Seminars and Colloquia by Series

Persistence of Invariant Objects under Delay Perturbations

Series
Dissertation Defense
Time
Thursday, May 6, 2021 - 16:00 for 1 hour (actually 50 minutes)
Location
ONLINE at https://bluejeans.com/137621769
Speaker
Jiaqi YangGeorgia Tech

 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 

A proof of the Erdős–Faber–Lovász conjecture

Series
School of Mathematics Colloquium
Time
Thursday, May 6, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Tom KellyUniversity of Birmingham

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.

Constructing non-bipartite $k$-common graphs

Series
Graph Theory Seminar
Time
Tuesday, May 4, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Fan WeiPrinceton University

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.

Normal surface theory and colored Khovanov homology

Series
Geometry Topology Seminar
Time
Monday, May 3, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Christine Ruey Shan LeeUniversity of South Alabama

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.

Global Constraints within the Developmental Program of the Drosophila Wing

Series
Mathematical Biology Seminar
Time
Friday, April 30, 2021 - 15:00 for 1 hour (actually 50 minutes)
Location
ONLINE
Speaker
Madhav ManiNorthwestern University

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

Steady waves in flows over periodic bottoms

Series
CDSNS Colloquium
Time
Friday, April 30, 2021 - 13:00 for 1 hour (actually 50 minutes)
Location
Zoom (see additional notes for link)
Speaker
Carlos Garcia AzpeitiaUNAM

Please Note: 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.

Macdonald and Schubert polynomials from Markov chains

Series
School of Mathematics Colloquium
Time
Thursday, April 29, 2021 - 11:00 for 1 hour (actually 50 minutes)
Location
https://us02web.zoom.us/j/87011170680?pwd=ektPOWtkN1U0TW5ETFcrVDNTL1V1QT09
Speaker
Lauren K. WilliamsHarvard University

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.

Maximum number of almost similar triangles in the plane

Series
Graph Theory Seminar
Time
Tuesday, April 27, 2021 - 15:45 for 1 hour (actually 50 minutes)
Location
https://us04web.zoom.us/j/77238664391. For password, please email Anton Bernshteyn (bahtoh ~at~ gatech.edu)
Speaker
Bernard LidickýIowa State University

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.

On the length of the shortest closed geodesic on positively curved 2-spheres.

Series
Geometry Topology Seminar
Time
Monday, April 26, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/579155918
Speaker
Franco Vargas PalleteYale University

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.

Some problems in point-set registration

Series
Applied and Computational Mathematics Seminar
Time
Monday, April 26, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
https://bluejeans.com/884917410
Speaker
Yuehaw KhooUniversity of Chicago

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.
 

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