suggestive of a bigger picture.
Link: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT09
Abstract: The Parameterisation Method is a powerful body of theory to compute the invariant manifolds of a dynamical system by looking for a parameterization of them in such a way that the dynamics on this manifold expressed in the coordinates of such parameterization writes as simply as possible. This methodology was foreseen by Guillamon and Huguet [SIADS, 2009 & J. Math. Neurosci, 2013] as a possible way of extending the domain of accuracy of the phase-reduction of periodic orbits. This fruitful approach, known as phase-amplitude reduction, has been fully developed during the last decade and provides an essentially complete understanding of deterministic oscillatory dynamics.
In this talk, we pursue the "simpler as possible" philosophy underlying the Parameterisation Method to develop an analogous phase-amplitude approach to stochastic oscillators. Main idea of our approach is to find a change of variables such that the system, when transformed to these variables, expresses in the mean as the deterministic phase-amplitude description. Then, we take advantage of the simplicity of this approach, to develop interesting objects with the aim of further clarifying the stochastic oscillation.
Zoom link: https://gatech.zoom.us/meeting/96948840253
Quadratic programming and semidefinite programming are vital tools in discrete and continuous optimization, with broad applications. A major challenge is to develop methodologies and algorithms to solve instances with special structures. For this purpose, we study some global relaxation techniques to quadratic and semidefinite programming, and prove theoretical properties about their qualities. In the first half we study the negative eigenvalues of $k$-locally positive semidefinite matrices, which are closely related to the sparse relaxation of semidefinite programming. In the second half we study aggregations of quadratic inequalities, a tool that can be leveraged to obtain tighter relaxation to quadratic programming than the standard Shor relaxation. In particular, our results on finiteness of aggregations can potentially lead to efficient algorithms for certain classes of quadratic programming instances with two constraints.
In this talk, I will present new stability results for non-degenerate centered self-decomposable laws with finite second moment and for non-degenerate symmetric alpha-stable laws with alpha in (1,2). These stability results are based on Stein's method and closed forms techniques. As an application, explicit rates of convergence are obtained for several instances of the generalized CLTs. Finally, I will discuss the standard Cauchy case.
Special time & day. Remote only.
►Presentation will be in hybrid format. Zoom link: https://gatech.zoom.us/j/99128737217?pwd=dllnNE1kSW1DZURrY1UycGxrazJtQT09
►Abstract: We study various averaging operators of discrete functions, inspired by number theory, in order to show they satisfy $\ell^p$ improving and maximal bounds. The maximal bounds are obtained via sparse domination results for $p \in (1,2)$, which imply boundedness on $\ell ^p (w)$ for $p \in (1, \infty )$, for all weights $w$ in the Muckenhoupt $A_p$ class.
We start by looking at averages along the integers weighted by the divisor function $d(n)$, and obtain a uniform, scale free $\ell^p$-improving estimate for $p \in (1,2)$. We also show that the associated maximal function satisfies $(p,p)$ sparse bounds for $p \in (1,2)$. We move on to study averages along primes in arithmetic progressions, and establish improving and maximal inequalities for these averages, that are uniform in the choice of progression. The uniformity over progressions imposes several novel elements on our approach. Lastly, we generalize our setting in the context of number fields, by considering averages over the Gaussian primes.
Finally, we explore the connections of our work to number theory: Fix an interval $\omega \subset \mathbb{T}$. There is an integer $N_\omega $, so that every odd integer $n$ with $N(n)>N_\omega $ is a sum of three Gaussian primes with arguments in $\omega $. This is the weak Goldbach conjecture. A density version of the strong Goldbach conjecture is proved, as well.
►Members of the committee:
· Michael Lacey (advisor)
· Chris Heil
· Ben Krause
· Doron Lubinsky
· Shahaf Nitzan
Zoom link (Meeting ID: 941 5991 7033, Passcode: 328576)
I will present two projects related to tropical divisors and multiplicities. First, my work with Philipp Jell on fully-faithful tropicalizations in 3-dimensions. Second, my work with Andreas Gross on algebraic and combinatorial multiplicities for multivariate polynomials over the tropical and sign hyperfields.
The first part is about using piecewise linear functions to describe tropical curves in 3 dimensions and how the changes in those slopes (a divisor) lift to non-Archimedean curves. These divisors give an embedding of a curve in a 3-dimensional toric variety whose tropicalization is isometric to the so-called extended skeleton of the curve.
In part two, I describe how Baker and Lorscheid's theory of multiplicities over hyperfields can be extended to multivariate polynomials. One key result is a new proof/view of the work of Itenburg and Roy who used patchworking to construct some lower bounds on the number of positive roots of a system of polynomials given a particular sign arrangement. Another result is a collection of upper bounds for the same problem.
Committee:
Zoom link. Meeting ID: 914 2801 6313, Passcode: 501018
We examine the relationship between Dupire's functional derivative and a variant of the functional derivative developed by Kim for analyzing functionals in systems with delay. Our findings demonstrate that if Dupire's space derivatives exist, differentiability in any continuous functional direction implies differentiability in any other direction, including the constant one. Additionally, we establish that co-invariant differentiable functionals can lead to a functional Ito formula in the Cont and Fournié path-wise setting under the right regularity conditions.
Next, our attention turns to functional extensions of the Meyer-Tanaka formula and the efforts made to characterize the zero-energy term for integral representations of functionals of semimartingales. Using Eisenbaum's idea for reversible semimartingales, we obtain an optimal integration formula for Lévy processes, which avoids imposing additional regularity requirements on the functional's space derivative, and extends other approaches using the stationary and martingale properties of Lévy processes.
Finally, we address the topic of integral representations for the delta of a path-dependent pay-off, which generalizes Benth, Di Nunno, and Khedher's framework for the approximation of functionals of jump-diffusions to cases where they may be driven by a process satisfying a path-dependent differential equation. Our results extend Jazaerli and Saporito's formula for the delta of functionals to the jump-diffusion case. We propose an adjoint formula for the horizontal derivative, hoping to obtain more tractable formulas for the Delta of strongly path-dependent functionals.
Committee
Many recent breakthroughs in additive combinatorics, such as results relating to Roth’s theorem or inverse sum set theorems, utilize a combination of Fourier analytical and physical methods. Physical methods refer to results relating to the physical space, such as almost-periodicity results regarding convolutions. This thesis focuses on the properties of convolutions.
Given a group G and sets A ⊆ G, we study the properties of the convolution for sum sets and difference sets, 1A ∗1A and 1A ∗1−A. Given x ∈ Gn, we study the set image of its sum set and difference set. We break down the study of set images into two cases, when x is independent, and when x is an arithmetic progression. In both cases, we provide some convexity result for the set image of both the sum set and difference set. For the case of the arithmetic progression, we prove convexity by first showing a recurrence relation for the distribution of the convolution.
Finally, we prove a smoothness property regarding 4-fold convolutions 1A ∗1A ∗1A ∗1A. We then construct different examples to better understand possible bounds for the smoothness property in the case of 2-fold convolutions 1A ∗ 1A.
Committee
Prof. Ernie Croot, Advisor
Prof. Michael Lacey
Prof. Josephine Yu
Prof. Anton Leykin
Prof. Will Perkins
Suppose you have a set S of integers from {1,2,...,N} that contains at least N / C elements. Then for large enough N, must S contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?
In 1953, Roth showed this is the case when C is roughly (log log N). Behrend in 1946 showed that C can be at most exp(sqrt(log N)). Since then, the problem has been a cornerstone of the area of additive combinatorics. Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on C to C = (log N)^(1+c) for some constant c > 0.
This talk will describe a new work showing that C can be as big as exp((log N)^0.08), thus getting closer to Behrend's construction. Based on joint work with Zander Kelley
Zoom Link: https://gatech.zoom.us/j/7776548887?pwd=SFEySmpVUW9FckxJVEZRY2hUbUVOQT09<br />
<br />
Committee Members:<br />
<br />
Matt Baker (Co-advisor)<br />
Oliver Lorscheid (Co-advisor)<br />
Anton Leykin <br />
Josephine Yu<br />
Xingxing Yu
I will talk about the application of algebra and algebraic geometry to matroid theory. Baker and Bowler developed the notions of weak and strong matroids over tracts. Later, Baker and Lorscheid developed the notion of foundation of a matroid, which characterize the representability of the matroid. I will introduce a variety of topics under this theme. First, I will talk about a condition which is sufficient to guarantee that the notions of strong and weak matroids coincide. Next, I will describe a software program that computes all representations of matroids over a field, based on the theory of foundations. Finally, I will define a notion of rank for matrices over tracts in order to get uniform proofs of various results about ranks of matrices over fields.
The study of contact and symplectic manifolds has relied heavily on understanding Legendrian and Lagrangian submanifolds in them -- both for constructing the manifolds using these submanifolds, and for computing invariants of the ambient space in terms of invariants of the submanifolds. This thesis explores the construction of Legendrian submanifolds in high dimensional contact manifolds (greater than 3) in two directions. In one, using open book decompositions, we generalise a doubling construction defined by Ekholm and show that the Legendrians obtained are trivial. In the second, which is joint work in progress with Hughes, we use the doubling and twist spun constructions to build a large family of Legendrians, compute their sheaf-theoretic invariants to distinguish them using techniques of Casals-Zaslow, and study their exact Lagrangian fillability properties.
Zoom link:
https://gatech.zoom.us/j/93109756512?pwd=Skljb0tVdjZVNEUvSm9tNnFHZFREUT09
Knots in contact manifolds are interesting objects to study. In this talk, I will focus on knots in overtwisted manifolds. There are two types of knots in an overtwisted manifold, one with overtwisted complement (known as loose) and one with tight complement (known as non-loose). Not very surprisingly, non-loose knots behave very mysteriously. They are interesting in their own right as we still do not understand them well. But also one might want to study them because surgery on them produces tight contact structures and understanding tight contact structures is a major problem in the contact world. I'll give a brief history on these knots and discuss some of their classification and structure problems and how these problems differ from the classification/ structure problems of knots in tight manifolds.
This seminar will be delivered in a hybrid Zoom format. The in-person version is held in Skiles 005 while the Zoom version is held at this link: https://gatech.zoom.us/j/99424341824
One-dimensional discrete-time population models, such as Logistic or Ricker growth, can exhibit periodic and chaotic dynamics. Incorporating epidemiological interactions through the addition of an infectious class causes an interesting complexity of new behaviors. Here, we examine a two-dimensional susceptible-infectious (SI) model with underlying Ricker population growth. In particular, the system with infection has a distinct bifurcation structure from the disease-free system. We use numerical bifurcation analysis to determine the influence of infection on the types and appearance of qualitatively distinct long-time dynamics. We find that disease-induced mortality leads to the appearance of multistability, such as stable four-cycles and chaos dependent upon the initial condition. Furthermore, previous work showed that infection that alters the ability to reproduce can lead to unexpected increases in total population size. A similar phenomenon is seen in some models where an increase in population size with a decreased growth rate occurs, known as the ‘hydra effect.’ Thus, we examine the appearance and extent of the hydra effect, particularly when infection is introduced during cyclic or chaotic population dynamics.
Contact 3-manifolds arise organically as boundaries of symplectic 4-manifolds, so it’s natural to ask: Given a contact 3-manifold Y, does there exist a symplectic 4-manifold X filling Y in a compatible way? Stein fillability is one such notion of compatibility that can be explored via open books: representations of a 3-manifold by means of a surface with boundary and its self-diffeomorphism, called a monodromy. I will discuss joint work with Andy Wand in which we exhibit first known Stein-fillable contact manifolds whose supporting open books of genus one have non-positive monodromies. This settles the question of correspondence between Stein fillings and positive monodromies for open books of all genera. Our methods rely on a combination of results of J. Conway, Lecuona and Lisca, and some observations about lantern relations in the mapping class group of the twice-punctured torus.
Say you’ve got an (orientable) surface S and you want to do geometry with it. Well, the complex plane C has dimension 2, so you might as well try to model S on C and see what happens. The objects you get from following this thought are called complex structures. It turns out that most surfaces have a rich but manageable amount of genuinely different complex structures. I’ll focus in this talk on how to think about comparing and deforming complex structures on S. I’ll explain the remarkable result that there are highly structured “best” maps between (marked) complex structures, and how this can be used to show the right space of complex structures on S is a finite-dimensional ball. This is known as Teichmüller’s theorem, and I’ll be following Bers’ proof.
This talk will be an introduction to the theory of surfaces, some tools we use to study surfaces, and some uses of surfaces in "real life". In particular, we will discuss the mapping class group and the curve complex. This talk will be aimed at an audience with a minimal background in low-dimensional topology.
The fine curve graph of a surface S was introduced by Bowden–Hensel–Webb in 2019 to study the diffeomorphism group of S. We consider a variant of this graph, called the fine 1-curve graph, whose vertices are essential simple closed curves and edges connect curves that intersect in at most one point. Building on the works of Long–Margalit–Pham–Verberne–Yao and Le Roux–Wolff, we show that the automorphism group of the fine 1-curve graph is isomorphic to the homeomorphism group of S. This is joint work with Katherine W. Booth and Daniel Minahan.
We study the global existence of classical solutions to the incompressible viscous MHD system without magnetic diffusion in 2D and 3D. The lack of resistivity or magnetic diffusion poses a major challenge to a global regularity theory even for small smooth initial data. However, the interesting nonlinear structure of the system not only leads to some significant challenges, but some interesting stabilization properties, that leads to the possibility of the theory of global existence of classical and/or strong solutions. This talk is based on joint works with Yi Zhou, Yi Zhu, Shijin Ding, Xiaoying Zeng, and Jingchi Huang.
I’ll present a quantitative version of a stability estimate
for the Sobolev Inequality improving previous results of Bianchi
and Egnell. The estimate has the correct dimensional dependence
which leads to a stability estimate for the Logarithmic Sobolev inequality.
This is joint work with Dolbeault, Esteban, Figalli and Frank.
In the early 80's, Freedman discovered that the Whitney trick could be performed in 4-dimensions which quickly led to a complete classification of closed, simply connected topological 4-manifolds. With gauge theory, Donaldson showed that 4-manifolds differ greatly from their higher dimensional counterparts which uncovered the stark differences between topological and smooth results in dimension 4. In this introductory talk, we will give a brief overview this classification and why dimension 4 is so unique. Then, we will describe handlebody decompositions of 4-manifolds and draw several Kirby pictures representing some basic 4-mfds.
Computing integrals against a high-dimensional posterior is the major computational bottleneck in Bayesian inference. A popular technique to reduce this computational burden is to use the Laplace approximation, a Gaussian distribution, in place of the true posterior. Despite its widespread use, the Laplace approximation's accuracy in high dimensions is not well understood. The body of existing results does not form a cohesive theory, leaving open important questions e.g. on the dimension dependence of the approximation rate. We address many of these questions through the unified framework of a new, leading order asymptotic decomposition of high-dimensional Laplace integrals. In particular, we (1) determine the tight dimension dependence of the approximation error, leading to the tightest known Bernstein von Mises result on the asymptotic normality of the posterior, and (2) derive a simple correction to this Gaussian distribution to obtain a higher-order accurate approximation to the posterior.
In this talk, we present recent results on the geometry of centrally-symmetric random polytopes generated by N independent copies of a random vector X. We show that under minimal assumptions on X, for N>Cn, and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector - namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. Moreover, by identifying the floating bodies associated with various random vectors we recover the estimates that have been obtained previously, and thanks to the minimal assumptions on X we derive estimates in cases that had been out of reach, involving random polytopes generated by heavy-tailed random vectors (e.g., when X is q-stable or when X has an unconditional structure). Finally, the structural results are used for the study of a fundamental question in compressive sensing - noise blind sparse recovery. This is joint work with Olivier Guédon (University of Paris-Est Marne La Vallée), Christian Kümmerle (UNC Charlotte), Shahar Mendelson (Sorbonne University Paris), and Holger Rauhut (LMU Munich).
Bio: Felix Krahmer received his PhD in Mathematics in 2009 from New York University under the supervision of Percy Deift and Sinan Güntürk. He was a Hausdorff postdoctoral fellow in the group of Holger Rauhut at the University of Bonn, Germany from 2009-2012. In 2012 he joined the University of Göttingen as an assistant professor for mathematical data analysis, where he has been awarded an Emmy Noether Junior Research Group. From 2015-2021 he was assistant professor for optimization and data analysis in the department of mathematics at the Technical University of Munich, before he was tenured and promoted to associate professor in 2021. His research interests span various areas at the interface of probability, analysis, machine learning, and signal processing including randomized sensing and reconstruction, fast random embeddings, quantization, and the computational sensing paradigm.
Due to linear superposition, solutions of a Linear Schrodinger Equation with a trapping potential, produce a discrete quasiperiodic part. When a nonlinear perturbation is turned on, it is known in principle, and proved in various situations, that at small energies there is a phenomenon of standing wave selection where, up to radiation, quasiperiodicity breaks down and there is convergence to a periodic wave. We will discuss this phenomenon in 1 D, where cubic nonlinearities are long range perturbations of the linear equations. Our aim is to show that a very effective framework to see these phenomena is provided by a combination of the dispersion theory of Kowalczyk, Martel and Munoz along with Maeda's notion of Refined Profile.
It is known for many years that various inequalities in convex geometry have information-theoretic analogues. The most well known example is the Entropy power inequality which corresponds to the Brunn-Minkowski inequality, but the theory of optimal transport allows to prove even better analogues.
At the same time, in recent years there is a lot of interest in the role of symmetry in Brunn-Minkowski type inequalities. There are many open conjectures in this direction, but also a few proven theorems such as the Gaussian Dimensional Brunn-Minkowski inequality. In this talk we will discuss the natural question — do the known information-theoretic inequalities similarly improve in the presence of symmetry? I will present some cases where the answer is positive together with some open problems.
Based on joint work with Gautam Aishwarya.
You are probably familiar with the concept of a knot in 3 space: a tangled string that can't be pushed and pulled into an untangled one. We briefly show how to prove mathematical knots are in fact knotted, and discuss some conditions which guarantee unknotting. We then give explicit examples of knotted 2-spheres in 4 space, and discuss 2-sphere version of the familiar theorems. A large part of the talk is practice with visualizing objects in 4 dimensional space. We will also prove some elementary facts to give a sense of what working with these objects feels like. Time permitting we will describe know knotted 2-spheres were used to give evidence for the smooth 4D Poincare conjecture, one of the guiding problems in the field.
Gaussian mixture block models are distributions over graphs that strive to model modern networks: to generate a graph from such a model, we associate each vertex with a latent feature vector sampled from a mixture of Gaussians, and we add edge if and only if the feature vectors are sufficiently similar. The different components of the Gaussian mixture represent the fact that there may be different types of nodes with different distributions over features---for example, in a social network each component represents the different attributes of a distinct community. Natural algorithmic tasks associated with these networks are embedding (recovering the latent feature vectors) and clustering (grouping nodes by their mixture component).
In this talk, we focus on clustering and embedding graphs sampled from high-dimensional Gaussian mixture block models, where the dimension of the latent feature vectors goes to infinity as the size of the network goes to infinity. This high-dimensional setting is most appropriate in the context of modern networks, in which we think of the latent feature space as being high-dimensional. We analyze the performance of canonical spectral clustering and embedding algorithms for such graphs in the case of 2-component spherical Gaussian mixtures and begin to sketch out the information-computation landscape for clustering and embedding in these models.
This is based on joint work with Tselil Schramm.
the Riemann-Roch theorem by Baker and Norine.
A CAT(0) space is a geodesic metric space where triangles are thinner than comparison triangles in a Euclidean plane. Prime examples of CAT(0) spaces are Cartan-Hadamard manifolds: complete simply connected Riemannian spaces with nonpositive curvature, which include Euclidean and Hyperbolic space as special cases. The triangle condition ensures that every pair of points in a CAT(0) space can be connected by a unique geodesic. A subset of a CAT(0) space is convex if it contains the geodesic connecting every pair of its points. We will give a quick survey of classical results in differential geometry on characterization of convex sets, such the theorems of Hadamard and of Chern-Lashof, and also cover other background from the theory of CAT(0) spaces and Alexandrov geometry, including the rigidity theorem of Greene-Wu-Gromov, which will lead to the new results in the second talk.
The principal minor map takes an n x n square matrix and maps it to the 2^n-length vector of its principal minors. In this talk, I will describe both the fiber and the image of this map. In 1986, Loewy proposed a sufficient condition for the fiber to be a single point up to diagonal equivalence. I will provide a necessary and sufficient condition for the fiber to be a single point. Additionally, I will describe the image of the space of complex matrices using a characterization of determinantal representations of multiaffine polynomials, based on the factorization of their Rayleigh differences. Using these techniques I will give equations and inequalities characterizing the images of the spaces of real and complex symmetric and Hermitian matrices. This is based on joint research with Cynthia Vinzant.
We show that in Cartan-Hadamard manifolds M^n, n≥ 3, closed infinitesimally convex hypersurfaces S bound convex flat regions, if curvature of M^n vanishes on tangent planes of S. This encompasses Chern-Lashof characterization of convex hypersurfaces in Euclidean space, and some results of Greene-Wu-Gromov on rigidity of Cartan-Hadamard manifolds. It follows that closed simply connected surfaces in M^3 with minimal total absolute curvature bound Euclidean convex bodies, as stated by M. Gromov in 1985. The proofs employ the Gauss-Codazzi equations, a generalization of Schur comparison theorem to CAT(0) spaces, and other techniques from Alexandrov geometry outlined by A. Petrunin, including Reshetnyak’s majorization theorem, and Kirszbraun’s extension theorem.
A lot of recent work in the theory of partial differential equations has focused on the existence and stability properties of special solutions for Hamiltonian PDE’s. We review some recent works (joint with Hakkaev and Stanislavova), for spatially periodic traveling waves and their stability properties. We concentrate on three examples, namely the Benney system, the Zakharov system and the KdV-NLS model. We consider several standard explicit solutions, given in terms of Jacobi elliptic functions. We provide explicit and complete description of their stability properties. Our analysis is based on the careful examination of the spectral properties of the linearized operators, combined with recent advances in the Hamiltonian instability index formalism.
Surface bundles lie in the intersection of many areas of math: algebraic topology, 2–4 dimensional topology, geometric group theory, algebraic geometry, and even number theory! However, we still know relatively little about surface bundles, especially compared to vector bundles. In this interactive talk, I will present the general (and beautiful) fiber bundle theory, including characteristic classes, as a starting point, and you the audience will get to specialize the general theory to surface bundles, with rewards! The talk aims to be accessible to anyone who had exposure to algebraic topology. This is also part one of three talks about surface bundles I will give this semester.
We discuss small-ball probability estimates of the smallest singular value of a rather general ensemble of random matrices which we call “inhomogeneous”. One of the novel ingredients of our family of universality results is an efficient discretization procedure, applicable under unusually mild assumption. Most of the talk will focus on explaining the ideas behind the proof of the first ingredient. Partially based on the joint work with Tikhomirov and Vershynin, and an ongoing joint work with Fernandez and Tatarko. We will also mention a related work on the cube minimal dispersion, joint with Litvak.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.
In this talk, we introduce a new method for minimizing analytic Morse functions over compact domains through the use of polynomial approximations. This is, in essence, an effective application of the Stone-Weierstrass Theorem, as we seek to extend a local method to a global setting, through the construction of polynomial approximants satisfying an arbitrary set precision in L-infty norm. The critical points of the polynomial approximant are computed exactly, using methods from computer algebra. Our Main Theorem states probabilistic conditions for capturing all local minima of the objective function $f$ over the compact domain. We present a probabilistic method, iterative on the degree, to construct the lowest degree possible least-squares polynomial approximants of $f$ which attains a desired precision over the domain. We then compute the critical points of the approximant and initialize local minimization methods on the objective function $f$ at these points, in order to recover the totality of the local minima of $f$ over the domain.
Elliptic surfaces are some of the most well-behaved families of smooth, simply-connected four-manifolds from the geometric and analytic perspective. Many of their smooth invariants are easily computable and they carry a fibration structure which makes it possible to modify them by various surgical operations. However, elliptic surfaces have large Euler characteristics which means even their simplest handle-decompositions appear to be quite complicated. In this seminar, we will learn how to draw several different handle diagrams of elliptic surfaces which show explicitly many of their nice properties. This will allow us to see many useful properties of elliptic surfaces combinatorially, and gives insight into the constructions of their exotic smooth structures.
Fintushel and Stern’s knot surgery constructions has been a central source of exotic 4-manifolds since its introduction in 1997. In the simply connected setting, it is known that there are also embedded corks in knot-surgered manifolds whose twists undo the knot surgery. This has been known abstractly since the construction was first given, but the explicit corks and embeddings have remained elusive. We will give an algorithmic process for transforming a generic Kirby diagram of a simply-connected knot surgered 4-manifold into one which contains an explicit cork whose twist undoes the surgery: answering the question. Along the way we will discuss $S^2\times S^2$-stable diffeomorphisms of knot-surgered 4-manifolds, and their relationship to the existence of corks.
This is a virtual seminar.<br />
<br />
Speaker Bio:<br />
Youngsoo is a computational math scientist in Center for Applied Scientific Computing (CASC) under Computing directorate at LLNL. His research focuses on developing efficient reduced order models for various physical simulations for time-sensitive decision-making multi-query problems, such as inverse problems, design optimization, and uncertainty quantification. His expertise includes various scientific computing disciplines. Together with his team and collaborators, he has developed powerful model order reduction techniques, such as machine learning-based nonlinear manifold, space–time reduced order models, and latent space dynamics identification methods for nonlinear dynamical systems. He has also developed the component-wise reduced order model optimization algorithm, which enables fast and accurate computational modeling tools for lattice-structure design. He is currently leading data-driven physical simulation team at LLNL, with whom he developed the open source codes, libROM (i.e., https://www.librom.net), LaghosROM (i.e., https://github.com/CEED/Laghos/tree/rom/rom), LaSDI (i.e., https://github.com/LLNL/LaSDI), gLaSDI (i.e., https://github.com/LLNL/gLaSDI), and GPLaSDI (i.e., https://github.com/LLNL/GPLaSDI). He earned his undergraduate degree in Civil and Environmental Engineering from Cornell University and his Ph.D. degree in Computational and Mathematical Engineering from Stanford University. He was a postdoctoral scholar at Sandia National Laboratories and Stanford University prior to joining LLNL in 2017.
A computationally expensive physical simulation is a huge bottleneck to advance in science and technology. Fortunately, many data-driven approaches have emerged to accelerate those simulations, thanks to the recent advancements in machine learning (ML) and artificial intelligence. For example, a well-trained 2D convolutional deep neural network can predict the solution of the complex Richtmyer–Meshkov instability problem with a speed-up of 100,000x [1]. However, the traditional black-box ML models do not incorporate existing governing equations, which embed underlying physics, such as conservation of mass, momentum, and energy. Therefore, the black-box ML models often violate important physics laws, which greatly concern physicists, and require big data to compensate for the missing physics information. Additionally, it comes with other disadvantages, such as non-structure-preserving, computationally expensive training phase, non-interpretability, and vulnerability in extrapolation. To resolve these issues, we can bring physics into the data-driven framework. Physics can be incorporated into different stages of data-driven modeling, i.e., the sampling stage and model-building stage. Physics-informed greedy sampling procedure minimizes the number of required training data for a target accuracy [2]. Physics-guided data-driven model better preserves the physical structure and is more robust in extrapolation than traditional black-box ML models. Numerical results, e.g., hydrodynamics [3,4], particle transport [5], plasma physics, and 3D printing, will be shown to demonstrate the performance of the data-driven approaches. The benefits of the data-driven approaches will also be illustrated in multi-query decision-making applications, such as design optimization [6,7].
Reference
[1] Jekel, Charles F., Dane M. Sterbentz, Sylvie Aubry, Youngsoo Choi, Daniel A. White, and Jonathan L. Belof. “Using Conservation Laws to Infer Deep Learning Model Accuracy of Richtmyer-meshkov Instabilities.” arXiv preprint arXiv:2208.11477 (2022).
[2] He, Xiaolong, Youngsoo Choi, William D. Fries, Jon Belof, and Jiun-Shyan Chen. “gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification.” arXiv preprint arXiv:2204.12005 (2022).
[3] Copeland, Dylan Matthew, Siu Wun Cheung, Kevin Huynh, and Youngsoo Choi. “Reduced order models for Lagrangian hydrodynamics.” Computer Methods in Applied Mechanics and Engineering 388 (2022): 114259.
[4] Kim, Youngkyu, Youngsoo Choi, David Widemann, and Tarek Zohdi. “A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder.” Journal of Computational Physics 451 (2022): 110841.
[5] Choi, Youngsoo, Peter Brown, William Arrighi, Robert Anderson, and Kevin Huynh. “Space–time reduced order model for large-scale linear dynamical systems with application to Boltzmann transport problems.” Journal of Computational Physics 424 (2021): 109845.
[6] McBane, Sean, and Youngsoo Choi. “Component-wise reduced order model lattice-type structure design.” Computer methods in applied mechanics and engineering 381 (2021): 113813.
[7] Choi, Youngsoo, Gabriele Boncoraglio, Spenser Anderson, David Amsallem, and Charbel Farhat. “Gradient-based constrained optimization using a database of linear reduced-order models.” Journal of Computational Physics 423 (2020): 109787.
Many data in real-world applications are in a high-dimensional space but exhibit low-dimensional structures. In mathematics, these data can be modeled as random samples on a low-dimensional manifold. I will talk about machine learning tasks like regression and classification, as well as PDE simulations. We consider deep learning as a tool to solve these problems. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. Our results demonstrate that deep neural networks can utilize low-dimensional geometric structures of data in machine learning and PDE simulations.
In 1999, Washington University in Saint Louis hosted a conference on Harmonic Analysis to celebrate the 70th birthday of G. Weiss. In his talk in flag singular integral operators, E. M. Stein asked “What is the Hardy space theory in the flag setting?” In our recent paper, we characterise completely a flag Hardy space on the Heisenberg group. It is a proper subspace of the classical one-parameter Hardy space of Folland and Stein that was studied by Christ and Geller. Our space is useful in several applications, including the endpoint boundedness for certain singular integrals associated with the Sub-Laplacian on Heisenberg groups, and representations of flag BMO functions.
Morse theory analyzes the topology of a smooth manifold by studying the behavior of its real-valued functions. From this, one obtains a well-behaved homology theory which provides further information about the manifold and places constraints on the smooth functions it admits. This idea has proven to be useful in approaching topological problems, playing an essential role in Smale's solution to the generalized Poincare conjecture in dimensions greater than 4. Morse theory has been adapted to study complex manifolds, and even algebraic varieties over more general fields, but the underlying principles remain the same. In this talk, we will define the basic notions of Morse theory and describe some of the fundamental results. Then we will define Morse homology and discuss some important corollaries and applications.
We study the Schur-Siegel-Smyth trace problem. We introduce a new linear programming problem that inclues Smyths' constraints, and we give an exact solution to it. This improves the best known lower bound on the Siegel trace problem which is based on Smyths' method. In a special case, we recover Siegel's original upper bound. Our method unifies Siegel's and Smyth's work under the same framework. This is joint work with Bryce Orloski.
We consider the Ising Curie-Weiss model on the complete graph constrained under a given $\ell_{p}$ norm. For $p=\infty$, it reduces to the classical Ising Curie-Weiss model. We prove that for all $p\ge 2$, there exists a critical inverse temperature $\beta_{c}(p)$ such that for $\beta<\beta_{c}(p)$, the magnetization is concentrated at zero and satisfies an appropriate Gaussian CLT. On the other hand, for $\beta>\beta_{c}(p)$, the magnetization is not concentrated at zero similar to the classical case. We further generalize the model for general symmetric spin distributions and prove similar phase transition. In this talk, we discuss a brief overview of classical Curie-Weiss model, a generalized Hubbard-Stratonovich transforms, and how we apply the transform to Curie-Weiss model under $\ell^p$ constraint. This is based on joint work with Partha Dey.
Atlanta Combinatorics Colloquium Hosted by Georgia Tech
A k-block in a graph is a set of k vertices every two of which are joined by k vertex disjoint paths. By a result of Weissauer, graphs with no k-blocks admit tree-decompositions with especially useful structure. While several constructions show that it is probably very difficult to characterize induced subgraph obstructions to bounded tree width, a lot can be said about graphs with no k-blocks. On the other hand, forbidding induced subgraphs places significant restrictions on the structure of a k-block in a graph. We will discuss this phenomenon and its consequences on the study of tree-decompositions in classes of graphs defined by forbidden induced subgraphs.
In this talk I will develop a parallel between the classical field theory of electromagnetism and geometric mechanics of animal locomotion. I will illustrate this parallel using some informative examples from the two disciplines. In the realm of electromagnetism, we will investigate the magnetic monopole, as classically as possible. In the realm of animal locomotion, we will investigate the aphorism that a cat dropped (from a safe height) upside-down always lands on her feet. It turns out that both of these phenomena are caused by the presence of non-trivial topology.
No prior knowledge of classical field theory will be assumed, and this talk may continue into a second session at a later date.
We develop nanopteron solutions for a coupled system of singularly perturbed ordinary differential equations. To leading order, one equation governs the traveling wave profile for the Korteweg-de Vries (KdV) equation, while the other models a simple harmonic oscillator whose small mass is the problem’s natural small parameter. A nanopteron solution consists of the superposition of an exponentially localized term and a small-amplitude periodic term. We construct two families of nanopterons. In the first, the periodic amplitude is fixed to be exponentially small but nonzero, and an auxiliary phase shift is introduced in the periodic term to meet a hidden solvability condition lurking within the problem. In the second, the phase shift is fixed as a (more or less) arbitrary value, and now the periodic amplitude is selected to satisfy the solvability condition. These constructions adapt different techniques due to Beale and Lombardi for related systems and is intended as the first step in a broader program uniting the flexible framework of Beale’s methods with the precision of Lombardi’s for applications to various problems in lattice dynamical systems. As a more immediate application, we use the results for the model problem to solve a system of coupled KdV-KdV equations that models the propagation of certain surface water waves.
This talk will include background information on contact structures and open book decompositions of 3-manifolds and the relationship between them. I will state the necessary definitions and include a number of concrete examples. I will also review some convex surface theory, which is the tool used in the main talk to investigate the contact structure – open book relationship.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.
Oriented matroids are matroids with extra sign data, and they are useful in the tropical study of real algebraic geometry. In order to study the topology of real algebraic hypersurfaces constructed from patchworking, Renaudineau and Shaw introduced an algebraically defined filtration of the tope space of an oriented matroid based on Quillen filtration. We will prove the equality between their filtration (together with the induced maps), the topologically defined Kalinin filtration, and the combinatorially defined Varchenko-Gelfand dual degree filtration over Z/2Z. We will also explain how the dual degree filtration can serve as a Z-coefficient version of the other two in this setting. This is joint work with Kris Shaw.
I will discuss recent work with K. Honda and Y. Huang on proving the Giroux correspondence between contact structures and open book decompositions. Though our work extends to all dimensions (with appropriate adjectives), this talk will focus on the 3-dimensional proof. I will first recall Giroux’s argument for existence of supporting open book decompositions, formulating it in the language adapted to our proof. The rest of the talk will be spent describing the proof of the stabilization correspondence.
The dynamic shortest path problem seeks to maintain the shortest paths/distances between pairs of vertices in a graph that is subject to edge insertions, deletions, or weight changes. The aim is to maintain that information more efficiently than naive recomputation via, e.g., Dijkstra's algorithm.
We present the first fully dynamic algorithm maintaining exact single source distances in unweighted graphs. This resolves open problems stated in [Demetrescu and Italiano, STOC'03], [Thorup SWAT'04], [Sankowski, COCOON 2005] and [vdBrand and Nanongkai, FOCS 2019].
In this talk, we will see how ideas from fine-grained complexity theory, computer algebra, and graph theory lead to insights for dynamic shortest paths problems.
This talk will present an overview of the behavior of the eigenvalues of the fractional Brownian matrix motion and other related matrix processes. We will do so by emphasizing the dynamics of the eigenvalues processes, the non-colliding property, the limit of the associated empirical process, as well as the free Brownian motion and the non commutative fractional Brownian motion.
In this talk, I will discuss my recent research on the asymptotic stability and inviscid damping of 2D monotone shear flows with non-constant density in inhomogeneous ideal fluids within a finite channel. More precisely, I proved that if the initial perturbations belong to the Gevrey-2- class, then linearly stable monotone shear flows in inhomogeneous ideal fluids are also nonlinear asymptotically stable. Furthermore, inviscid damping is proved to hold, meaning that the perturbed velocity converges to a shear flow as time approaches infinity.
A conversation with Stephen Young, 2008 GT ACO PhD and Senior Research Mathematician at Pacific Northwest National Laboratory, on opportunities for mathematicians in the unique combination of business/industry/government afforded by the DOE national labs.
(Coffee will be available at 3:30 following this discussion and before the speaker's ACO Alumni Lecture at 4pm.)
Abstract: While spectral methods provide far-ranging insights on graph structure, there remain significant challenges in their application to real data. Most notably, spectral methods do not incorporate information that maybe available beyond adjacency. A common approach to incorporating such additional information is encode this information in an ad-hoc manner into weights associated with the edges. Not only does this have limited expressivity, but is also restricted by graph structure: if two vertices are not adjacent, then edge weights cannot capture any closeness implied by metadata.
We address this issue by introducing the inner product Hodge Laplacian for an arbitrary simplicial complex. Within this framework we prove generalizations of foundational results in spectral graph theory, such as the Cheeger inequality and the expander mixing lemma, and show our framework recovers the usual combinatorial and normalized Laplacians as special cases. Our framework allows for the principled synthesis of combinatorial approaches in network science with more metadata driven approaches by using latent space encodings of the metadata to define an inner product both the vertices and the edges.
(Coffee will be available at 3:30 before this talk, following the speaker's Professional Development Seminar at 2:30pm.)
Zoom link: https://gatech.zoom.us/j/94868589860
The Teichmueller space parametrizes Riemann surfaces of fixed topological type and is fundamental in various contexts of mathematics and physics. It can be defined as a component of the moduli space of flat G=PSL(2,R) connections on the surface. Higher Teichmüller spaces extend this notion to appropriate higher rank classical Lie groups G. Other generalizations are given by the super-Teichmueller spaces, describing Riemann surfaces enhanced by odd or anti-commuting coordinates (known as super Riemann surfaces). The super-Teichmueller spaces arise naturally as higher Teichmueller spaces, corresponding to supergroups, which play an important role in geometric topology, algebraic geometry, and mathematical physics, where the anti-commuting variables correspond to Fermions.
After introducing these spaces, I will explain the solution to the long-standing problem of describing the counterpart of Penner coordinates on the super-Teichmueller space and its higher analogues. The importance of these coordinates is justified by two remarkable properties: the action of the mapping class group is rational, and the Weil-Petersson form is given by a simple explicit formula. From the algebraic and combinatorial perspectives, their transformations lead to an important generalization of cluster algebras.
In the end, I will discuss some recent applications of this construction.
We study the problem of estimating the left and right singular subspaces for a collection of heterogeneous random graphs with a shared common structure. We analyze an algorithm that first estimates the orthogonal projection matrices corresponding to these subspaces for each individual graph, then computes the average of the projection matrices, and finally finds the matrices whose columns are the eigenvectors corresponding to the d largest eigenvalues of the sample averages. We show that the algorithm yields an estimate of the left and right singular vectors whose row-wise fluctuations are normally distributed around the rows of the true singular vectors. We then consider a two-sample hypothesis test for the null hypothesis that two graphs have the same edge probabilities matrices against the alternative hypothesis that their edge probabilities matrices are different. Using the limiting distributions for the singular subspaces, we present a test statistic whose limiting distribution converges to a central chi-square (resp. non-central chi-square) under the null (resp. alternative) hypothesis. Finally, we adapt the theoretical analysis for multiple networks to the setting of distributed PCA; in particular, we derive normal approximations for the rows of the estimated eigenvectors using distributed PCA when the data exhibit a spiked covariance matrix structure.
We consider a numerical method to approximate the solution operator for evolutional partial differential equations (PDEs). By employing a general reduced-order model, such as a deep neural network, we connect the evolution of a model's parameters with trajectories in a corresponding function space. Using the Neural Ordinary Differential Equations (NODE) technique we learn a vector field over the parameter space such that from any initial starting point, the resulting trajectory solves the evolutional PDE. Numerical results are presented for a number of high-dimensional problems where traditional methods fail due to the curse of dimensionality.
This talk will present a new online algorithm for sequential detection of change points in state-space models. The algorithm is computationally fast, and sensitive to changes in model parameters (including observation and evolution variances), as well as model structure. We consider change point detection in a sequential way, when observations are received one by one, or in batches, with a (possibly soft) restart after each detected change point. We provide the theoretical foundation of the algorithm, and study its performance in different state space models used to model the growth of epidemics over time, using simulated data and the recent COVID-19 dataset. This work is joint work with Ruyu Tan.
This seminar is in a Hybrid format. The in-person version is on campus at Georgia Tech in Skiles 005. The virtual version will be at: https://gatech.zoom.us/j/92952024862
Let H be a graph. We show that if r is large enough as a function of H,
then the r-partite Turán graph maximizes the number of copies of H among
all Kr+1-free graphs on a given number of vertices. This confirms a
conjecture of Gerbner and Palmer.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.
Recent research trends have explored curious analogies between the theory of graphs and Riemann surfaces. To each graph we can associate a real metric torus, known as its Jacobian. It was previously known that isomorphisms of graph Jacobians yield isomorphisms of the associated graphic matroids, partially mirroring a classical algebraic geometry result known as the Torelli theorem. However, the result on graphs is not as strong as a direct analogue of the Riemann surface result would be, nor does it use as much data. I will discuss how the graph Torelli theorem can be refined to incorporate additional data as with Riemann surfaces, in which case it produces isomorphisms of graphs. If time permits, I will describe further recent work in this direction.
We consider the Bayesian approach to the linear Gaussian inference problem of inferring the initial condition of a linear dynamical system from noisy output measurements taken after the initial time. In practical applications, the large dimension of the dynamical system state poses a computational obstacle to computing the exact posterior distribution. Model reduction offers a variety of computational tools that seek to reduce this computational burden. In particular, balanced truncation is a control-theoretic approach to model reduction which obtains an efficient reduced-dimension dynamical system by projecting the system operators onto state directions which trade off the reachability and observability of state directions. We define an analogous balanced truncation procedure for the Bayesian inference setting based on the trade off between prior uncertainty and data information. The resulting reduced model inherits desirable theoretical properties for both the control and inference settings: numerical demonstrations on two benchmark problems show that our method can yield near-optimal posterior covariance approximations with order-of-magnitude state dimension reduction.
We consider a mechanical system consisting of a rotator and a pendulum, subject to a small, conformally symplectic perturbation. The resulting system has energy dissipation. We provide explicit conditions on the dissipation parameter, so that the resulting system exhibits Arnold diffusion. More precisely, we show that there are diffusing orbits along which the energy of the rotator grows by an amount independent of the smallness parameter. The fact that Arnold diffusion may play a role in systems with small dissipation was conjectured by Chirikov. Our system can be viewed as a simplified model for an energy harvesting device, in which context the energy growth translates into generation of electricity.
Joint work with S.W. Akingbade and T-M. Seara.
We will start by introducing the Teichmüller space of a surface, which parametrizes the possible conformal structures it supports. By defining this space analytically, we can equip it with the Lp metrics, of which the Teichmüller and Weil-Petersson metrics are special cases. We will discuss the incompleteness of the Lp metrics on Teichmüller space and what we know about their completions.
Fox et al introduced the model of c-closed graphs, a distribution-free model motivated by one of the most universal signatures of social networks, triadic closure. Even though enumerating maximal cliques in an arbitrary network can run in exponential time, it is known that for c-closed graph, enumerating maximal cliques and maximal complete bipartite graphs is always fast, i.e., with complexity being polynomial in the number of vertices in the network. In this work, we investigate further by enumerating maximal blow-ups of an arbitrary graph H in c-closed graphs. We prove that given any finite graph H, the number of maximal blow-ups of H in any c-closed graph on n vertices is always at most polynomial in n. When considering maximal induced blow-ups of a finite graph H, we provide a characterization of graphs H when the bound is always polynomial in n. A similar general theorem is also proved when H is infinite.
It is known that if $\{x_n\}$ is a frame for a separable Hilbert space, then there exist some sequences $\{y_n\}$ such that $x= \sum x_n$, and this sum converges in the norm of H. This equation is called the reconstruction formula of x. In this talk, we will talk about the existence of frames that admit absolutely convergent and unconditionally convergent reconstruction formula. Some characterizations of such frames will also be presented. Finally, we will present an extension of this problem about the unconditional convergence of Gabor expansion in Modulation spaces.
Abstract: Motivated by applications in Reinforcement Learning (RL), this talk focuses on the Stochastic Appproximation (SA) method to find fixed points of a contractive operator. First proposed by Robins and Monro, SA is a popular approach for solving fixed point equations when the information is corrupted by noise. We consider the SA algorithm for operators that are contractive under arbitrary norms (especially the l-infinity norm). We present finite sample bounds on the mean square error, which are established using a Lyapunov framework based on infimal convolution and generalized Moreau envelope. We then present our more recent result on concentration of the tail error, even when the iterates are not bounded by a constant. These tail bounds are obtained using exponential supermartingales in conjunction with the Moreau envelop and a novel bootstrapping approach. Our results immediately imply the state-of-the-art sample complexity results for a large class of RL algorithms.
Bio: Siva Theja Maguluri is Fouts Family Early Career Professor and Associate Professor in the H. Milton Stewart School of Industrial and Systems Engineering at Georgia Tech. He obtained his Ph.D. and MS in ECE as well as MS in Applied Math from UIUC, and B.Tech in Electrical Engineering from IIT Madras. His research interests span the areas of Control, Optimization, Algorithms and Applied Probability and include Reinforcement Learning theory and Stochastic Networks. His research and teaching are recognized through several awards including the Best Publication in Applied Probability award, NSF CAREER award, second place award at INFORMS JFIG best paper competition, Student best paper award at IFIP Performance, CTL/BP Junior Faculty Teaching Excellence Award, and Student Recognition of Excellence in Teaching: Class of 1934 CIOS Award.
We discuss some recent results in graph coloring that show somewhat stronger conclusions in a similar parameter range to traditional coloring theorems. We consider the standard setup of list coloring but ask for a decomposition of the lists into pairwise-disjoint list colorings. The area is new and we discuss many open problems.
A major question in dynamical systems is to understand the mechanisms driving global instability in the 3 Body Problem (3BP), which models the motion of three bodies under Newtonian gravitational interaction. The 3BP is called restricted if one of the bodies has zero mass and the other two, the primaries, have strictly positive masses $m_0, m_1$. In the region of the phase space where the massless body is far from the primaries, the problem can be studied as a (fast) periodic perturbation of the 2 Body Problem (2BP), which is integrable.
We prove that the restricted 3BP exhibits topological instability: for any values of the masses $m_0, m_1$ (except $m_0 = m_1$), we build orbits along which the angular momentum of the massless body (conserved along the flow of the 2BP) experiences an arbitrarily large variation. In order to prove this result we show that a degenerate Arnold diffusion mechanism takes place in the restricted 3BP. Our work extends previous results by Delshams, Kaloshin, De la Rosa and Seara for the a priori unstable case $m_1< 0$, where the model displays features of the so-called a priori stable setting. This is joint work with Marcel Guardia and Tere Seara.
In this talk, we will give background on Lefschetz fibrations and their relationship to symplectic 4-manifolds. We will then discuss results on their fundamental groups. Genus-2 Lefschetz fibrations are of particular interest because of how much we know and don't know about them. We will see what fundamental groups a genus-2 Lefschetz fibration can have and what questions someone might ask when studying these objects.
Let P be a set of points and L be a set of lines in the plane, what can we say about the number of incidences between P and L, I(P, L):= |\{ (p, l)\in P\times L, p\in L\}| ?
The problem changes drastically when we consider a thickening version, i.e. when P is a set of unit balls and L is a set of tubes of radius 1. Furstenberg set conjecture can be viewed as an incidence problem for tubes. It states that a set containing an s-dim subset of a line in every direction should have dimension at least (3s+1)/2 when s>0.
We will survey a sequence of results by Orponen, Shmerkin and a recent joint work with Ren that leads to the solution of this conjecture
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am-11:30 am in Skiles 006.
We present an algorithm that generates sets of size equal to the degree of a given projective variety. The steps of this "CCAR" algorithm are individually well-known, but we argue that when combined they form a versatile and under-used method for studying problems in computational algebraic geometry. The latter part of the talk will focus on applying the CCAR algorithm to examples from Schubert calculus.
I will survey recent progress toward Khovanov homology for links in general 3-manifolds based on categorification of $q$-series invariants labeled by Spin$^c$ structures. Much of the talk will focus on the $q$-series invariants themselves. In particular, I hope to explain how to compute them for a general 3-manifold and to describe some of their properties, e.g. relation to other invariants labeled by Spin or Spin$^c$ structures, such as Turaev torsion, Rokhlin invariants, and the "correction terms'' of the Heegaard Floer theory. There are many problems to work on in this relatively new research area. If time permits, I will outline some of them, and, in the context of plumbed 3-manifolds, comment on the relation to lattice cohomology proposed by Akhmechet, Johnson, and Krushkal.
Suppose you have a set $A$ of integers from $\{1, 2, …, N\}$ that contains at least $N / C$ elements.
Then for large enough $N$, must $A$ contain three equally spaced numbers (i.e., a 3-term arithmetic progression)?
In 1953, Roth showed that this is indeed the case when $C \approx \log \log N$, while Behrend in 1946 showed that $C$ can be at most $2^{\sqrt{\log N}}$ by giving an explicit construction of a large set with no 3-term progressions.
Since then, the problem has been a cornerstone of the area of additive combinatorics.
Following a series of remarkable results, a celebrated paper from 2020 due to Bloom and Sisask improved the lower bound on $C$ to $C = (\log N)^{1 + c}$, for some constant $c > 0$.
This talk will describe a new work which shows that the same holds when $C \approx 2^{(\log N)^{1/12}}$, thus getting closer to Behrend's construction.
Based on a joint work with Raghu Meka.
In 2016, Hutchings introduced a knot filtration on the embedded contact homology (ECH) chain complex in order to estimate the linking of periodic orbits of the Reeb vector field, with an eye towards applications to dynamics on the disk. Since then, the knot filtration has been computed for certain lens spaces by myself, and the "action-linking" relationship has been studied for generic contact forms on general three-manifolds by Bechara Senior-Hryniewicz-Salomao. In joint work with Jo Nelson, we study dynamics on surfaces with one boundary component by computing the knot filtration on the ECH chain complex of positive torus knots in S^3. This requires us to understand the contact form as both a prequantization orbibundle and as a periodic open book with positive fractional Dehn twist coefficient. We will focus on the latter point of view to describe how the computation works and the prospects for extending it to more general open books.
The Acyclic Edge Coloring Conjecture, posed independently by Fiam\v{c}ik in 1978 and Alon, Sudakov and Zaks in 2001, asserts that every graph can be properly edge colored with $\Delta+2$ colors such that there is no bicolored cycle. Over the years, this conjecture has attracted much attention. We prove that the conjecture holds asymptotically, that is $(1+o(1))\Delta$ colors suffice. This is joint work with Michelle Delcourt and Luke Postle.
This talk will be about the convergence problem in mean field control (MFC), i.e. the challenge of rigorously justifying the convergence of certain "symmetric" N-particle control problems towards their mean field counterparts. On the one hand, this convergence problem is already well-understood from a qualitative perspective, thanks to powerful probabilistic techniques based on compactness. On the other hand, quantitative results (i.e. rates of convergence) are more difficult to obtain, in large part because the value function of the mean field problem (which is also the solution to a certain Hamilton-Jacobi equation on the Wasserstein space) may fail to be C^1, even if all the data is smooth. After giving an overview of the convergence problem, I will discuss the results of two recent joint works with Cardaliaguet, Daudin, Delarue, and Souganidis, in which we use some ideas from the theory of viscosity solutions to overcome this lack of regularity and obtain rates of convergence of the N-particle value functions towards the value function of the corresponding MFC problem.
Let ${\mathcal F}L^q_s ({\mathbf R}^2)$ denote the set of all tempered distributions $f \in {\mathcal S}^\prime ({\mathbf R}^2)$ such that the norm $ \| f \|_{{\mathcal F}L^q_s} = (\int_{{\mathbf R}^2}\, ( |{\mathcal F}[f](\xi)| \,( 1+ |\xi| )^s )^q\, d \xi )^{ \frac{1}{q} }$ is finite, where ${\mathcal F}[f]$ denotes the Fourier transform of $f$. We investigate the spectral synthesis for the unit circle $S^1 \subset {\mathbf R}^2$ in ${\mathcal F}L^q_s ({\mathbf R}^2)$ with $1\frac{2}{q^\prime}$, where $q^\prime$ denotes the conjugate exponent of $q$. This is joint work with Prof. Sato (Yamagata University).
The talk is based on my joint works with Maxim Prasolov and Vladimir Shastin, where we studied the relation between rectangular diagrams of links and Legendrian links. This relation allows for a complete classification of exchange classes of rectangular diagrams in terms of equivalence classes of Legendrian links and their symmetry groups. Since all rectangular diagrams of given complexity can be searched, this yields a method to algorithmically compare Legendrian links. Of course, the general algorithm has too high complexity for a practical implementation, but in some situations, the most time consuming parts can be bypassed, which allows us to confirm the non-equivalence of Legendrian knots in several previously unresolved cases.
In this talk I will continue to develop a parallel between the classical field theory of electromagnetism and geometric mechanics of animal locomotion. The focus of the previous talk was on electromagnetism, and the focus of this talk will be on the geometric mechanics of animal locomotion. We will investigate the aphorism that a cat dropped (from a safe height) upside-down always lands on her feet. I will explain how non-trivial topology of the configuration space of the cat can act as a "source" of locomotion.
No prior knowledge of classical field theory will be assumed. I will rely on some results from part 1, but I will review the relevant definitions.
This talk will delve into a method specifically designed for
constructing high-order normal forms in Poincaré maps with high-order
precision and without any major assumption or structure of the
dynamical system itself. We will use the result to generate explicit
twist maps, calculating invariant tori, and determining the flying
time expansions around an elliptic fixed point of a Poincaré map. In
particular, this approach is able to check some non-degenerate
conditions in perturbation theory.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.
Positroid varieties are subvarieties of the Grassmannian that arise in the study of total positivity. Knutson, Lam, and Speyer described a certain type of Gröbner degeneration called the Hodge degeneration as projections of order complexes of intervals in the Bruhat order, but their description does not give an explicit Gröbner basis nor initial ideal. We give an explicit, combinatorial description of the Gröbner basis and initial ideal corresponding to the Hodge degeneration for an arbitrary positroid variety. As an application, we show that promotion on rectangular-shaped semistandard tableaux gives a bijection between standard monomials of a positroid variety and its cyclic shifts. This is joint work with Shiliang Gao (UIUC) and Daoji Huang (Minnesota).
Inverse problems involve the reconstruction of hidden objects from possibly noisy indirect measurements and are ubiquitous in a variety of scientific and engineering applications. This kind of problems have two main features that make them interesting yet challenging to solve. First, they tend to be ill-posed: the reconstruction is very sensitive to perturbations in the measurements. Second, real-world applications are often large-scale: resulting in computationally demanding tasks. In this talk I will focus on discrete linear problems: giving a general overview of the well-established class of solvers called Krylov subspace methods and its regularizing properties; as well as flexible variants that make them suitable to solve more challenging optimization tasks. I will show results and examples in different imaging applications.
We consider the algorithmic problem of finding large balanced independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the bipartition. In a bipartite graph it is trivial to find an independent set of density at least half (take one of the partition classes). In contrast, in a random bipartite graph of average degree d, the largest balanced independent sets (containing equal number of vertices from each class) are typically of density (2 + od(1)) log d/d . Can we find such large balanced independent sets in these graphs efficiently? By utilizing the overlap gap property and the low-degree algorithmic framework, we prove that local and low-degree algorithms (even those that know the bipartition) cannot find balanced independent sets of density greater than (1 + ε) log d/d for any ε > 0 fixed and d large but constant.
In recent decades there has been much interest and measured progress in the study of moments of the Riemann zeta-function and, more generally, of various L-functions. Despite a great deal of effort spanning over a century, asymptotic formulas for moments of L-functions remain stubbornly out of reach in all but a few cases. In this talk, we consider the problem for the family of all Dirichlet L-functions of even primitive characters of bounded conductor. I will outline how to harness the asymptotic large sieve to prove an asymptotic formula for the general 2kth moment of an approximation to this family. The result, which assumes the generalized Lindelöf hypothesis for large values of k, agrees with the prediction of Conrey, Farmer, Keating, Rubenstein, and Snaith. Moreover, it provides the first rigorous evidence beyond the so-called “diagonal terms” in their conjectured asymptotic formula for this family of L-functions
If the formal square root of an abelian surface over Q looks like an elliptic curve, it has to be an elliptic curve."
We discuss what such a proposition might mean, and prove the most straightforward version where the precise condition is simply that the L-function of the abelian surface possesses an entire holomorphic square root. The approach follows the Diophantine principle that algebraic numbers or zeros of L-functions repel each other, and is in some sense similar in spirit to the Gelfond--Linnik--Baker solution of the class number one problem.
We discuss furthermore this latter connection: the problems that it raises under a hypothetical presence of Siegel zeros, and a proven analog over finite fields. The basic remark that underlies and motivates these researches is the well-known principle (which is a consequence of the Deuring--Heilbronn phenomenon, to be taken with suitable automorphic forms $f$ and $g$): an exceptional character $\chi$ would cause the formal $\sqrt{L(s,f)L(s,f \otimes \chi)}L(s,g)L(s, g \otimes \chi)$ to have a holomorphic branch on an abnormally big part of the complex plane, all the while enjoying a Dirichlet series formal expansion with almost-integer coefficients. This leads to the kind of situation oftentimes amenable to arithmetic algebraization methods. The most basic (qualitative) form of our main tool is what we are calling the "integral converse theorem for GL(2)," and it is a refinement of a recent Unbounded Denominators theorem that we proved jointly with Frank Calegari and Yunqing Tang.
Mapping class groups of surfaces in general have cohomology that is hard to compute. Meanwhile, within something called the cohomologically-stable range, a family of characteristic classes called the MMM classes (of surface bundles) is enough to generate this cohomology and thus plays an important role for understanding both the mapping class group and surface bundles. Moreover, constructing the so-called Atiyah-Kodaira manifold provides the setting to prove that these MMM classes are non-trivial. Most of this beginner-friendly talk will be dedicated to proving the non-triviality of the first MMM class. Maybe as a side quest, we will also give a crash course on the geometric viewpoint of (co)homology and then apply this viewpoint to understand the constructions and the proofs.
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no inclusion-maximal clique is monochromatic (ignoring isolated vertices). For the binomial random graph G_{n,p} the clique chromatic number has been studied in a number of works since 2016, but for sparse edge-probabilities in the range n^{-2/5} \ll p \ll 1 even the order of magnitude remained a technical challenge.
Resolving open problems of Alon and Krivelevich as well as Lichev, Mitsche and Warnke, we determine the clique chromatic number of the binomial random graph G_{n,p} in most of the missing regime: we show that it is of order (\log n)/p for edge-probabilities n^{-2/5+\eps} \ll p \ll n^{-1/3} and n^{-1/3+\eps} \ll p \ll 1, for any constant \eps > 0. Perhaps surprisingly for a result about random graphs, a key ingredient in the proof is an application of the probabilistic method (that hinges on careful counting and density arguments).
This talk is based on joint work with Lutz Warnke.
The hybrid version of this talk will be available at: https://gatech.zoom.us/j/92357952326
Discrete delay population models are often considered as a compromise between single-species models and more advanced age-structured population models, C.W. Clark, J. Math. Bio. 1976. This talk is based on a recent work (S. Streipert and G.S.K. Wolkowicz, 2023), where we provide a procedure for deriving discrete population models for the size of the adult population at the beginning of each breeding cycle and assume only adult individuals reproduce. This derivation technique includes delay to account for the number of breeding cycles a newborn individual remains immature and does not contribute to reproduction. These models include a survival probability (during the delay period) for the immature individuals, since these individuals have to survive to reach maturity and become members of what we consider the adult population. We discuss properties of this class of discrete delay population models and show that there is a critical delay threshold. The population goes extinct if the delay exceeds this threshold. We apply this derivation procedure to two well-known population models, the Beverton–Holt and the Ricker population model. We analyze their dynamics and compare it to existing delay models.
I will give an overview of recent work, joint with Jacques Verstraete, where we gave an improved lower bound for the off-diagonal Ramsey number $r(4,t)$, solving a long-standing conjecture of Erd\H{o}s. Our proof has a strong non-probabilistic component, in contrast to previous work. This approach was generalized in further work with David Conlon, Dhruv Mubayi and Jacques Verstraete to off-diagonal Ramsey numbers $r(H,t)$ for any fixed graph $H$. We will go over of the main ideas of these proofs and indicate some open problems.
In the early 60’s J. B. Keller and D. Levy discovered a fundamental property: the instability tongues in Mathieu-type equations lose sharpness with the addition of higher-frequency harmonics in the Mathieu potentials. Twenty years later, V. Arnold discovered a similar phenomenon on the sharpness of Arnold tongues in circle maps (and rediscovered the result of Keller and Levy). In this paper we find a third class of object where a similar type of behavior takes place: area-preserving maps of the cylinder. loosely speaking, we show that periodic orbits of standard maps are extra fragile with respect to added drift (i.e. non-exactness) if the potential of the map is a trigonometric polynomial. That is, higher-frequency harmonics make periodic orbits more robust with respect to “drift". This observation was motivated by the study of traveling waves in the discretized sine-Gordon equation which in turn models a wide variety of physical systems. This is a joint work with Mark Levi.
There will be a pre-seminar from 11am to 11:30am (aimed toward grad students and postdocs) in Skiles 006.
The connections between representations of complex semisimple Lie algebras and the geometry of the corresponding flag manifolds have a long history. Moreover, combinatorics plays an important role in the related computations. My talk is devoted to new aspects of this story. On the Lie algebra side, I consider certain modules for quantum affine algebras. I discuss their relationship with Macdonald polynomials, which generalize the irreducible characters of simple Lie algebras. On the geometric side, I consider the quantum K-theory of flag manifolds, which is a K-theoretic generalization of quantum cohomology. A new combinatorial model, known as the quantum alcove model, is also presented. The talk is based on joint work with S. Naito, D. Sagaki, A. Schilling, and M. Shimozono.
The Burau representation is a kind of homological representation of braid groups that has been around for around a century. It remains mysterious in many ways and is of particular interest because of its relation to quantum invariants of knots and links such as the Jones polynomial. In recent work, I came across a relationship between this representation and a moduli space of Euclidean cone metrics on spheres (think e.g. convex polyhedra) first examined by Thurston. After introducing the relevant definitions, I'll explain a bit about this connection and how I used the geometric structure on this moduli space to exactly identify the kernel of the Burau representation after evaluating its formal parameter at complex roots of unity. There will be many pictures!
For every natural number n, if we start with sufficiently many points in R^d in general position there will always exist n points in convex position. The problem of determining quantitative bounds for this statement is known as the Erdős-Szekeres problem, and is one of the oldest problems in Ramsey theory. We will discuss some of its history, along with the recent developments in the plane and in higher dimensions.
In this talk I will present some recent results in collaboration with Jessica Trespalacios where we consider Einstein-Belinski-Zakharov spacetimes and prove local and global existence, long time behavior of possibly large solutions and some applications to gravisolitons of Kasner type.
Legendrian knots are an important kind of knot in contact topology. One of their invariants, the Thurston-Bennequin number, has an upper bound for any given knot type, called max-tb. Using convex surface theory, we will compute the max-tb of positive torus knots and show that two max-tb positive torus knots are Legendrian isotopic. If time permits, we will show that any non max-tb positive torus knot is obtained from the unique max-tb positive torus knot by a sequence of stabilizations.
The fractal uncertainty principle (FUP) roughly says that a function and its Fourier transform cannot both be concentrated on a fractal set. These were introduced to harmonic analysis in order to prove new results in quantum chaos: if eigenfunctions on hyperbolic manifolds concentrated in unexpected ways, that would contradict the FUP. Bourgain and Dyatlov proved FUP over the real numbers, and in this talk I will discuss an extension to higher dimensions. The bulk of the work is constructing certain plurisubharmonic functions on C^n.
Given a Brauer class on a K3 surface over a number field, we prove that there exists infinitely many primes where the reduction of the Brauer class vanishes, under some mild assumptions. This answers a question of Frei--Hassett--Várilly-Alvarado. The proof uses Arakelov intersection theory on GSpin Shimura varieties. If time permits, I will explain some applications to rationality questions. The results in this talk are joint work with Davesh Maulik.
In the past decade, deep learning has made astonishing breakthroughs in various real-world applications. It is a common belief that deep neural networks are good at learning various geometric structures hidden in data sets, such as rich local regularities, global symmetries, or repetitive patterns. One of the central interests in deep learning theory is to understand why deep neural networks are successful, and how they utilize low-dimensional data structures. In this talk, I will present some statistical learning theory of deep neural networks where data exhibit low-dimensional structures, such as lying on a low-dimensional manifold. The learning tasks include regression, classification, feature representation and operator learning. When data are sampled on a low-dimensional manifold, the sample complexity crucially depends on the intrinsic dimension of the manifold instead of the ambient dimension of the data. These results demonstrate that deep neural networks are adaptive to low-dimensional geometric structures of data sets.
Higher-order multiway data is ubiquitous in machine learning and statistics and often exhibits community-like structures, where each component (node) along each different mode has a community membership associated with it. In this talk we propose the tensor mixed-membership blockmodel, a generalization of the tensor blockmodel positing that memberships need not be discrete, but instead are convex combinations of latent communities. We first study the problem of estimating community memberships, and we show that a tensor generalization of a matrix algorithm can consistently estimate communities at a rate that improves relative to the matrix setting, provided one takes the tensor structure into account. Next, we study the problem of testing whether two nodes have the same community memberships, and we show that a tensor analogue of a matrix test statistic can yield consistent testing with a tighter local power guarantee relative to the matrix setting. If time permits we will also examine the performance of our estimation procedure on flight data. This talk is based on two recent works with Anru Zhang.
A fully nonlinear surface dynamics of the time dependent potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. We also consider a generalized hydrodynamics for two components of superfluid Helium which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement.
In 1973, Nambu published an article entitled "Generalized Hamiltonian dynamics". For that purpose, he constructed multilinear brackets - equivalent to Poisson brackets - with some interesting properties reminiscent of the Jacobi identity.
These brackets found some applications in fluid mechanics, plasma physics and mathematical physics with superintegrable systems.
In this seminar, I will recall some basic elements on Nambu mechanics in finite dimension with an n-linear Nambu bracket in dimension larger than n. I will discuss all possible Nambu brackets and compare them with all possible Poisson brackets. I will conclude that Nambu mechanics can hardly be considered a generalization of Hamiltonian mechanics.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.
Speaker will present in person
In many applications across science and engineering it is common to have access to disparate types of data or models with different levels of fidelity. In general, low-fidelity data are easier to obtain in greater quantities, but it may be too inaccurate or not dense enough to accurately train a machine learning model. High-fidelity data is costly to obtain, so there may not be sufficient data to use in training, however, it is more accurate. A small amount of high-fidelity data, such as from measurements or simulations, combined with low fidelity data, can improve predictions when used together. The important step in such constructions is the representation of the correlations between the low- and high-fidelity data. In this talk, we will present two frameworks for multifidelity machine learning. The first one puts particular emphasis on operator learning, building on the Deep Operator Network (DeepONet). The second one is inspired by the concept of model reduction. We will present the main constructions along with applications to closure for multiscale systems and continual learning. Moreover, we will discuss how multifidelity approaches fit in a broader framework which includes ideas from deep learning, stochastic processes, numerical methods, computability theory and renormalization of complex systems.
We consider classical scalar fields in dimension 1+1 with a
self-interaction potential being a symmetric double-well. Such a model
admits non-trivial static solutions called kinks and antikinks. A kink
cluster is a solution approaching, for large positive times, a
superposition of alternating kinks and antikinks whose velocities
converge to 0 and mutual distances grow to infinity. Our main result is
a determination of the asymptotic behaviour of any kink cluster at the
leading order.
Our results are partially inspired by the notion of "parabolic motions"
in the Newtonian n-body problem. I will present this analogy and mention
its limitations. I will also explain the role of kink clusters as
universal profiles for formation of multi-kink configurations.
This is a joint work with Andrew Lawrie.
The well-known tree packing conjecture of Gyárfás from 1976 says that, given any sequence of n trees in which the ith tree has i vertices, the trees can be packed edge-disjointly into the complete n-vertex graph. Packing even just the largest trees in such a sequence has proven difficult, with Bollobás drawing attention to this in 1995 by conjecturing that, for each k, if n is sufficiently large then the largest k trees in any such sequence can be packed. This has only been shown for k at most 5, by Zak, despite many partial results and much related work on the full tree packing conjecture.
I will discuss a result which proves Bollobás's conjecture by showing that, moreover, a linear number of the largest trees can be packed in the tree packing conjecture. This is joint work with Barnabás Janzer.
Vassiliev knot invariants, or finite-type invariants, are a broad class of knot invariants resulting from extending usual invariants to knots with transverse double points. We will show that the Conway and Jones polynomials are fully described by Vassiliev invariants. We will discuss the fundamental theorem of Vassiliev invariants, relating them to the algebra of chord diagrams and weight systems. Time permitting, we will also discuss the Kontsevich integral, the universal Vassiliev invariant.
Langhede and Thomassen conjectured in 2020 that there exists a positive constant c such that every planar graph G with 5-correspondence assignment (L,M) has at least 2^{c v(G)} distinct (L,M)-colourings. I will discuss a proof of this conjecture (which relies on the hyperbolicity of a certain family of graphs), a generalization of this result to some other embedded graphs (again, relying on a hyperbolicity theorem), and a few open problems in the area. Everything presented is joint work with Luke Postle.
In this talk, we study the long-term behaviour of Random Dynamical Systems (RDSs) conditioned upon staying in a region of the space. We use the absorbing Markov chain theory to address this problem and define relevant dynamical systems objects for the analysis of such systems. This approach aims to develop a satisfactory notion of ergodic theory for random systems with escape.
There will be a pre-seminar from 11am to 11:30am (aimed toward grad students and postdocs) in Skiles 006.
A toric vector bundle is a vector bundle over a toric variety equipped with a linear action by the torus of the base. Toric vector bundles pf rank r were famously classified by Klyachko (1989) using certain combinatorial data of compatible filtrations in an r-dimensional vector space E. This data can be thought of as a higher rank generalization of an (integer-valued) piecewise linear function. In this talk, we give an interpretation of Klyachko data as a "piecewise linear map" to a tropical linear space. This point of view leads us to introduce the notion of a "matroidal vector bundle", a generalization of toric vector bundles to (possibly non-representable) matroids. As a special case and by-product of this construction, one recovers the tautological classes of matroids introduced by Berget, Eur, Spink and Tseng. This is a work in progress with Chris Manon (Kentucky).
Mean field game models have been developed in different application areas. We discuss here inverse problems to mean field game models where we are interested in reconstructing missing information from observed data. We present a few different scenarios where differential data allows for the unique reconstruction of model parameters in various forms. The talk is mainly based on recent joint works with Nathan Soedjak and Kewei Wang.
Locally conformal symplectic (LCS) geometry is a variant of symplectic geometry in which the symplectic form is locally only defined up to positive scale. For example, for the symplectization R x Y of a contact manifold Y, translation in the R direction are symplectomorphisms up to scale, and hence the quotient (R/Z) x Y is naturally an LCS manifold. The importation of symplectic techniques into LCS geometry is somewhat subtle because of this ambiguity of scale. In this talk, we define a notion of product for LCS manifolds, in which the underlying manifold of an LCS product is not simply the smooth product of the underlying manifolds, but which nonetheless appears to fill the same role in LCS geometry as the standard symplectic product does in standard symplectic geometry. As a proof of concept, with input from an LCS result of Chantraine and Murphy, we use the LCS product to prove that C^0 small Hamiltonian isotopies have a lower bound on the number of fixed points given by the rank Morse-Novikov homology. This is a natural generalization of the classical symplectic proof of the analogous result by Laudenbach and Sikorav which uses the graph of a Hamiltonian diffeomorphism in the product manifold. These results are joint work in progress with Baptiste Chantraine.
Let $S$ be a subset of $1 ,…, N$ avoiding the nontrivial progressions $x, x+y^2-1, x+2(y^2-1)$. We prove that $|S| < N/\log \log \cdots \log(N)$, where we have a fixed constant number of logarithms. This answers a question of Green, and is the first effective polynomial Szemerédi result over the integers where the polynomials involved are not homogeneous of the same degree and the underlying pattern has linear complexity. Joint work with Sarah Peluse and Mehtaab Sawhney.
I will discuss a mixture of results and conjectures related to the Khovanov homology and Knot Floer homology for ribbon knots. We will explore relationships with fusion numbers (a measure of complexity on ribbon disks) and particular families of symmetric unions (ribbon knots given by particular diagrams). This is joint work with Nathan Dunfield, Sherry Gong, Tom Hockenhull, and Marco Marengon.
In 1975 Bollobas, Erdos, and Szemeredi asked the following question: given positive integers $n, t, r$ with $2\le t\le r$, what is the largest minimum degree among all $r$-partite graphs G with parts of size $n$ and which do not contain a copy of $K_t$? The $r=t$ case has attracted a lot of attention and was fully resolved by Haxell and Szabo, and Szabo and Tardos in 2006. In this talk we discuss recent progress on the $r>t$ case and related extremal results on multipartite graphs.
I will begin by describing the ideas involved in the Nash iterative constructions of solutions to the Euler equations. These were introduced by De Lellis and Szekelyhidi (and developed by many authors) in order to tackle the flexible side of the Onsager conjecture. Then, I will describe Isett’s proof of the conjecture in the 3D case, and highlight the simple reason for which the strategy will not work in 2D. Finally, I will describe a construction of non-conservative solutions that works also in 2D (this is joint work with Vikram Giri).
The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. More precisely, we show that the operator
$$
H_C(f_1, f_2, f_3)(x):=p.v. \int_{\mathbb R} f_1(x-t)f_2(x+t^2)f_3(x+t^3) \frac{dt}{t}, \quad x \in \mathbb R
$$
is bounded from $L^{p_1}(\mathbb R) \times L^{p_2}(\mathbb R) \times L^{p_3}(\mathbb R}$ into $L^r(\mathbb R)$ within the Banach H\"older range $\frac{1}{p_1}+\frac{1}{p_2}+\frac{1}{p_3}=\frac{1}{r}$ with $1
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
2). a structural analysis of suitable maximal "joint Fourier coefficients";
3). a level set analysis with respect to the time-frequency correlation set.
This is a joint work with my postdoc advisor Victor Lie from Purdue.
This talk is a summary of a summary. We will be going over Jen Hom's 2024 Levi L. Conant Prize Winning Article "Getting a handle on the Conway knot," which discusses Lisa Piccirillo's renowned 2020 paper proving the Conway knot is not slice. In this presentation, we will go over what it means for a knot to be slice, past attempts to classify the Conway knot with knot invariants, and Piccirillo's approach of constructing a knot with the same knot trace as the Conway knot. This talk is designed for all audiences and NO prior knowledge of topology or knot theory is required. Trust me, I'm (k)not a topologist.
Join us as we define a whole new algebraic structure, starting from the axioms of the projective plane. This seminar will be aimed at students who have never seen this material and will focus on hands-on constructions of classic (and new!) algebraic structures that can arise from a projective plane. The goal of this seminar is to expose you to Desargues's theorem and hopefully even examine non-Desarguesian planes.
In this talk, we consider a finite-horizon optimal control problem of stochastic reaction-diffusion equations. First, we apply the spike variation method which relies on introducing the first and second order adjoint state. We give a novel characterization of the second order adjoint state as the solution to a backward SPDE. Using this representation, we prove the maximum principle for controlled SPDEs.
In the second part, we present a numerical algorithm that allows the efficient approximation of optimal controls in the case of stochastic reaction-diffusion equations with additive noise by first reducing the problem to controls of feedback form and then approximating the feedback function using finitely based approximations. Numerical experiments using artificial neural networks as well as radial basis function networks illustrate the performance of our algorithm.
This talk is based on joint work with Wilhelm Stannat and Alexander Vogler. Talk will also be streamed: https://gatech.zoom.us/j/93808617657?pwd=ME44NWUxbk1NRkhUMzRsK3c0ZGtvQT09
The notion of $S$-labeling, where $S$ is a subset of the symmetric group, is a common generalization of signed $k$-coloring, signed $\mathbb{Z}_k$-coloring, DP (or Correspondence) coloring, group coloring, and coloring of gained graphs that was introduced in 2019 by Jin, Wong, and Zhu. In this talk we use a well-known theorem of Alon and F\"{u}redi to present an algebraic technique for bounding the number of colorings of an $S$-labeled graph from below. While applicable in the broad context of counting colorings of $S$-labeled graphs, we will focus on the case where $S$ is a symmetric group, which corresponds to DP-coloring (or, correspondence coloring) of graphs, and the case where $S$ is a set of linear permutations which is applicable to the coloring of signed graphs, etc.
This technique allows us to prove exponential lower bounds on the number of colorings of any $S$-labeling of graphs that satisfy certain sparsity conditions. We apply these to give exponential lower bounds on the number of DP-colorings (and consequently, number of list colorings, or usual colorings) of families of planar graphs, and on the number of colorings of families of signed (planar) graphs. These lower bounds either improve previously known results, or are first known such results.
This joint work with Samantha Dahlberg and Jeffrey Mudrock.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.
Determining the computational complexity of matrix multiplication has been one of the central open problems in theoretical computer science ever since in 1969
Strassen presented an algorithm for multiplication of n by n matrices requiring only O(n^2.81) arithmetic operations. The data describing this method is
equivalently an expression to write the structure tensor of the 2 by 2 matrix algebra as a sum of 7 decomposable tensors. Any such decomposition of an n by n
matrix algebra yields a Strassen type algorithm, and Strassen showed that such algorithms are general enough to determine the exponent of matrix multiplication. Bini later showed all of the above remains true when we allow the decomposition to depend on a parameter and take limits.
Joint {Applied & Computational Math Seminar} and {School of Math Colloquium}.<br />
Speaker will present in person.
Geometry arises in myriad ways within machine learning and related areas. I this talk I will focus on settings where geometry helps us understand problems in machine learning, optimization, and sampling. For instance, when sampling from densities supported on a manifold, understanding geometry and the impact of curvature are crucial; surprisingly, progress on geometric sampling theory helps us understand certain generalization properties of SGD for deep-learning! Another fascinating viewpoint afforded by geometry is in non-convex optimization: geometry can either help us make training algorithms more practical (e.g., in deep learning), it can reveal tractability despite non-convexity (e.g., via geodesically convex optimization), or it can simply help us understand existing methods better (e.g., SGD, eigenvector computation, etc.).
Ultimately, I hope to offer the audience some insights into geometric thinking and share with them some new tools that help us design, understand, and analyze models and algorithms. To make the discussion concrete I will recall a few foundational results arising from our research, provide several examples, and note some open problems.
––
Bio: Suvrit Sra is a Alexander von Humboldt Professor of Artificial Intelligence at the Technical University of Munich (Germany), and and Associate Professor of EECS at MIT (USA), where he is also a member of the Laboratory for Information and Decision Systems (LIDS) and of the Institute for Data, Systems, and Society (IDSS). He obtained his PhD in Computer Science from the University of Texas at Austin. Before TUM & MIT, he was a Senior Research Scientist at the Max Planck Institute for Intelligent Systems, Tübingen, Germany. He has held visiting positions at UC Berkeley (EECS) and Carnegie Mellon University (Machine Learning Department) during 2013-2014. His research bridges mathematical topics such as differential geometry, matrix analysis, convex analysis, probability theory, and optimization with machine learning. He founded the OPT (Optimization for Machine Learning) series of workshops, held from OPT2008–2017 at the NeurIPS conference. He has co-edited a book with the same name (MIT Press, 2011). He is also a co-founder and chief scientist of Pendulum, a global AI+logistics startup.
Joint {School of Math Colloquium} and {Applied & Computational Math Seminar}. Note: *special time*.<br />
Speaker will present in person.<br />
Geometry arises in myriad ways within machine learning and related areas. I this talk I will focus on settings where geometry helps us understand problems in machine learning, optimization, and sampling. For instance, when sampling from densities supported on a manifold, understanding geometry and the impact of curvature are crucial; surprisingly, progress on geometric sampling theory helps us understand certain generalization properties of SGD for deep-learning! Another fascinating viewpoint afforded by geometry is in non-convex optimization: geometry can either help us make training algorithms more practical (e.g., in deep learning), it can reveal tractability despite non-convexity (e.g., via geodesically convex optimization), or it can simply help us understand existing methods better (e.g., SGD, eigenvector computation, etc.).
Ultimately, I hope to offer the audience some insights into geometric thinking and share with them some new tools that help us design, understand, and analyze models and algorithms. To make the discussion concrete I will recall a few foundational results arising from our research, provide several examples, and note some open problems.
––
Bio: Suvrit Sra is a Alexander von Humboldt Professor of Artificial Intelligence at the Technical University of Munich (Germany), and and Associate Professor of EECS at MIT (USA), where he is also a member of the Laboratory for Information and Decision Systems (LIDS) and of the Institute for Data, Systems, and Society (IDSS). He obtained his PhD in Computer Science from the University of Texas at Austin. Before TUM & MIT, he was a Senior Research Scientist at the Max Planck Institute for Intelligent Systems, Tübingen, Germany. He has held visiting positions at UC Berkeley (EECS) and Carnegie Mellon University (Machine Learning Department) during 2013-2014. His research bridges mathematical topics such as differential geometry, matrix analysis, convex analysis, probability theory, and optimization with machine learning. He founded the OPT (Optimization for Machine Learning) series of workshops, held from OPT2008–2017 at the NeurIPS conference. He has co-edited a book with the same name (MIT Press, 2011). He is also a co-founder and chief scientist of Pendulum, a global AI+logistics startup.
Stochastic flows of an advective-diffusive nature are ubiquitous in biology and the physical sciences. Of particular interest is the problem to reconcile observed marginal distributions with a given prior posed by E. Schroedinger in 1932/32 and known as the Schroedinger Bridge Problem (SBP). It turns out that Schroedinger’s problem can be viewed as a problem in large deviations, a modeling problem, as well as a control problem. Due to the fundamental significance of this problem, interest in SBP and in its deterministic (zero-noise limit) counterpart of Optimal Transport (OT) has in recent years enticed a broad spectrum of disciplines, including physics, stochastic control, computer science, and geometry. Yet, while the mathematics and applications of SBP/OT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention; the problem to interpolate between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations, in an adhoc manner, chiefly driven by applications in image registration. Nevertheless, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schroedinger’s dictum; that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated law represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this talk is to present recent results on stochastic evolutions with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of Schroedinger, given a prior law that allows for losses, we explore the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman-Kac multiplicative reweighing of the reference measure which, as we argue, is far from Schroedinger’s quest. We develop a suitable Schroedinger system of coupled PDEs' for this problem, an iterative Fortet-IPF-Sinkhorn algorithm for computations, and finally formulate and solve a related fluid-dynamic control problem for the flow of one-time marginals where both the drift and the new killing rate play the role of control variables. Joint work with Tryphon Georgiou and Michele Pavon.
Seminar is in-person. Zoom link available: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT0... />
Continuum theories of physics are traditionally described by local partial differential equations (PDEs). In this talk I will discuss the Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) algorithm: a general algorithm combining the weak formulation, symmetry covariance, and sparse regression to discover quantitatively accurate and qualitatively simple PDEs directly from data. This method is applied to simulated 3D turbulence and experimental 2D active turbulence. A complete mathematical model is found in both cases.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.
Chebyshev polynomials offer a natural basis for solving polynomial equations. When we switch from monomials to Chebyshev polynomials, we can replace toric varieties with Chebyshev varieties. We will introduce these objects and discuss their main properties, including equations, dimension, and degree. This is an ongoing project with Zaïneb Bel-Afia and Simon Telen.
Generative machine learning models, including variational auto-encoders (VAE), normalizing flows (NF), generative adversarial networks (GANs), diffusion models, have dramatically improved the quality and realism of generated content, whether it's images, text, or audio. In science and engineering, generative models can be used as powerful tools for probability density estimation or high-dimensional sampling that critical capabilities in uncertainty quantification (UQ), e.g., Bayesian inference for parameter estimation. Studies on generative models for image/audio synthesis focus on improving the quality of individual sample, which often make the generative models complicated and difficult to train. On the other hand, UQ tasks usually focus on accurate approximation of statistics of interest without worrying about the quality of any individual sample, so direct application of existing generative models to UQ tasks may lead to inaccurate approximation or unstable training process. To alleviate those challenges, we developed several new generative diffusion models for various UQ tasks, including diffusion-model-assisted supervised learning of generative models, a score-based nonlinear filter for recursive Bayesian inference, and a training-free ensemble score filter for tracking high dimensional stochastic dynamical systems. We will demonstrate the effectiveness of those methods in various UQ tasks including density estimation, learning stochastic dynamical systems, and data assimilation problems.
This talk concerns improving sum-product exponents for sets of integers under the condition that each element of has no more than prime factors. The argument combines combinatorics, harmonic analysis and number theory.
Turán-type problems ask for the densest-possible structure which avoids a fixed substructure H. Ramsey-type problems ask for the largest possible "complete" structure which can be decomposed into a fixed number of H-free parts. We discuss some of these problems in the context of vector spaces over finite fields. In the Turán setting, Furstenberg and Katznelson showed that any constant-density subset of the affine space AG(n,q) must contain a k-dimensional affine subspace if n is large enough. On the Ramsey side of things, a classical result of Graham, Leeb, and Rothschild implies that any red-blue coloring of the projective space PG(n-1,q) must contain a monochromatic k-dimensional projective subspace, for n large. We highlight the connection between these results and show how to obtain new bounds in the latter (projective Ramsey) problem from bounds in the former (affine Turán) problem. This is joint work with Liana Yepremyan.
A sequence of remarkable results in recent decades have shown that for a surface group H there are many Lie groups G and connected components C of Hom(H,G) consisting of discrete and faithful representations. These are known as higher Teichmüller spaces. With two exceptions, all known constructions of higher Teichmüller spaces work only for surface groups. This is an expository talk on the remarkable paper Convexes Divisibles III (Benoist ‘05), in which the first construction of higher Teichmüller spaces that works for some non-surface-groups was discovered. The paper implies the fundamental group H’ of any closed hyperbolic n-manifold has a higher Teichmüller space C’ in PGL(n+1,R). This is proved by showing any element of C’ preserves a convex domain in RP^n with a group-invariant tiling.
Permutation limit theory arises by viewing a permutation as a probability measure on the unit square. Using the theory of permutation limits (permutons), we can compute limiting properties of various permutation statistics for random permutations, such as number of fixed points, number of small cycles, pattern counts, and degree distribution of permutation graphs. We can also derive LDPs for random permutations. Our results apply to many non uniform distributions on permutations, including the celebrated Mallows model, and mu-random permutations. This is based on joint work with Jacopo Borga, Sayan Das and Peter Winkler.
Come learn about chip firing games! While simple to define, these games provide surprisingly strong combinatorial tools for studying algebraic curves. Fueling this theory is a strong analogy between algebraic curves and finite graphs. In ways we will make more precise, many of the features of algebraic curves can be studied in graphs, however certain parts of the theory don’t make it through intact. In this talk we will focus on a central question in this analogy: which graphs are the best models for algebraic curves? We will set up the background needed to ask this question as well as the tools and techniques used to study such graphs. No prior knowledge of chip-firing or algebraic geometry needed.
A successful trend in modern extremal/probabilistic combinatorics is the investigation of how well classical theorems, like those of Ramsey, Turán, and Szemerédi, hold in sparse random contexts. Graph and hypergraph container methods have played a big role in improving our knowledge of these sparse structures. I will present joint work with Jozsef Balogh and Haoran Luo on a random version of the Erdős-Ko-Rado Theorem and Sperner's Theorem, giving the flavor of some graph container techniques.
Continuum theories of physics are traditionally described by local partial differential equations (PDEs). In this talk I will discuss the Sparse Physics-Informed Discovery of Empirical Relations (SPIDER) algorithm: a general algorithm combining the weak formulation, symmetry covariance, and sparse regression to discover quantitatively accurate and qualitatively simple PDEs directly from data. This method is applied to simulated 3D turbulence and experimental 2D active turbulence. A complete mathematical model is found in both cases.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11 am to 11:30 am in Skiles 006.
The homotopy continuation is a widely recognized method for finding solutions to polynomial systems by tracking the homotopy paths of solutions. However, the current implementation of homotopy continuation relies on heuristics, and hence it requires certification to verify its correctness. We discuss two modalities of certification in algebraic geometry exploiting interval arithmetic. The first is certified homotopy tracking using the Krawczyk method which guarantees correct tracking without path jumping. The second is Smale’s alpha theory over regions for faster certification. We discuss experimental results to demonstrate the effectiveness of these new methods. This talk is a preliminary report of two separate ongoing works.
The sparse solution obtained from greedy-based optimization approach such as orthogonal matching pursuit can be very useful and have many applications in different directions. In this talk, I will present two research projects, one is about semi-supervised local clustering, and the other is about function approximation, which make use of the sparse solution technique. We will show that the target cluster can be effectively retrieved in the local clustering task and the curse of dimensionality can be overcome for a dense subclass of the space of continuous functions via Kolmogorov superposition theorem. Both the theoretical and numerical results will be discussed.
In her thesis, Maryam Mirzakhani counted the number of simple closed geodesics of bounded length on a (real) hyperbolic surface. This breakthrough theorem and the subsequent explosion of related results use techniques and draw inspiration from Teichmüller theory, symplectic geometry, surface topology, and homogeneous dynamics. In this talk, I’ll discuss some of these connections and a qualitative strengthening of her theorem, describing what these curves, and their complements, actually (generically) look like. This is joint work with Francisco Arana-Herrera.
https://gatech.zoom.us/j/94328087718
How complicated can successive manifolds get in a tower of covering
spaces? Specifically, how large can the dimension of the first
cohomology get? We will begin with a tour of possible behaviors for
low-dimensional spaces, and then focus on arithmetic manifolds.
Specifically, for towers of complex-hyperbolic manifolds, I will
describe how to bound the rates of growth using known instances of
Langlands functoriality.
Sampling from the Gibbs distribution is a long-standing problem studied across various fields. Among many sampling algorithms, Langevin dynamics plays a crucial role, particularly for high-dimensional target distributions. In practical applications, accelerating sampling dynamics is always desirable. It has long been studied that adding an irreversible component to reversible dynamics, such as Langevin, can accelerate convergence. Concrete constructions of irreversible components have also been explored in specific scenarios. However, a general strategy for such construction is still elusive. In this talk, I will introduce the concept of leveraging irreversibility to accelerate general dynamics, along with the quantification of irreversible dynamics. Our theory is mainly based on designing a modified entropy functional originally developed for linear kinetic equations (Dolbeault et al., 2015).
A $k \times n$ partial Latin rectangle is \textit{$C$-sparse} if the number of nonempty entries in each row and column is at most $C$ and each symbol is used at most $C$ times. We prove that the probability a uniformly random $k \times n$ Latin rectangle, where $k < (1/2 - \alpha)n$, contains a $\beta n$-sparse partial Latin rectangle with $\ell$ nonempty entries is $(\frac{1 \pm \varepsilon}{n})^\ell$ for sufficiently large $n$ and sufficiently small $\beta$. Using this result, we prove that a uniformly random order-$n$ Latin square asymptotically almost surely has no Latin subsquare of order greater than $c\sqrt{n\log n}$ for an absolute constant $c$. This is joint work with Tom Kelly, Camille Kennedy, and Jasdeep Sidhu.
How many critical points does a random function from R^N to R have for large N? Such functions appear naturally in probability, data science, and mathematical physics. Questions like this one, which have attracted longstanding interest from both physicists and mathematicians, can help explain both physical phase transitions and algorithmic thresholds. I will give an overview of this "landscape complexity" program, its motivations, and recent progress coming from random matrices.
The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.
The Poincaré constant of a body, or more generally a probability density, in $\mathbb R^n$ measures how "spread out" the body is - for instance, this constant controls how long it takes heat to flow from an arbitrary point in the body to any other. It's thus intuitively reasonable that convolving a "sufficiently nice" measure with a Gaussian, which tends to flatten and smooth out the measure, would increase its Poincaré constant ("spectral monotonicity"). We show that this is true if the original measure is log-concave, via two very different strategies - a dynamic variant of Bakry-Émery's $\Gamma$-calculus, and a mass-transportation argument. Moreover, we show that the dynamic $\Gamma$-calculus argument can also be extended to the discrete setting of measures on $\mathbb Z$, and that spectral monotonicity holds in this setting as well. Some results joint with B. Klartag.
A fundamental problem in analytic number theory is to calculate the maximal value of L-functions at a given point. For L-functions associated to quadratic Dirichlet characters at s = 1, the upper bounds and Ω-results of Littlewood differ by a factor of 2, and it is a long-standing (and still unsolved) problem to find out which one is closer to the truth. The important work of Granville and Soundararajan, who model the distribution of L(1, χ) by the distribution of random Euler products L(1, X) for random variables X(p) attached to each prime, shed more light to the question. We use similar techniques to study the distribution of L(1, χ) for cubic Dirichlet characters. Unlike the quadratic case, there is an asymmetry between lower and upper bounds for the cubic case, and small values are less probable than large values. This is a joint work with P. Darbar, M. Lalin and A. Lumley.
Link to join via Zoom: https://gatech.zoom.us/j/93394018195?pwd=MGJZaWIwQUhVYW9ZZDFoWWFOc29EZz0... />
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Meeting ID: 933 9401 8195<br />
Passcode: SoM
We define and motivate the Poisson point process, which is, informally, a “maximally random” scattering of points in some locally compact, second countable space. We introduce the ideal Poisson--Voronoi tessellation (IPVT), a new random object with intriguing geometric properties when considered on a semisimple symmetric space (the hyperbolic plane, for example). In joint work with Mikolaj Fraczyk and Sam Mellick, we use the IPVT to prove the minimal number of generators of a torsion-free lattice in a higher rank, semisimple Lie group is sublinear in the co-volume of the lattice. We give some intuition for the proof. No prior knowledge on Poisson point processes or symmetric spaces will be assumed.
Note special time, due to time zone difference from Japan.<br />
Joint with SIAM GT Student Chapter Seminar
Part I (SAM as an Optimal Relaxation of Bayes) Dr. Thomas Moellenhoff
Sharpness-aware minimization (SAM) and related adversarial deep-learning methods can drastically improve generalization, but their underlying mechanisms are not yet fully understood. In this talk, I will show how SAM can be interpreted as optimizing a relaxation of the Bayes objective where the expected negative-loss is replaced by the optimal convex lower bound, obtained by using the so-called Fenchel biconjugate. The connection enables a new Adam-like extension of SAM to automatically obtain reasonable uncertainty estimates, while sometimes also improving its accuracy.
Part II (Lie Group updates for Learning Distributions on Machine Learning Parameters) Dr. Eren Mehmet Kıral
I will talk about our recent paper https://arxiv.org/abs/2303.04397 with Thomas Möllenhoff and Emtiyaz Khan, and other related results. Bayesian Learning learns a distribution over the model parameters, allowing for different descriptions of the same data. This is (contrary to classical learning which "bets-it-all" on a single set of parameters in describing a given dataset and making predictions. We focus on classes of distributions which have a transitive Lie group action on them given by pushforwards of an action on the parameter space. I will also specialize to a few concrete Lie groups and show distinct learning behavior.
This colloquium will also be the staring talk for the 2023 Tech Topology Conference.
This talk will give an elementary introduction to my joint work with Kyler Siegel that shows how cuspidal curves in a symplectic manifold X such as the complex projective plane determine when an ellipsoid can be symplectically embedded into X.
Abstract: In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k \ge 2$, consider the set of $k$-tuples of reduced fractions $\frac{a_1}{q_1}, \dots, \frac{a_k}{q_k} \in I$, where $I$ is an interval around $0$.
How many $k$-tuples are there with $\sum_i \frac{a_i}{q_i} \in \mathbb Z$?
When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from ``diagonal'' terms, where the fractions $\frac{a_i}{q_i}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In ongoing work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.
Abstract: Fundamental to our understanding of Teichm\"uller space T(S) of a closed oriented genus $g \geq 2$ surface S are two different perspectives: one as connected component in the PSL(2,\R) character variety \chi(\pi_1S, PSL(2,\R)) and one as the moduli space of marked hyperbolic structures on S. The latter can be thought of as a moduli space of (PSL(2,\R), \H^2) -structures. The G-Hitchin component, denoted Hit(S,G), for G a split real simple Lie group, is a connected component in \chi(\pi_1S, G) that is a higher rank generalization of T(S). In this talk, we discuss a new geometric structures (i.e., (G,X)-structures) interpretation of Hit(S, G_2'), where G_2' is the split real form of the exceptional complex simple Lie group G_2.
After discussing the motivation and background, we will present some of the main ideas of the theorem, including a family of almost-complex curves
that serve as bridge between the geometric structures and representations.
This lecture concerns the metric Riemannian geometry of Einstein manifolds, which is a central theme in modern differential geometry and is deeply connected to a large variety of fundamental problems in algebraic geometry, geometric topology, analysis of nonlinear PDEs, and mathematical physics. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics. My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. Such problems constitute the most challenging part in the metric geometry of Einstein manifolds. We will introduce recent major progress in the field. If time permits, I will propose several important open questions.
Krylov subspace methods (KSMs) are among the most widely used algorithms for a number of core linear algebra tasks. However, despite their ubiquity throughout the computational sciences, there are many open questions regarding the remarkable convergence of commonly used KSMs. Moreover, there is still potential for the development of new methods, particularly through the incorporation of randomness as an algorithmic tool. This talk will survey some recent work on the analysis of the well-known Lanczos method for matrix functions and the design of new KSMs for low-rank approximation of matrix functions and approximating partial traces and reduced density matrices.
Algebraic geometry studies solution sets of polynomial equations. For instance, over the complex numbers, one may examine the topology of the solution set, whereas over a finite field, one may count its points. For polynomials with integer coefficients, these two fundamental invariants are intimately related via cohomological comparison theorems and trace formulas for the action of Frobenius. I will discuss the general framework relating point counting over finite fields to topology of complex algebraic varieties and also present recent applications to the cohomology of moduli spaces of curves that resolve longstanding questions in algebraic geometry and confirm more recent predictions from the Langlands program.
The complete flag variety is a fundamental object at the confluence of algebraic geometry, representation theory, and algebra. It is defined to be the space parametrizing certain chains of vector subspaces, and is intimately linked to Grassmannians, incidence varieties, and other important geometric objects of a representation-theoretic flavor. The problem of computing the cohomology of any line bundle on a flag variety in characteristic 0 was solved in the 1950's, culminating in the celebrated Borel--Weil--Bott theorem. The situation in positive characteristic is wildly different, and remains a wide-open problem despite many decades of study. After surveying this topic, I will speak about recent progress on a characteristic-free analogue of the Borel--Weil--Bott theorem through the lens of representation stability and the theory of polynomial functors. This "stabilization" of cohomology, combined with certain universal categorifications of the Jacobi-Trudi identity, has opened the door to concrete computational techniques whose applications include effective vanishing results for Koszul modules, yielding an algebraic counterpart for the failure of Green's conjecture for generic curves in arbitrary characteristic.
The existence of global solutions for the Schrödinger equation
i\partial_t u + \Delta u = P_d(u),
with nonlinearity $P_d$ homogeneous of degree $d$, has been extensively studied. Most results focus on the case with gauge invariant nonlinearity, where the solution satisfies several conservation laws. However, the problem becomes more complicated as we consider a general nonlinearity $P_d$. So far, global well-posedness for small data is known for $d$ strictly greater than the Strauss exponent. In dimension $3$, this Strauss exponent is $2$, making NLS with quadratic nonlinearity an interesting topic.
In this talk, I will present a result that shows the global existence and scattering for systems of quadratic NLS for small, localized data. To tackle the challenge presented by the $u\Bar{u}$-type nonlinearity, we require an $\epsilon$ regularization for the terms of this type in the system.
The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface. A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3. We will explain work of Ershov—He and Church—Ershov—Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1. Time permitting, we will also discuss some extensions of these ideas. In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.
This talk will describe connections between algebraic geometry, convex geometry and algebraic topology. We will be discussing linear projections of the special orthogonal group and when they are convex (in the sense that every pair of points in the image of the projection are connected by a line segment contained in the projection). In particular, I'll give a proof of the fact that the image of SO(n) under any linear map to R^2 is convex using some elementary homotopy theory. These kinds of question are not only geometrically interesting but are also useful in solving some optimization problems involved in space travel.
Symmetry is prevalent in a variety of machine learning and scientific computing tasks, including computer vision and computational modeling of physical and engineering systems. Empirical studies have demonstrated that machine learning models designed to integrate the intrinsic symmetry of their tasks often exhibit substantially improved performance. Despite extensive theoretical and engineering advancements in symmetry-preserving machine learning, several critical questions remain unaddressed, presenting unique challenges and opportunities for applied mathematicians.
Firstly, real-world symmetries rarely manifest perfectly and are typically subject to various deformations. Therefore, a pivotal question arises: Can we effectively quantify and enhance the robustness of models to maintain an “approximate” symmetry, even under imperfect symmetry transformations? Secondly, although empirical evidence suggests that symmetry-preserving models require fewer training data to achieve equivalent accuracy, there is a need for more precise and rigorous quantification of this reduction in sample complexity attributable to symmetry preservation. Lastly, considering the non-convex nature of optimization in modern machine learning, can we ascertain whether algorithms like gradient descent can guide symmetry-preserving models to indeed converge to objectively better solutions compared to their generic counterparts, and if so, to what degree?
In this talk, I will provide an overview of my research addressing these intriguing questions. Surprisingly, the answers are not as straightforward as one might assume and, in some cases, are counterintuitive. My approach employs an interesting blend of applied probability, harmonic analysis, differential geometry, and optimization. However, specialized knowledge in these areas is not required.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:30 am to noon in Skiles 005.
A neural code C on n neurons is a collection of subsets of {1,2,...,n} which is used to encode the intersections of subsets U_1, U_2,...,U_n of some topological space. The study of neural codes reveals the ways in which geometric or topological properties can be encoded combinatorially. A prominent example is the property of max-intersection completeness: if a code C contains every possible intersection of its maximal codewords, then one can always find a collection of open convex U_1, U_2,..., U_n for which C is the code. In this talk I will answer a question posed by Curto et al. (2018), which asks if there is a way of determining max-intersection completeness from examination of the neural ideal, an algebraic counterpart to the neural code.
In this presentation, I will discuss my research in the field of data science, specifically in two areas: improving autoencoder interpolations and accelerating federated learning algorithms. My work combines advanced mathematical concepts with practical machine learning applications, contributing to both the theoretical and applied aspects of data science. The first part of my talk focuses on image sequence interpolation using autoencoders, which are essential tools in generative modeling. The focus is when there is only limited training data. By introducing a novel regularization term based on dynamic optimal transport to the loss function of autoencoder, my method can generate more robust and semantically coherent interpolation results. Additionally, the trained autoencoder can be used to generate barycenters. However, computation efficiency is a bottleneck of our method, and we are working on improving it. The second part of my presentation focuses on accelerating federated learning (FL) through the application of Anderson Acceleration. Our method achieves the same level of convergence performance as state-of-the-art second-order methods like GIANT by reweighting the local points and their gradients. However, our method only requires first-order information, making it a more practical and efficient choice for large-scale and complex training problems. Furthermore, our method is theoretically guaranteed to converge to the global minimizer with a linear rate.
It has been known since the pioneering work of J.L. Lions in 1960s that 3D hyper-dissipative (HD) Navier-Stokes (NS) system exhibits global-in-time regularity as long as the hyper-diffusion exponent is greater or equal to 5/4. One should note that at 5/4, the system is critical, i.e., the energy level and the scaling -invariant level coincide. What happens in the super-critical regime, the hyper-diffusion exponent being strictly between 1 and 5/4 remained a mystery.
The goal of this talk is to demonstrate that as soon as the hyper-diffusion exponent is greater than 1, a class of monotone blow-up scenarios consistent with the analytic structure of the flow (prior to the possible singular time) can be ruled out (a sort of 'runaway train' scenario). The argument is in the spirit of the regularity theory of the 3D HD NS system in 'turbulent scenario' (in the super-critical regime) developed by Grujic and Xu, relying on 'dynamic interpolation' – however, it is much shorter, tailored to the class of blow-up profiles in view. This is a joint work with Aseel Farhat.
Title: Topology, geometry and adaptivity in soft and living matter
Abstract:
Topology and adaptivity play fundamental roles in controlling the dynamics of biological and physical systems, from chromosomal DNA and biofilms to cilia carpets and worm collectives. How topological rules govern the self-adaptive dynamics of living matter remains poorly understood. Here we investigate the interplay between topology, geometry and reconfigurability in knotted and tangled matter. We first identify topological counting rules which predict the relative mechanical stability of human-designed knots, by developing a mapping between elastic knots and long-range ferromagnetic spin systems. Building upon this framework, we then examine the adaptive topological dynamics exhibited by California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments controls their emergent topological state. To further understand how tangled matter, along with more general biological networks, adapt to their surroundings, we introduce a theory of adaptive elastic networks which can learn mechanical information. By identifying how topology and adaptivity produce stable yet responsive structures, these results have applications in understanding broad classes of adaptive, self-optimizing biological systems.
Zoom: https://gatech.zoom.us/j/93619173236?pwd=ZGNRZUZ2emNJbG5pRzgzMnlFL1dzQT09
Over the past decade, a rich theory of existence for the isoperimetric problem in spaces of nonnegative curvature has been established by multiple authors.
We will briefly review this theory, with a special focus on the reasons why one may expect the isoperimetric problem to have a solution in any nonnegatively curved space: it is true for large enough volumes, it is true if the ambient is 2-dimensional, and it is true under appropriate assumptions on the ambient space at infinity.
The main topic of the talk will be the presentation of a counterexample to this "intuition": a 3-dimensional manifold of positive sectional curvature without isoperimetric sets for small volumes.
This is a joint work with G. Antonelli.
Braid groups are relatively simple to describe, but they have deep and intricate connections to many different areas of math. We will discuss three specific instances where the braid group on 3 strands arises in geometry and knot theory. In exploring connections between these perspectives, we will take a detour into the world of elliptic curves and their moduli space. As a result, we will see that these three perspectives are actually the same. Time permitting, we will explore generalizations of this to the braid group on n strands for n > 3.
Zoom link: https://gatech.zoom.us/j/98245747313?pwd=RmFtcmlWYjBncXJTOU00NFMvSVNsZz0... />
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Meeting ID: 982 4574 7313<br />
Passcode: SoM
Abstract: Classifying the invariant measures for a given dynamical system represents a fundamental challenge.
Every graph is associated to a symmetric function constructed from proper colorings of the graph. The Stanley-Stembridge conjecture posits that the expansion of the chromatic symmetric function into the elementary symmetric functions has positive coefficients for a certain class of graphs. We explore a potential new approach to the Stanley-Stembridge Conjecture using combinatorial objects called "special rim hooks" and connect this to the "chromatic quasisymmetric functions" introduced by Shareshian and Wachs as a generalization of chromatic symmetric functions. This is joint work with Meagan Hodge.
A central theme in 4-dimensional topology is the search for exotic 4-manifolds, i.e. families of smooth manifolds that are homeomorphic not diffeomorphic. We will survey some basic results in this area.
There is no pre-seminar this time.
The classical theorems of Perron and Frobenius, which explore spectral properties of nonnegative matrices, have been extensively examined and generalized from various perspectives, including a cone-theoretic (geometric) viewpoint. Concurrently, in the past decade, there has been a notable effort to fuse the techniques of algebraic geometry and combinatorics in an exploration of Lorentzian polynomials by Brändén and Huh, also known as completely log-concave polynomials (CLC) by Anari et.al. or strongly log-concave polynomials by Gurvits.
In this talk, I will discuss my ongoing joint work with Greg Blekherman regarding the class of polynomials with Lorentzian signature (PLS) defined over closed convex cones. This class encompasses various special polynomials, including Lorentzian polynomials over the nonnegative orthant and hyperbolic polynomials over hyperbolicity cones. We establish a compelling connection between PLS over a self-dual cone K and the generalized Perron Frobenius theorem over K. This connection enables us to provide an alternative necessary and sufficient condition to characterize the Lorentzian polynomials.
We discuss new methods for using the Heegaard Floer homology of hypersurfaces to distinguish between smooth closed 4-manifolds that are homeomorphic but non-diffeomorphic. Specifically, for a 4-manifold X with b_1(X)=1, the minimum rank of the reduced Heegaard Floer homology of any embedded 3-manifold X representing a generator of H_1(X) gives a diffeomorphism invariant of X. We use this invariant to distinguish certain infinite families of exotic 4-manifolds that cannot be distinguished by previously known techniques. Using related ideas, we also provide the first known examples of (non-simply-connected) exotic 4-manifolds with negative definite intersection form. This is joint work with Tye Lidman and Lisa Piccirillo.
In recent years, algebraization theorems arising from model theory, in particular o-minimality, have been a crucial ingredient in several breakthroughs in arithmetic geometry and Hodge theory. In this talk, I'll present some of my recent work on p-adic versions of these model theoretic algebraization criteria, with a focus on two different applications of this circle of ideas. The first being an algebraization theorem in the context of Shimura varieties, which are vaguely speaking moduli spaces of Hodge structures. The second being in the context of non-abelian Hodge theory, in the setting of moduli spaces of flat connections and local systems.
Zoom: https://gatech.zoom.us/j/95425627723
This will be an expository talk which aims to introduce some problems in harmonic analysis and geometric measure theory concerning the geometry of a measure for which an associated integral operator is well behaved. As an example, we shall prove a result of Mattila and Preiss concerning the relationship between the rectifiability of a measure and the existence of the Riesz transform in the sense of principle value.
The Johnson filtration is a filtration of the mapping class group induced by the action of the mapping class group on the lower central series of the fundamental group of a surface. A theorem of Johnson tells us that the first term of this filtration, called the Torelli group, is finitely generated for surfaces of genus at least 3. We will explain work of Ershov-He and Church-Ershov-Putman, which uses Johnson's result to show that the kth term of the Johnson filtration is finitely generated for surfaces of genus g at least 2k - 1. Time permitting, we will also discuss some extensions of these ideas. In particular, we will explain how to show that the terms of the Johnson filtration are finitely presented assuming the Torelli group is finitely presented.
This special algebra seminar will feature short talks by our very own May Cai and Matt Baker, who will speak on the following topics:
May Cai: The completion problem asks one to take a partial observation of some underlying object, and try to recover the original observation. Concretely, we have some object of interest, and a point in the image of that object under a projection map, and want to understand the fiber of this point under this map. In particular, for log-linear models, which are the restrictions of toric varieties to the probability simplex, under certain mild conditions, when this fiber is finite it turns out to have exactly either one or two entries. This is joint work with Cecilie Olesen Recke and Thomas Yahl.
Matt Baker: The determinant of a skew-symmetric matrix has a canonical square root given by the Pfaffian. Similarly, the resultant of two reciprocal polynomials of even degree has a canonical square root given by their reciprocant. Computing the reciprocant of two cyclotomic polynomials yields a short and elegant proof of the Law of Quadratic Reciprocity.
Many numerical methods have been developed in the past years for computing weak solutions (with shock waves) to nonlinear hyperbolic conservation laws. My research, specifically, concerns the design of well-balanced numerical algorithms that preserve certain key structure of these equations in various applications, including for problems involving moving phase boundaries and other scale-dependent interfaces. In particular, in this lecture, I will focus on the evolution of a compressible fluid in spherical symmetry on a Schwarzschild curved background, for which I have designed a class of well-balanced numerical algorithms up to third-order of accuracy. Both the relativistic Burgers-Schwarzschild model and the relativistic Euler-Schwarzschild model were considered, and the proposed numerical algorithm took advantage of the explicit or implicit forms available for the stationary solutions of these models. The schemes follow the finite volume methodology and preserve the stationary solutions and, most importantly, allow us to investigate the global asymptotic behavior of such flows and determine the asymptotic behavior of the mass density and velocity field of the fluid. Blog: philippelefloch.org
We prove that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a (smooth) submanifold within the space of complex projective structures on that surface.
In the realm of mathematical fluid dynamics, a formidable challenge lies in establishing inviscid limits from the Navier-Stokes equations to the Euler equations. The pursuit of solving this intricate problem, particularly concerning singular solutions, persists in both compressible and incompressible scenarios. In particular, compressible Euler equations are a typical system of hyperbolic conservation laws, whose solution forms shock waves in general.
In this talk, we will discuss the recent proof on the unique vanishing viscosity limit from Navier-Stokes equations to the BV solution of compressible Euler equations, for the general Cauchy Problem. Moreover, we extend our findings by establishing the well-posedness of such solutions within the broader class of inviscid limits of Navier-Stokes equations with locally bounded energy initial values. This is a joint work with Kang and Vasseur, which can be found on arXiv:2401.09305.
The uniqueness and L2 stability of Euler equations, done by Chen-Krupa-Vasseur, will also be discussed in this talk.
We show new lower bounds for sphere packings in high dimensions and for independent sets in graphs with not too large co-degrees. For dimension d, this achieves a sphere packing of density (1 + o(1)) d log d / 2^(d+1). In general dimension, this provides the first asymptotically growing improvement for sphere packing lower bounds since Roger's bound of c*d/2^d in 1947. The proof amounts to a random (very dense) discretization together with a new theorem on constructing independent sets on graphs with not too large co-degree. Both steps will be discussed, and no knowledge of sphere packings will be assumed or required. This is based on joint work with Marcelo Campos, Matthew Jenssen and Julian Sahasrabudhe.
The concept of holonomy arises in many areas of mathematics, especially control theory. This concept is also related to the broader program of geometrization of forces in physics. In order to understand holonomy, we need to understand principal (fiber) bundles. In this talk I will explain U(1)-principal bundles by example. This explanation will be from the point-of-view of a geometer, but I will introduce the terminology of control theory. Finally, we will do a holonomy computation for a famous example of Aharonov and Bohm.
James Anderson: Odd coloring (resp, PCF coloring) is a stricter form of proper coloring in which every nonisolated vertex is required to have a color in its neighborhood with odd multiplicity (resp, with multiplicity 1). Using the discharging method, and a new tool which we call the Forb-Flex method, we improve the bounds on the odd and PCF chromatic number of planar graphs of girth 10 and 11, respectively.
Sean Kafer: Many classical combinatorial optimization problems (e.g. max weight matching, max weight matroid independent set, etc.) have formulations as linear programs (LPs) over 0/1 polytopes on which LP solvers could be applied. However, there often exist bespoke algorithms for these problems which, by merit of being tailored to a specific domain, are both more efficient and conceptually nicer than running a generic LP solver on the associated LP. We will discuss recent results which show that a number of such algorithms (e.g. the shortest augmenting path algorithm, the greedy algorithm, etc.) can be "executed" by the Simplex method for solving LPs, in the sense that the Simplex method can be made to generate the same sequence of solutions when applied to the appropriate corresponding LP.
Tantan Dai: There has been extensive research on Latin Squares. It is simple to construct a Latin Square with n rows and n columns. But how do we define a Latin Triangle? What are the rows? When does a Latin Triangle exist? How can we construct them? In this talk, I will discuss two types of Latin Triangles and the construction of a countable family of Latin Triangles.
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This will be an introduction to Legendrian contact homology (LCH), a version of Floer homology that's important in contact topology, for the setting of Legendrian knots in R^3 with the standard contact structure. LCH is the homology of a differential graded algebra that can be defined combinatorially in terms of a diagram for the knot. We'll explore this combinatorial definition, with examples, and discuss some auxiliary invariants derived from LCH. No background about contact manifolds or Legendrian knots will be assumed.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30am in Skiles 005.
Hilbert's Nullstellensatz about zero sets of polynomials is one of the most fundamental correspondences between algebra and geometry. More recently, there has been an emerging interest in polynomial equations and inequalities in several matrix variables, prompted by developments in control systems, quantum information theory, operator algebras and optimization. The arising problems call for a suitable version of (real) algebraic geometry in noncommuting variables; with this in mind, the talk considers matricial sets where noncommutative polynomials attain singular values, and their algebraic counterparts.
Given a polynomial f in noncommuting variables, its free (singularity) locus is the set of all matrix tuples X such that f(X) is singular. The talk focuses on the interplay between geometry of free loci (irreducible components, inclusions, eigenlevel sets, smooth points) and factorization in the free algebra. In particular, a Nullstellensatz for free loci is given, as well as a noncommutative variant of Bertini's irreducibility theorem and its consequences.
Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information about the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to design transferable neural feature spaces for the shallow neural networks from purely function approximation perspectives without using PDE information. The construction of the feature space involves the re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. We use the proposed feature space as the predetermined feature space of a random feature model, and use existing least squares solvers to obtain the weights of the output layer. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.
For the past 25 years, a key player in contact topology has been the Floer-theoretic invariant called Legendrian contact homology. I'll discuss a package of new invariants for Legendrian knots and links that builds on Legendrian contact homology and is derived from rational symplectic field theory. This includes a Poisson bracket on Legendrian contact homology and a symplectic structure on augmentation varieties. Time permitting, I'll also describe an unexpected connection to cluster theory for a family of Legendrian links associated to positive braids. Parts of this are joint work in progress with Roger Casals, Honghao Gao, Linhui Shen, and Daping Weng.
I will discuss a nonlinear elliptic system of partial differential equations arising in Riemannian geometry and General Relativity. Specifically, I will present recent advances on the analysis of asymptotically Euclidean, initial data sets for Einstein’s field equations. In collaboration with Bruno Le Floch (Sorbonne University) I proved that solutions to the Einstein constraints can be glued together along possibly nested conical domains. The constructed solutions may have arbitrarily low decay at infinity, while enjoying (super-)harmonic estimates within possibly narrow cones at infinity. Importantly, our localized seed-to-solution method, as we call it, leads to a proof of a conjecture by Alessandro Carlotto and Richard Schoen on the localization problem at infinity, and generalize P. LeFloch and Nguyen’s theorem on the asymptotic localization problem. This lecture will be based on https://arxiv.org/abs/2312.17706
Sidorenko's conjecture can be formulated as "Let $H$ be a bipartite graph, and $\rho\in [0,1]$. Of all the graphs with edge density $\rho$, the graph(-limit) obtained by picking edges uniformly at random minimizes the homomorphism density of $H$." This conjecture, first formulated in 1991 by Sidorenko, has received considerable attention over the last decades, and yet remains open in the general case.
It was shown recently [Blekherman, Raymond, Singh, Thomas, 2020] that sums-of-squares in Razborov's flag algebra are not strong enough to prove even small, known cases of the conjecture. To circumvent this, we introduce a novel kind of derivation of flags. Due to their combinatoric nature, we can use them to systematically gain knowledge on global minimizers of problems in extremal graph theory. We combine them with the flag algebra method to find new proofs for various cases of Sidorenko's conjecture.
Available on zoom at:<br />
https://gatech.zoom.us/j/98258240051
We shall discuss the quantum dynamics associated with ergodic Schroedinger operators. Anderson localization (pure point spectrum with exponentially decaying eigenfunctions) has been obtained for a variety of ergodic operator families, but it is well known that Anderson localization is highly unstable and can also be destroyed by generic rank one perturbations. For quasiperiodic operators, it also sensitively depends on the arithmetic properties of the phase (a seemingly irrelevant parameter from the point of view of the physics of the problem) and doesn’t hold generically. These instabilities are also present for the physically relevant notion of dynamical localization.
In this talk we will introduce the notion of the transport exponent, explain its stability, and explain how logarithmic upper bounds may be obtained in the quasi-periodic setting for all relevant parameters. This is based on joint work with S. Jitomirskaya.
The fine curve graph of a surface is a graph whose vertices are essential simple closed curves in the surface and whose edges connect disjoint curves. Following a rich history of hyperbolicity in various graphs based on surfaces, the fine curve was shown to be hyperbolic by Bowden–Hensel–Webb. Given how well-studied the curve graph and the case of “up to isotopy” is, we ask: what about the mysterious part of the fine curve graph not captured by isotopy classes? In this talk, we introduce the result that the subgraph of the fine curve graph spanned by curves in a single isotopy class is not hyperbolic; indeed, it contains a flat of EVERY dimension. Along the way, we will discuss how to not prove this theorem as we explore proofs of hyperbolicity of related complexes. This work is joint with Ryan Dickmann.
Unicritical polynomials, typically written in the form $z^d+c$, have been widely studied in arithmetic and complex dynamics and are characterized by their one finite critical point. The behavior of a map's critical points under iteration often determines the dynamics of the entire map. Rational maps where the critical points have a finite forward orbit are called post-critically finite (PCF), and these are of great interest in arithmetic dynamics. They are viewed as a dynamical analogue of abelian varieties with complex multiplication and often display interesting dynamical behavior. The family of (single-cycle normalized) dynamical Belyi polynomials have two fixed critical points, so they are PCF by construction, and these maps provide a new testing ground for conjectures and ideas related to post-critically finite polynomials. Using this family, we can begin to explore properties of polynomial maps with two critical points. In this talk we will discuss applications of this family in arithmetic dynamics; in particular, how this family can be used to determine more general reduction properties of PCF polynomials.
Motivated by problems in data science, we study the following questions:
(1) Given a Hilbert space V and a group G of linear isometries, does there exist a bilipschitz embedding of the quotient metric space V/G into a Hilbert space?
(2) What are necessary and sufficient conditions for such embeddings?
(3) Which embeddings minimally distort the metric?
We answer these questions in a variety of settings, and we conclude with several open problems.
Empirical Bayes provides a powerful approach to learning and adapting to latent structure in data. Theory and algorithms for empirical Bayes have a rich literature for sequence models, but are less understood in settings where latent variables and data interact through more complex designs.
In this work, we study empirical Bayes estimation of an i.i.d. prior in Bayesian linear models, via the nonparametric maximum likelihood estimator (NPMLE). We introduce and study a system of gradient flow equations for optimizing the marginal log-likelihood, jointly over the prior and posterior measures in its Gibbs variational representation using a smoothed reparametrization of the regression coefficients. A diffusion-based implementation yields a Langevin dynamics MCEM algorithm, where the prior law evolves continuously over time to optimize a sequence-model log-likelihood defined by the coordinates of the current Langevin iterate.
We show consistency of the NPMLE under mild conditions, including settings of random sub-Gaussian designs under high-dimensional asymptotics. In high noise, we prove a uniform log-Sobolev inequality for the mixing of Langevin dynamics, for possibly misspecified priors and non-log-concave posteriors. We then establish polynomial-time convergence of the joint gradient flow to a near-NPMLE if the marginal negative log-likelihood is convex in a sub-level set of the initialization.
This is joint work with Leying Guan, Yandi Shen, and Yihong Wu.
Quasi-stationary distributions (QSDs) are those almost invariant to a diffusion process over exponentially long time. Representing important transient stochastic dynamics, they arise frequently in applications especially in chemical reactions and population systems admitting extinction states. This talk will present some rigorous results on the existence, uniqueness, concentration, and convergence of QSDs along with their connections to the spectra of the Fokker-Planck operators.
I will talk about various notions of equivalence for manifolds and morphisms and the relationships between them. Questions, interruptions, and detours are strongly encouraged!
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30 am in Skiles 005.
Given a matroid and a group of its matroid automorphisms, we study the induced group action on the Chow ring of the matroid. This turns out to always be a permutation action. Work of Adiprasito, Huh and Katz showed that the Chow ring satisfies Poincar\'e duality andthe Hard Lefschetz theorem. We lift these to statements about this permutation action, and suggest further conjectures in this vein.
In dimension 4, there exist simply connected manifolds which are homeomorphic but not diffeomorphic; the difference between the distinct smooth structures can be localized using corks. Similarly, there exist diffeomorphisms of simply connected 4-manifolds which are topologically but not smoothly isotopic to the identity. In this talk, I will discuss some preliminary results towards an analogous localization of this phenomena using corks for diffeomorphisms. This project is joint work with Slava Krushkal, Anubhav Mukherjee, and Mark Powell.
It is known that the symplectic property is preserved by the mean curvature flow in a K\"ahler-Einstein surface which is called "symplectic mean curvature flow". It was proved that there is no finite time Type I singularities for the symplectic mean curvature flow. We will talk about recent progress on an important Type II singularity of symplectic mean curvature flow-symplectic translating soliton. We will show that a symplectic translating soliton must be a plane under some natural assumptions which are necessary by investigating some examples.
The study of Ramsey properties of the binomial random graph G_{n,p} was initiated in the 80s by Frankl & Rödl and Łuczak, Ruciński & Voigt. In this area we are often interested in establishing what function f(n) governs G_{n,p} having a particular Ramsey-like property P or not, i.e. when p is sufficiently larger than f(n) then G_{n,p} a.a.s. has P and when p is sufficiently smaller than f(n) then G_{n,p} a.a.s. does not have P (the former we call a 1-statement, the latter a 0-statement). I will present recent results on this topic from two different papers.
In the first, we almost completely resolve an outstanding conjecture of Kohayakawa and Kreuter on asymmetric Ramsey properties. In particular, we reduce the 0-statement to a necessary colouring problem which we solve for almost all pairs of graphs. Joint work with Candy Bowtell and Robert Hancock.
In the second, we prove similar results concerning so-called anti- and constrained-Ramsey properties. In particular, we (essentially) completely resolve the outstanding parts of the problem of determining the threshold function for the constrained-Ramsey property, and we reduce the anti-Ramsey problem to a necessary colouring problem which we prove for a specific collection of graphs. Joint work with Natalie Behague, Robert Hancock, Shoham Letzter and Natasha Morrison.
When studying symplectic 4-manifolds, it is useful to consider Lefschetz fibrations over the 2-sphere due to their one-to-one correspondence uncovered by Freedman and Gompf. Lefschetz fibrations of genera 0 and 1 are well understood, but for genera greater than or equal to 2, much less is known. However, some Lefschetz fibrations with monodromies that respect the hyperelliptic involution of a genus-g surface have stronger properties which make their invariants easier to compute. In this talk, we will explore Terry Fuller's results from the late 90's which explore two-fold branched covers of hyperelliptic genus-g Lefschetz fibrations. We will look at his proof of why a Lefschetz fibration with only nonseparating vanishing cycles is a two-fold cover of $S^2 \times S^2$ branched over an embedded surface. The talk will include definitions, constructions, and Kirby pictures of branched covers in 4 dimensions. If time, we will discuss his results on hyperelliptic genus-g Lefschetz fibration which contain at least one separating vanishing cycles.
In this talk, I will present some joint works with Tom Courtade on characterizing probability measures that optimize the constant in a given functional inequalitiy via integration by parts formulas, and how Stein's method can be used to prove quantitative bounds on how close almost-optimal measures are to true optimizers. I will mostly discuss Poincaré inequalities and Gaussian optimizers, but also some other examples if time allows it.
Note unusual date and length for the seminar!
It is well known that all contact 3-manifolds can be obtained from the standard contact structure on the 3-sphere by contact surgery on a Legendrian link. Hence, an interesting and much studied question asks what properties (e.g. tightness, fillability, vanishing or non-vanishing of various Floer or symplectic homology classes) of contact structures are preserved under various types of contact surgeries. The case for the negative contact surgeries is fairly well understood. The case of positive contact surgeries much more subtle. In this talk, extending an earlier work of the speaker with Conway and Etnyre, I will discuss some new results about symplectic fillability of positive contact surgeries, and in particular we will provide a necessary and sufficient condition for contact (positive) integer surgery along a Legendrian knot to yield a fillable contact manifold. When specialized to knots in the three sphere with its standard tight structure, this result can be rather efficient to find many examples of fillable surgeries along with various obstructions and surprising topological applications. This will report on joint work with T. Mark.
Designing effective positional encodings for graphs is key to building powerful graph transformers and enhancing message-passing graph neural networks’ expressive power. However, since there lacks a canonical order of nodes in the graph-structure data, the choice of positional encodings for graphs is often tricky. For example, Laplacian eigenmap is used as positional encodings in many works. However, it faces two fundamental challenges: (1) Non-uniqueness: there are many different eigen-decompositions of the same Laplacian, and (2) Instability: small perturbations to the Laplacian could result in completely different eigenvectors, leading to unpredictable changes in positional encoding. This is governed by the Davis-Kahan theorem, which further negatively impacts the model generalization. In this talk, we are to introduce some ideas on building stable positional encoding and show their benefits in model out-of-distribution generalization. The idea can be extended to some other types of node positional encodings. Finally, we evaluate the effectiveness of our method on molecular property prediction, link prediction, and out-of-distribution generalization tasks, finding improved generalization compared to existing positional encoding methods.
I will mainly talk about three papers:
1. Distance Encoding: Design Provably More Powerful Neural Networks for Graph Representation Learning,NeurIPS20, Pan Li, Yanbang Wang, Hongwei Wang, Jure Leskovec
2. Equivariant and Stable Positional Encoding for More Powerful Graph Neural Networks, ICLR22 Haorui Wang, Haoteng Yin, Muhan Zhang, Pan Li
3. On the Stability of Expressive Positional Encodings for Graphs, ICLR24 Yinan Huang, William Lu, Joshua Robinson, Yu Yang, Muhan Zhang, Stefanie Jegelka, Pan Li
In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Generalizations of this criterion, under additional hypotheses, have been proven for other toric varieties, for instance by Eisenbud-Erman-Schreyer for products of projective spaces, by Schreyer for Segre-Veronese varieties, and Ottaviani for Grassmannians and quadrics. This talk is about a splitting criterion for arbitrary smooth projective toric varieties, as well as an algorithm for finding indecomposable summands of sheaves and modules in the more general setting of Mori dream spaces.
In this presentation the analytical background of nonlinear observers based on minimal energy estimation is discussed. Originally, such strategies were proposed for the reconstruction of the state of finite dimensional dynamical systems by means of a measured output where both the dynamics and the output are subject to white noise. Our work aims at lifting this concept to a class of partial differential equations featuring deterministic perturbations using the example of a wave equation with a cubic defocusing term in three space dimensions. In particular, we discuss local regularity of the corresponding value function and consider operator Riccati equations to characterize its second spatial derivative.
A graph is called k-critical if its chromatic number is k but any proper subgraph has chromatic number less than k. There have been extensive reseaches on k-critical graphs over the past decades, yet several basic problems remain widely open. One of such problems is to determine the maximum number of edges in an n-vertex k-critical graph. In this talk, we will discuss some recent results on extremal aspects of k-critical graphs, including improvments on the extremal number of edges/cliques/critical subgraphs in k-critical graphs. This is based on some joint works with Jun Gao, Cong Luo and Tianchi Yang.
The curve graph provides a combinatorial perspective to study surfaces. Classic work of Ivanov showed that the automorphisms of this graph are naturally isomorphic to the mapping class group. By dropping isotopies, more recent work of Long-Margalit-Pham-Verberne-Yao shows that there is also a natural isomorphism between the automorphisms of the fine curve graph and the homeomorphism group of the surface. Restricting this graph to smooth curves might appear to be the appropriate object for the diffeomorphism group, but it is not. In this talk, we will discuss why this doesn’t work and some progress towards describing the group of homeomorphisms that is naturally isomorphic to automorphisms of smooth fine curve graphs.
Large-scale parallel-processing infrastructures such as data centers and cloud networks form the cornerstone of the modern digital environment. Central to their efficiency are resource management policies, especially load balancing algorithms (LBAs), which are crucial for meeting stringent delay requirements of tasks. A contemporary challenge in designing LBAs for today's data centers is navigating data locality constraints that dictate which tasks are assigned to which servers. These constraints can be naturally modeled as a bipartite graph between servers and various task types. Most LBA heuristics lean on the mean-field approximation's accuracy. However, the non-exchangeability among servers induced by the data locality invalidates this mean-field framework, causing real-world system behaviors to significantly diverge from theoretical predictions. From a foundational standpoint, advancing our understanding in this domain demands the study of stochastic processes on large graphs, thus needing fundamental advancements in classical analytical tools.
In this presentation, we will delve into recent advancements made in extending the accuracy of mean-field approximation for a broad class of graphs. In particular, we will talk about how to design resource-efficient, asymptotically optimal data locality constraints and how the system behavior changes fundamentally, depending on whether the above bipartite graph is an expander, a spatial graph, or is inhomogeneous in nature.
An edge-coloured graph is said to be rainbow if it uses no colour more than once. Extremal problems involving rainbow objects have been a focus of much research as they capture the essence of a number of interesting problems in a variety of areas. A particularly intensively studied question due to Keevash, Mubayi, Sudakov and Verstraëte from 2007 asks for the maximum possible average degree of a properly edge-coloured graph on n vertices without a rainbow cycle. Improving upon a series of earlier bounds, Tomon proved an upper bound of (log n)^(2+o(1)) for this question. Very recently, Janzer-Sudakov and Kim-Lee-Liu-Tran independently removed the o(1) term in Tomon's bound. We show that the answer to the question is equal to (log n)^(1+o(1)).
Joint work with: Noga Alon, Lisa Sauermann, Dmitrii Zakharov and Or Zamir.
In this talk, we'll sketch how one might hope to construct spaces (or spectra) from Floer theories, including framed flow categories and finite-dimensional approximation. If time allows, we'll talk about some questions Floer spaces (or spectra) can be useful for.
There will be a pre-seminar from 11 am to 11:30 am in Skiles 005.
We describe an algorithm to compute Whitney stratifications of real algebraic varieties, and of polynomial maps between them, by exploiting the algebraic structure of certain conormal spaces. One of the map stratification algorithms described here yields a new method for solving the real root classification problem. We also explore applications of this new map stratification algorithm to the study of the singularities of Feynman integrals; understanding and evaluating these integrals is a fundamental component in a wide variety of problems arising in quantum field theory.
We'll give a short description of what exactly monopole Floer spectra are, and then explain how to calculate them for AR plumbings, a class of 3-manifolds including Seifert spaces. This is joint work with Irving Dai and Hirofumi Sasahira.
Speaker will present in person.
Diffusion models have become ubiquitous for image generation and are increasingly being used for scientific applications. To date, many flavors of diffusion models have been developed by varying the stochastic process that noises data, but also the domain on which these processes act. Typically, generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact analysis. We relate the theory to the existing continuous-state Gaussian diffusion in discrete and continuous time. This approach is validated using a simple stochastic decay process, in which the reverse process generates images from a single all-black image, rather than a noisy prior distribution.
An optimal control problem in the space of probability measures, and the viscosity solutions of the corresponding dynamic programming equations defined using the intrinsic linear derivative are studied. The value function is shown to be Lipschitz continuous with respect to a novel smooth Fourier Wasserstein metric. A comparison result between the Lipschitz viscosity sub and super solutions of the dynamic programming equation is proved using this metric, characterizing the value function as the unique Lipschitz viscosity solution. This is joint work with Prof. H. Mete Soner.
For a graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap percolation process on $G$ is the process which starts with $G$ and, at every time step, adds any missing edges on the vertices of $G$ that complete a copy of $H$. This process eventually stabilises and we are interested in the extremal question raised by Bollob\'as, of determining the maximum \emph{running time} (number of time steps before stabilising) of this process, over all possible choices of $n$-vertex graph $G$. We initiate a systematic study of this parameter, denoted $M_H(n)$, and its dependence on properties of the graph $H$. In a series of works we determine the precise running time for cycles and asymptotic running time for several other important classes. In general, we study necessary and sufficient conditions on $H$ for fast, i.e. sublinear or linear $H$-bootstrap percolation, and in particular explore the relationship between running time and minimum vertex degree and connectivity. Furthermore we also obtain the running time of the process for typical $H$ and discover several graphs exhibiting surprising behavior. The talk represents joint work with David Fabian and Patrick Morris.
In recent years, the few classical results in large deviations for random matrices have been complemented by a variety of new ones, in both the math and physics literatures, whose proofs leverage connections with Harish-Chandra/Itzykson/Zuber integrals. We present one such result, focusing on extreme eigenvalues of deformed sample-covariance and Wigner random matrices. This confirms recent formulas of Maillard (2020) in the physics literature, precisely locating a transition point whose analogue in non-deformed models is not yet fully understood. Joint work with Jonathan Husson.
This talk has two goals. The first is to talk through Keynes-Newton’s construction of minimal non-uniquely ergodic interval exchange transformations. The second is to explain why I’m talking about this in the student topology seminar.
The game of HEX has deep mathematical underpinnings despite its simple rules. What could this game possibly have to do with coffee?! And how does that connection, once identified, lead to consideration of ferromagnetism and even to the melting polar ice caps? Join Hugo Duminil-Copin, Professor of Mathematics at IHES and the University of Geneva, for an exploration of the way in which mathematical thinking can help us make some truly surprising connections.
The Ising model is one of the most classical lattice models of statistical physics undergoing a phase transition. Initially imagined as a model for ferromagnetism, it revealed itself as a very rich mathematical object and a powerful theoretical tool to understand cooperative phenomena. Over one hundred years of its history, a profound understanding of its critical phase has been obtained. While integrability and mean-field behavior led to extraordinary breakthroughs in the two-dimensional and high-dimensional cases respectively, the model in three and four dimensions remained mysterious for years. In this talk, we will present recent progress in these dimensions based on a probabilistic interpretation of the Ising model relating it to percolation models.
This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series).
Corrine Yap: The Ising model is a classical model originating in statistical physics; combinatorially it can be viewed as a probability distribution over 2-vertex-colorings of a graph. We will discuss a fixed-magnetization version—akin to fixing the number of, say, blue vertices in every coloring—and a natural Markov chain sampling algorithm called the Kawasaki dynamics. We show some surprising results regarding the existence and location of a fast/slow mixing threshold for these dynamics. (joint work with Aiya Kuchukova, Marcus Pappik, and Will Perkins)
Changxin Ding: For trees on a fixed number of vertices, the path and the star are two extreme cases. Many graph parameters attain its maximum at the star and its minimum at the path among these trees. A trivial example is the number of leaves. I will introduce more interesting examples in the mini talk.
Jing Yu: We show that all simple outerplanar graphs G with minimum degree at least 2 and positive Lin-Lu-Yau Ricci curvature on every edge have maximum degree at most 9. Furthermore, if G is maximally outerplanar, then G has at most 10 vertices. Both upper bounds are sharp.
A theorem of Bertini says that an irreducible algebraic variety remains irreducible after intersecting with a generic hyperplane. We will discuss toric Bertini theorems for intersections with generic algebraic subtori (defined by generic binomial equations) instead of hyperplanes. As an application, we obtain a tropical Bertini theorem and a strengthening of the Structure Theorem of tropical algebraic geometry, by showing that irreducible tropical varieties remain connected through codimension one even after removing some facets. As part of the proof of the Toric Bertini over prime characteristics, we constructed a new algebraically closed field containing the multivariate rational functions, which is smaller than previously known constructions. This is based on joint works with Diane Maclagan, Francesca Gandini, Milena Hering, Fatemeh Mohammadi, Jenna Rajchgot, and Ashley Wheeler.
The aim of my talk is to discuss the following result, its variations and its connections with a no-dimensional Tverberg theorem. For any n red and n blue points in the Euclidean d-space, there exists a perfect red-blue matching M such that the balls whose diameters are edges of M share a common point.
(Joint works with O. Pirahmad, A. Vasilevskii, and P. Barabanshchikova.)
The oscillation of a Laplacian eigenfunction gives a great deal of information about the manifold on which it is defined. This oscillation can be encoded in the nodal deficiency, an important geometric quantity that is notoriously hard to compute, or even estimate. Here we compare two recently obtained formulas for the nodal deficiency, one in terms of an energy function on the space of equipartitions of the manifold, and the other in terms of a two-sided Dirichlet-to-Neumann map defined on the nodal set. We relate these two approaches by giving an explicit formula for the Hessian of the equipartition energy in terms of the Dirichlet-to-Neumann map. This allows us to compute Hessian eigenfunctions, and hence directions of steepest descent, for the equipartition energy in terms of the corresponding Dirichlet-to-Neumann eigenfunctions. Our results do not assume bipartiteness, and hence are relevant to the study of spectral minimal partitions. This is joint work with Greg Berkolaiko, Yaiza Canzani and Graham Cox.
Available online at: https://gatech.zoom.us/j/98258240051
Consider the following question of interest to cryptographers: A message is encoded in a binary string of length n. Consider a set of scrambling operations S (a proper subset of permutations on n bits). If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations look like a random permutation on all the bits? This question asks for the convergence time for a random walk on the permutation group. Replace the binary string with a quantum state and scrambling operations in S with Haar random quantum channels on two bits (qudits) at a time. Broadly speaking, we study the following question: If a scrambling operation is applied uniformly at random from S at each step, then after how many steps will the composition of scrambling operations (quantum channels) look like a Haar random channel on all qudits? This question asks about the convergence time for a random walk on the unitary group. Various protocols in quantum computing require Haar random channels, which motivates us to understand the number of operations one would require to approximately implement that channel.
Virtual knot theory is a variant of classical knot theory in which one allows a new type of crossing called a "virtual" crossing. It was originally developed by Louis Kauffman in order to study the Jones polynomial but has since developed into its own field and has genuine significance in low dimensional topology. One notable interpretation is that virtual knots are equivalent to knots in thickened surfaces. In this talk we'll introduce virtual knots and show why they are a natural extension of classical knots. We will then discuss what virtual knot theory can tell us about the both the classical Jones polynomial and its potential extensions to knots in arbitrary 3-manifolds. An important tool we will use throughout the talk is the knot quandle, a classical knot invariant which is complete up to taking mirror images.
We present a unified methodology for obtaining rates of estimation of optimal transport maps in general function spaces. Our assumptions are significantly weaker than those appearing in the literature: we require only that the source measure P satisfy a Poincare inequality and that the optimal map be the gradient of a smooth convex function that lies in a space whose metric entropy can be controlled. As a special case, we recover known estimation rates for Holder transport maps, but also obtain nearly sharp results in many settings not covered by prior work. For example, we provide the first statistical rates of estimation when P is the normal distribution, between log-smooth and strongly log-concave distributions, and when the transport map is given by an infinite-width shallow neural network. (joint with Vincent Divol and Aram-Alexandre Pooladian.)
This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series).
The classical Borsuk--Ulam theorem states that for any continuous map from the sphere to Euclidean space, $f\colon S^d\to R^d$, there is a pair of antipodal points that are identified, so $f(x)=f(-x)$. We prove a colorful generalization of the Borsuk–Ulam theorem. The classical result has many applications and consequences for combinatorics and discrete geometry and we in turn prove colorful generalizations of these consequences such as the colorful ham sandwich theorem, which allows us to prove a recent result of B\'{a}r\'{a}ny, Hubard, and Jer\'{o}nimo on well-separated measures as a special case, and Brouwer's fixed point theorem, which allows us to prove an alternative between KKM-covering results and Radon partition results. This is joint work with Florian Frick.
The dynamics of a passive scalar, such as temperature or concentration, transported by an incompressible flow can be modeled by the advection-diffusion equation. Advection often results in the formation of complicated, small-scale structures and can result in solutions relaxing to equilibrium at a rate much faster than the corresponding heat equation in regimes of weak diffusion. This phenomenon is typically referred to as enhanced diffusion. In this talk, I will discuss a joint work with Tarek Elgindi and Jonathan Mattingly in which we construct an example of a divergence-free velocity field on the two-dimensional torus that results in optimal enhanced diffusion. The flow consists of time-periodic, alternating piece-wise linear shear flows. The proof is based on the probabilistic representation formula for the advection-diffusion equation, a discrete time approximation, and ideas from hyperbolic dynamics.
There will be a pre-seminar (aimed toward grad students and postdocs) from 11:00 am to 11:30 am in Skiles 005.
In this talk, we give an overview of recent work in gradient elasticity. We first give a friendly introduction to gradient elasticity—a mathematical framework for understanding three-dimensional bodies that do not dissipate a form of energy during deformation. Compared to classical elasticity theory, gradient elasticity incorporates higher spatial derivatives that encode certain microstructural information and become significant at small spatial scales. We then discuss a recently introduced theory of three-dimensional Green-elastic bodies containing gradient elastic material boundary surfaces. We then indicate how the resulting model successfully eliminates pathological singularities inherent in classical linear elastic fracture mechanics, presenting a new and geometric alternative theory of fracture.
Many problems in rigidity theory and matrix completion boil down to finding a nice combinatorial description of some matroid supported on the edge set of a complete (bipartite) graph. In this talk, I will give many such examples. My goal is to convince you that a general theory of matroids supported on graphs is needed and to give you a sense of what that could look like.
This presentation is devoted to studying matrix solutions of the cubic Szegő equation, leading to the matrix Szegő equation on the 1-d torus and on the real line. The matrix Szegő equation enjoys a Lax pair structure, which is slightly different from the Lax pair structure of the cubic scalar Szegő equation introduced in Gérard-Grellier [arXiv:0906.4540]. We can establish an explicit formula for general solutions both on the torus and on the real line of the matrix Szegő equation. This presentation is based on the works Sun [arXiv:2309.12136, arXiv:2310.13693].
Clifford algebra was first developed to describe Maxwell's equations, but the subject has found applications in quantum mechanics, computer graphics, robotics, and even machine learning, way beyond its original purpose. In topology and geometry, Clifford algebra appears in the proofs of the celebrated Atiyah-Singer Index Theorem and Bott Periodicity; it is fundamental to the understanding of spin structures on Riemannian manifolds. Despite its algebraic nature, it somehow gives us the power to understand and manipulate geometry. What a marvelous machine offered by the devil! In this talk, we will investigate the unreasonable effectiveness of Clifford algebra by exploring its algebraic structure and constructing the Pin and Spin groups. If time permits, we will prove that Spin(p,q) is a double cover of SO(p,q), complementing the belt trick talk of Sean Eli.
I will describe some connections between arithmetic geometry of abelian varieties, non-archimedean/tropical geometry, and combinatorics. For example, we give formulas for (non-archimedean) canonical local heights in terms of tropical invariants. Our formula extends a classical computation of local height functions due to Tate (involving Bernoulli polynomials).
Based on ongoing work with Robin de Jong.
In 1870 Jordan explained how Galois theory can be applied to problems from enumerative geometry, with the group encoding intrinsic structure of the problem. Earlier Hermite showed the equivalence of Galois groups with geometric monodromy groups, and in 1979 Harris initiated the modern study of Galois groups of enumerative problems. He posited that a Galois group should be `as large as possible' in that it will be the largest group preserving internal symmetry in the geometric problem.
I will describe this background and discuss some work of many to compute, study, and use Galois groups of geometric problems, including those that arise in applications of algebraic geometry.
When performing regression analysis, researchers often face the challenge of selecting the best single model from a range of possibilities. Traditionally, this selection is based on criteria evaluating model goodness-of-fit and complexity, such as Akaike's AIC and Schwartz's BIC, or on the model's performance in predicting new data, assessed through cross-validation techniques. In this talk, I will show that a linear combination of a large number of these possible models can have better predictive accuracy than the best single model among them. Algorithms and theoretical guarantees will be discussed, which involve interesting connections to constrained optimization and shrinkage in statistics.
Recent advancements in operator-type neural networks, such as Fourier Neural Operator (FNO) and Deep Operator Network (DeepONet), have shown promising results in approximating the solutions of spatial-temporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new operator learning framework to address these issues. The proposed paradigm leverages the traditional wisdom from numerical PDE theory and techniques to refine the pipeline of existing operator neural networks. Specifically, the proposed architecture initiates the training for a single or a few epochs for the operator-type neural networks in consideration, concluding with the freezing of the model parameters. The latter are then fed into an error correction scheme: a single parametrized linear spectral layer trained with a convex loss function defined through a reliable functional-type a posteriori error estimator.This design allows the operator neural networks to effectively tackle low-frequency errors, while the added linear layer addresses high-frequency errors. Numerical experiments on a commonly used benchmark of 2D Navier-Stokes equations demonstrate improvements in both computational time and accuracy, compared to existing FNO variants and traditional numerical approaches.
A hereditary class $\mathcal C$ of graphs is said to have the Erdős–Hajnal property if every $n$-vertex graph in $\mathcal C$ has a clique or stable set of size at least $n^c$. We discuss a proof of a conjecture of Chernikov–Starchenko–Thomas and Fox–Pach–Suk that for every $d\ge1$, the class of graphs of VC-dimension at most $d$ has the Erdős–Hajnal property. Joint work with Alex Scott and Paul Seymour.
Can you hear the shape of a drum? A classical inverse problem in mathematical physics is to determine the shape of a membrane from the resonant frequencies at which it vibrates. This problem is very much still open for smooth, strictly convex planar domains and one tool in that is often used in this context is the wave trace, which contains information on the asymptotic distribution of eigenvalues of the Laplacian on a Riemannian manifold. It is well known that the singular support of the wave trace is contained in the length spectrum, which allows one to infer geometric information only under a length spectral simplicity or other nonresonance type condition. In a recent work together with Vadim Kaloshin and Illya Koval, we construct large families of domains for which there are multiple geodesics of a given length, having different Maslov indices, which interfere destructively and cancel arbitrarily many orders in the wave trace. This shows that there are potential obstacles in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different objects, at least insofar as the wave trace is concerned.
This defense will also be on zoom at: https://gatech.zoom.us/j/99428720697
In this defense we describe three topics in tropical and toric positivity and completion. In the first part, we describe the finite completability of a partial point to a log-linear statistical model: a toric variety restricted to the probability simplex. We show when a generic point in some projection of a log-linear model has finite preimage, and the exact number of preimages in such a case. In the second part, we describe the tropical variety of symmetric tropical rank 2 matrices. We give a description of the tropical variety as a coarsening of the simplicial complex of a type of bicolored trees, and show that the tropical variety is shellable. Finally, we discuss two tropical notions of positivity, and give results on the positive part of certain tropical determinantal varieties.
Committee:
Josephine Yu, Georgia Institute of Technology (Advisor)
Matt Baker, Georgia Institute of Technology
Greg Blekherman, Georgia Institute of Technology,
Kaie Kubjas, Aalto University
Anton Leykin, Georgia Institute of Technology
Thesis draft:
Link
Speaker will present in person.<br />
<br />
Bio: Yuanqi Du is a PhD student at the Department of Computer Science, Cornell University studying AI and its intersection with Scientific Discovery advised by Prof. Carla P. Gomes. His research interests include Geometric Deep Learning, Probabilistic Machine Learning, Sampling, Optimization, and AI for Science (with a focus on molecular discovery). Aside from his research, he is passionate about education and community building. He leads the organization of a series of events such as the Learning on Graphs conference and AI for Science, Probabilistic Machine Learning workshops at ML conferences and an educational initiative (AI for Science101) to bridge the AI and Science community.
Recent advancements in machine learning have paved the way for groundbreaking opportunities in the realm of molecular discovery. At the forefront of this evolution are improved computational tools with proper inductive biases and efficient optimization. In this talk, I will delve into our efforts around these themes from a geometry, sampling and optimization perspective. I will first introduce how to encode symmetries in the design of neural networks and the balance of expressiveness and computational efficiency. Next, I will discuss how generative models enable a wide range of design and optimization tasks in molecular discovery. In the third part, I will talk about how the advancements in stochastic optimal control, sampling and optimal transport can be applied to find transition states in chemical reactions.
Gromov's non-squeezing theorem established symplectic rigidity and is widely regarded as one of the most important theorems in symplectic geometry. In contrast, in the contact setting, a standard ball of any radius can be contact embedded into an arbitrarily small neighborhood of a point. Despite this flexibility, Eliashberg, Kim, and Polterovich discovered instances of contact rigidity for pre-quantized balls in $\mathbb R^{2n} \times S^1$ under a more restrictive notion of contact squeezing. In particular, in 2006 they applied holomorphic techniques to show that for any {\it integer} $R \geq 1$, there does not exist a contact squeezing of the pre-quantized ball of capacity $R$ into itself; this result was reproved by Sandon in 2011 as an application of the contact homology groups she defined using the generating family technique. Around 2016, Chiu applied the theory of microlocal sheaves to obtain the stronger result that squeezing is impossible for all $R \geq 1$. Very recently, Fraser, Sandon, and Zhang, gave an alternate proof of Chiu’s nonsqueezing result by developing an equivariant version of Sandon’s generating family contact homology groups. I will explain another proof of Chiu’s nonsqueezing, one that uses a persistence module viewpoint to extract new obstructions from the contact homology groups as defined by Sandon in 2011. This is joint work in progress with Maia Fraser.
A knot K in S^3 is (smoothly) slice if K is the boundary of a properly embedded disk D in B^4, and K is ribbon if this disk can be realized without any local maxima with respect to the radial Morse function on B^4. In dimension three, a knot K with nice topology – that is, a fibered knot – bounds a unique fiber surface up to isotopy. Thus, it is natural to wonder whether this sort of simplicity could extend to the set of ribbon disks for K, arguably the simplest class of surfaces bounded by a knot in B^4. Surprisingly, we demonstrate that the square knot, one of the two non-trivial ribbon knots with the lowest crossing number, bounds infinitely many distinct ribbon disks up to isotopy. This is joint work with Jeffrey Meier.
Available online at: https://gatech.zoom.us/j/98258240051
I read Benoist's paper Convexes Divisibles IV (2006, Invent. Math.), and will talk about it. The main result is a striking structural theorem for triangles in the boundaries of 3-dimensional properly convex divisible domains O that are not strictly convex (which exist). These bound "flats" in O. Benoist shows that every Z^2 subgroup of the group G preserving O preserves a unique such triangle. Conversely, all such triangles are disjoint and any such triangle descends to either a torus or Klein bottle in the quotient M = O/G (and so must have many symmetries!). Furthermore, this "geometrizes" the JSJ decomposition of M, in the sense that cutting along these tori and Klein bottles gives an atoroidal decomposition of M.
The goal of this talk is to discuss the Lp boundedness of the trilinear Hilbert transform along the moment curve. We show that it is bounded in the Banach range.
The main difficulty in approaching this problem(compared to the classical approach to the bilinear Hilbert transform) is the lack of absolute summability after we apply the time-frequency discretization(which is known as the LGC-methodology introduced by V. Lie in 2019). To overcome such a difficulty, we develop a new, versatile approch -- referred to as Rank II LGC (which is also motived by the study of the non-resonant bilinear Hilbert-Carleson operator by C. Benea, F. Bernicot, V. Lie, and V. Vitturi in 2022), whose control is achieved via the following interdependent elements:
1). a sparse-uniform deomposition of the input functions adapted to an appropriate time-frequency foliation of the phase-space;
2). a structural analysis of suitable maximal "joint Fourier coefficients";
3). a level set analysis with respect to the time-frequency correlation set.
This is a joint work with my postdoc advisor Victor Lie from Purdue.
Weighted Poincar\'e inequalities known for various laws such as the exponential or Cauchy ones are shown to follow from the "usual" Poincar\'e inequality involving the non-local gradient. A key ingredient in showing so is a covariance representation and Hardy's inequality.
The framework under study is quite general and comprises infinitely divisible laws as well as some log-concave ones. This same covariance representation is then used to obtain quantitative concentration inequalities of exponential type, recovering in particular the Gaussian results.
Joint Work with Benjamin Arras.
The goal of this talk is to explore curve graphs, which are combinatorial tools that encode topological information about surfaces. We focus on variants of the fine curve graph of a surface. The fine curve graph has its vertices essential simple closed curves on the surface and its edges connect pairs of curves that are disjoint. We will mention a sampling of related theorems which were proven in collaboration with various coauthors and then prove several results regarding the finitary curve graph, which has as its vertices essential simple closed curves while its edges connect pairs of curves that intersect at finitely many points.
In this talk, we will prove that the finitary curve graph has diameter 2 (geometry), that the flag complex induced by the finitary curve graph is contractible (topology), and that the automorphism group of the finitary curve graph is naturally isomorphic to the homeomorphism group of the surface (combinatorics).
Work mentioned in the talk will be a subset of independent work and of collaborations with Katherine Booth, Ryan Dickmann, Dan Minahan, and Alex Nolte. The talk will be aimed at a non-expert audience.
This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series).
Atlanta Combinatorics Colloquium Hosted by Georgia Tech
Abstract: Many tools have been developed in combinatorics to study global structure in finite graphs. One such tool is called Szemerédi's regularity lemma, which gives a structural decomposition for any large finite graph. Beginning with work of Alon–Fischer–Newman, Lovász–Szegedy, and Malliaris–Shelah, it has been shown over the last 15 years that regularity lemmas can be used to detect structural dichotomies in graphs, and that these dichotomies have deep connections to model theory. In this talk, I present extensions of this type of result to arithmetic regularity lemmas, which are analogues of graph regularity lemmas, tailored to the study of combinatorial problems in finite groups. This work uncovered tight connections between tools from additive combinatorics, and ideas from the model theoretic study of infinite groups.
There will be a pre-seminar in Skiles 005 at 11 am.
Independent component analysis (ICA) is a classical data analysis method to study mixtures of independent sources. An ICA model is said to be identifiable if the mixing can be recovered uniquely. Identifiability is known to hold if and only if at most one of the sources is Gaussian, provided the number of sources is at most the number of observations. In this talk, I will discuss our work to generalize the identifiability of ICA to the overcomplete setting, where the number of sources can exceed the number of observations.The underlying problem is algebraic and the proof studies linear spaces of rank one symmetric matrices. Based on joint work with Anna Seigal https://arxiv.org/abs/2401.14709
Diffusion models, particularly score-based generative models (SGMs), have emerged as powerful tools in diverse machine learning applications, spanning from computer vision to modern language processing. In the first part of this talk, we delve into the generalization theory of SGMs, exploring their capacity for learning high-dimensional distributions. Our analysis show that SGMs achieve a dimension-free generation error bound when applied to a class of sub-Gaussian distributions characterized by certain low-complexity structures. In the second part of the talk, we consider the application of diffusion models in solving partial differential equations (PDEs). Specifically, we present the development of a physics-guided diffusion model designed for reconstructing high-fidelity solutions from their low-fidelity counterparts. This application showcases the adaptability of diffusion models and their potential to scientific computation.
In this talk, we will discuss mathematical construction of self-similar solutions exhibiting implosion arising in gas dynamics and gaseous stars, with focus on self-similar converging-diverging shock wave solutions to the non-isentropic Euler equations and imploding solutions to the Euler-Poisson equations describing gravitational collapse. The talk is based on joint works with Guo, Hadzic, Liu and Schrecker.
For integers $k>\ell\ge0$, a graph $G$ is $(k,\ell)$-stable if $\alpha(G-S)\geq \alpha(G)-\ell$ for every
$S\subseteq V(G)$ with $|S|=k$. A recent result of Dong and Wu [SIAM J.
Discrete Math. 36 (2022) 229--240] shows that every $(k,\ell)$-stable
graph $G$ satisfies $\alpha(G) \le \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$. A $(k,\ell)$-stable graph $G$ is tight if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell$; and $q$-tight for some integer $q\ge0$ if $\alpha(G) = \lfloor ({|V(G)|-k+1})/{2}\rfloor+\ell-q$.
In this talk, we first prove that for all $k\geq 24$, the only tight $(k, 0)$-stable graphs are $K_{k+1}$ and $K_{k+2}$, answering a question of Dong and Luo [arXiv: 2401.16639]. We then prove that for all nonnegative integers $k, \ell, q$ with $k\geq 3\ell+3$, every $q$-tight $(k,\ell)$-stable graph has at most $k-3\ell-3+2^{3(\ell+2q+4)^2}$ vertices, answering a question of Dong and Luo in the negative. \\
This is joint work with Xiaonan Liu and Zhiyu Wang.
Available via zoom at: https://gatech.zoom.us/j/98258240051
This presentation is dedicated to extending both defocusing and focusing Calogero–Moser–Sutherland derivative nonlinear Schrödinger equations (CMSdNLS), which are introduced in Abanov–Bettelheim–Wiegmann [arXiv:0810.5327], Gérard-Lenzmann [arXiv:2208.04105] and R. Badreddine [arXiv:2303.01087, arXiv:2307.01592], to a system of two matrix-valued variables. This new system is an integrable extension and perturbation of the original CMSdNLS equations. Thanks to the conjugation acting method, I can establish the explicit expression for general solutions on the torus and on the real line in my work [hal-04227081].
We prove a wavelet T(1) theorem for compactness of multilinear Calderón -Zygmund (CZ) operators. Our approach characterizes compactness in terms of testing conditions and yields a representation theorem for compact CZ forms in terms of wavelet and paraproduct forms that reflect the compact nature of the operator. This talk is based on joint work with Walton Green and Brett Wick.
The asymptotic behavior of closed geodesic on negatively curved spaces occupies a central place in Riemannian geometry. Minimal surfaces are higher dimensional analogies of geodesics and I will talk about some recent developments regarding the growth rate of minimal surfaces in negatively curved manifolds.
We discuss a general scheme that allows to realize certain geometric functional inequalities as statements about convexity of some functionals, and, inspired by the work of Bobkov and Ledoux, we obtain various interesting inequalities as their realizations. For example, we draw a link between Ehrhard’s inequality and an interesting extension of Bobkov’s inequality, and several new and more general inequalities are discussed as well. In this talk we discuss a joint project with Barthe, Cordero-Erausquin and Ivanisvili, and also mention briefly some results from a joint project with Cordero-Erausquin and Rotem.
I will present numerical methods for solving the optimal transport (OT) problems in three settings. Firstly, I will discuss discrete OT problems from the perspective of linear programming and assignment problems. Additionally, I will provide a solution for a discrete problem with an obstacle in the domain.
Next, I will consider and compare several different numerical methods to solve the classic continuous OT problem with the squared Euclidean cost function. I will compare two numerical methods used for the fluid dynamics formulation with a direct discretization of the Monge-Ampère PDE. Furthermore, I will introduce a new class of problems called separable, for which very accurate methods can be devised.
Lastly, I propose a novel implementation of Newton's method for solving semi-discrete OT problems for cost functions that are a positive combination of $p$-norms, $1
This series of talks will discuss connections between Riemannian geometry and contact topology. Both structures have deep connections to the topology of 3-manifolds, but there has been little study of the interactions between them (at least the implications in contact topology). We will see that there are interesting connections between curvature and properties of contact structures. The talks will give a brief review of both Riemannian geometry and contact topology and then discuss various was one might try to connect them. There will be many open problems discussed (probably later in the series).
How many different ways can we arrange n convex sets in R^d? One answer is provided by counting the number of d-representable complexes on vertex set [n]. We show that there are exp(Theta(n^d log n))-many such complexes, and provide bounds on the constants involved. As a consequence, we show that d-representable complexes comprise a vanishingly small fraction of the class of d-collapsible complexes. In the case d = 1 our results are more precise, and improve the previous best estimate for the number of interval graphs.
In 1989, Eremenko investigated the set of points that escape to infinity under iteration of a transcendental entire function, the so-called escaping set. He proved that every component of the closure of the escaping set is unbounded and conjectured that all the components of the escaping set are unbounded. Much of the recent work on the iteration of entire functions is involved in investigating properties of the escaping set, motivated by Eremenko's conjecture. We will begin by introducing many of the basic dynamical properties of iterates of an analytic function, and finally discuss constructing a transcendental entire function with a point connected component of the escaping set, providing a counterexample to Eremenko's conjecture. This is joint work with David Martí-Pete and Lasse Rempe.
Generative Adversarial Networks (GANs) are powerful tools for creating new content, but they face challenges such as sensitivity to starting conditions and mode collapse. To address these issues, we propose a deep generative model that utilizes the Gromov-Monge embedding (GME). It helps identify the low-dimensional structure of the underlying measure of the data and then map it, while preserving its geometry, into a measure in a low-dimensional latent space, which is then optimally transported to the reference measure. We guarantee the preservation of the underlying geometry by the GME and c-cyclical monotonicity of the generative map, where c is an intrinsic embedding cost employed by the GME. The latter property is a first step in guaranteeing better robustness to initialization of parameters and mode collapse. Numerical experiments demonstrate the effectiveness of our approach in generating high-quality images, avoiding mode collapse, and exhibiting robustness to different starting conditions.
The seminar has been rescheduled from Monday to Tuesday.
Galois groups embody the structure of algebraic equations arising in both enumerative geometry and various scientific applications where such equations must be solved. I will describe a line of work that aims to elucidate the role of Galois groups in applications where data taken from multiple images are used to reconstruct a 3D scene. From this perspective, I will revisit two well-known solutions to camera pose estimation problems, which originate from classical photogrammetry and are still heavily used within modern 3D reconstruction systems. I will then discuss some less-classical problems, for which the insight we gleaned from computing Galois groups led to significant practical improvements over previous solutions. A key ingredient was the use of numerical homotopy continuation methods to (heuristically) compute monodromy permutations. Time-permitting, I will explain how such methods may also be used to automatically recover certain symmetries underlying enumerative problems.
Given graphs G and H and a positive integer q, an (H,q)-coloring of G is an edge-coloring in which each copy of H receives at least q colors. Erdős and Shelah raised the question of determining the minimum number of colors, f(G,H,q), which are required for an (H,q)-coloring of G. Determining f(K_n,K_p,2) for all n and p is equivalent to determining the classical multicolor Ramsey numbers. Recently, Mubayi and Joos introduced the use of a new method for proving upper bounds on these generalized Ramsey numbers; by finding a “conflict-free" matching in an appropriate auxiliary hypergraph, they determined the values of f(K_{n,n},C_4,3) and f(K_n,K_4,5). In this talk, we will show how to generalize their approach to give bounds on the generalized Ramsey numbers for several families of graphs. This is joint work with Deepak Bal, Patrick Bennett, and Shira Zerbib.
We present new results concerning characterizations of the spaces $C^{1,\alpha}$ and “$LI_{\alpha+1}$” for $0<\alpha<1$. The space $LI_{\alpha +1}$ is the space of Lipschitz functions with $\alpha$-order fractional derivative having bounded mean oscillation. These characterizations involve geometric square functions which measure how well the graph of a function is approximated by a hyperplane at every point and scale. We will also discuss applications of these results to higher-order rectifiability.
The Fisher-KPP equation was introduced in 1937 to model the spread of an advantageous gene through a spatially distributed population. Remarkably precise information on the traveling front has been obtained via a connection with branching Brownian motion, beginning with works of McKean and Bramson in the 70s. I will discuss an extension of this probabilistic approach to the Road-Field Model: a reaction-diffusion PDE system introduced by H. Berestycki et al. to describe enhancement of biological invasions by a line of fast diffusion, such as a river or a road. Based on joint work with Amir Dembo.
In this talk, we introduce and survey continuous-time deep learning approaches based on neural ordinary differential equations (neural ODEs) arising in supervised learning, generative modeling, and numerical solution of high-dimensional optimal control problems. We will highlight theoretical advantages and numerical benefits of neural ODEs in deep learning and their use to solve otherwise intractable PDE problems.
There will be a pre-seminar at 11am in Skiles 005.
Given a basis for a linear subspace U of nxn matrices, we study the problem of either producing a rank-one matrix in U, or certifying that none exist. While this problem is NP-Hard in the worst case, we present a polynomial time algorithm to solve this problem in the generic setting under mild conditions on the dimension of U. Our algorithm is based on Hilbert’s Nullstellensatz and a “lifted” adaptation of the simultaneous diagonalization algorithm for tensor decompositions. We extend our results to the more general setting in which the set of rank-one matrices is replaced by an algebraic set. Time permitting, we will discuss applications to quantum separability testing and tensor decompositions. This talk is based on joint work with Harm Derksen, Nathaniel Johnston, and Aravindan Vijayaraghavan.
There are lots of ways to measure the complexity of a knot. Some come from knot diagrams, and others come from topological or geometric quantities extracted from some auxiliary space. In this talk, I’ll describe a geometry property, which we call “twist positivity”, that often puts strong restrictions on how the braid and bridge index are related. I’ll describe some old and new results about twist positivity, as well as some new applications towards knot concordance. In particular, I’ll describe how using a suite of numerical knot invariants (including the braid index) in tandem allows one to prove that there are infinitely many positive braid knots which all represent distinct smooth concordance classes. This confirms a prediction of the slice-ribbon conjecture. Everything I’ll discuss is joint work with Hugh Morton. I will assume very little background about knot invariants for this talk – all are welcome!
The “hard direction” of the Giroux correspondence states that any two open books representing the same contact structure is related by a sequence of positive stabilisations and destabilisations. We give a proof of this statement using convex surface theory. This is a joint work with Joan Licata.
I will give essentially matching upper and lower bounds for the expected max-sliced 1-Wasserstein distance between a probability measure on a separable Hilbert space and its empirical distribution from n samples. A version of this result for Banach spaces will also be presented. From this, we will derive an upper bound for the expected max-sliced 2-Wasserstein distance between a symmetric probability measure on a Euclidean space and its symmetrized empirical distribution.
Flow-based models are widely used in generative tasks, including normalizing flow, where a neural network transports from a data distribution P to a normal distribution. This work develops a flow-based model that transports from P to an arbitrary Q (which can be pre-determined or induced as the solution to an optimization problem), where both distributions are only accessible via finite samples. We propose to learn the dynamic optimal transport between P and Q by training a flow neural network. The model is trained to optimally find an invertible transport map between P and Q by minimizing the transport cost. The trained optimal transport flow subsequently allows for performing many downstream tasks, including infinitesimal density ratio estimation (DRE) and distribution interpolation in the latent space for generative models. The effectiveness of the proposed model on high-dimensional data is demonstrated by strong empirical performance on high-dimensional DRE, OT baselines, and image-to-image translation.
Zoom link to attend remotely: https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT0... />
In this talk I will present a computer-assisted method to study solutions vanishing at infinity in differential equations on R^n. Such solutions arise naturally in various models, in the form of traveling waves or localized patterns for instance, and involve multiple challenges to address both on the numerical and on the analytical side. Using spectral techniques, I will explain how Fourier series can serve as an approximation of the solution as well as an efficient mean for the construction of a fixed-point operator for the proof. To illustrate the method, I will present applications to the constructive proof of localized patterns in the 2D Swift-Hohenberg equation and in the Gray-Scott model. The method extends to non-local equations and proofs of solitary travelling waves in the (capillary-gravity) Whitham equation will be exposed.
Speaker will present in person.
In this talk, we discuss generative modeling algorithms motivated by the time reversal and reflection properties of diffusion processes. Score-based diffusion models (SBDM) have recently emerged as state-of-the-art approaches for image generation. We develop SBDMs in the infinite-dimensional setting, that is, we model the training data as functions supported on a rectangular domain. Besides the quest for generating images at ever higher resolution, our primary motivation is to create a well-posed infinite-dimensional learning problem so that we can discretize it consistently at multiple resolution levels. We demonstrate how to overcome two shortcomings of current SBDM approaches in the infinite-dimensional setting by ensuring the well-posedness of forward and reverse processes, and derive the convergence of the approximation of multilevel training. We illustrate that approximating the score function with an operator network is beneficial for multilevel training.
In the second part of this talk, we propose the Reflected Schrodinger Bridge algorithm: an entropy-regularized optimal transport approach tailored for generating data within diverse bounded domains. We derive reflected forward-backward stochastic differential equations with Neumann and Robin boundary conditions, extend divergence-based likelihood training to bounded domains, and demonstrate its scalability in constrained generative modeling.
We discuss recent improved bounds for Szemerédi’s Theorem. The talk will seek to provide a gentle introduction to higher order Fourier analysis and recent quantitative developments. In particular, the talk will provide a high level sketch for how the inverse theorem for the Gowers norm enters the picture and the starting points for the proof of the inverse theorem. Additionally, the talk (time permitting) will discuss how recent work of Leng on equidistribution of nilsequences enters the picture and is used. No background regarding nilsequences will be assumed.
Based on joint work with James Leng and Ashwin Sah.
A fundamental conjecture in number theory is the Riemann hypothesis, which implies the prime number theorem with an optimally strong error term. While a proof remains elusive, many results in number theory can nonetheless be proved using weaker inputs. I will discuss how one such weaker input, subconvexity, can be used to prove strong results on the equidistribution of geometric objects such as lattice points on the sphere. If time permits, I will also discuss how various proofs of subconvexity reduce to understanding period integrals of automorphic forms.
Zoom link for streaming the talk: <br />
<br />
https://gatech.zoom.us/j/91390791493?pwd=QnpaWHNEOHZTVXlZSXFkYTJ0b0Q0UT0... />
A conjecture of Buchanan and Lillo states that all nontrivial oscillatory solutions of
\begin{equation*}
x'(t)=p(t)x(t-\tau(t)),
\end{equation*}
with $0\leq p(t)\leq 1,0\leq \tau(t)\leq 2.75+\ln2 \approx 3.44$ tend to a known function $\varpi$, which is antiperiodic:
\begin{equation*}
\varpi(t+T/2)\equiv - \varpi(t)
\end{equation*}
where $T$ is its minimal period. We discuss recent developments on this question, focusing on the periodic solutions characterizing the threshold case. We consider the case of positive feedback ($0\leq p(t)\leq 1$) with $\sup\tau(t)= 2.75+\ln2$, the well-known $3/2$-criterion corresponding to negative feedback ($0\leq -p(t)\leq 1$) with $\sup\tau(t)=1.5$, as well as higher order equations.
We investigate the behavior of the threshold periodic solutions under perturbation together with the symmetry (antiperiodicity) which characterizes them. This problem is set within the broader background of delay effects on stability for autonomous and nonautonomous equations, taking into account the fundamental relation between oscillation speed and dynamics of delay equations. We highlight the crucial role of symmetry in both the intuitions behind this vein of research, as well as the relevant combinatorial-variational problems.
