Monday, March 13, 2017 - 14:00 , Location: Skiles 005 , Prof. Yao Li , University of Massachusetts Amherst , email@example.com , Organizer: Molei Tao
In this talk I will present my recent result about the ergodic properties of nonequilibrium steady-state (NESS) for a stochastic energy exchange model. The energy exchange model is numerically reduced from a billiards-like deterministic particle system that models the microscopic heat conduction in a 1D chain. By using a technique called the induced chain method, I proved the existence, uniqueness, polynomial speed of convergence to the NESS, and polynomial speed of mixing for the stochastic energy exchange model. All of these are consistent with the numerical simulation results of the original deterministic billiards-like system.
Monday, March 13, 2017 - 14:05 , Location: Skiles 006 , Kristen Hendricks , Michigan State University , Organizer: Jennifer Hom
We use the conjugation symmetry on the Heegaard Floer complexes to define a three-manifold invariant called involutive Heegaard Floer homology, which is meant to correspond to Z_4-equivariant Seiberg-Witten Floer homology. From this we obtain two new invariants of homology cobordism, explicitly computable for surgeries on L-space knots and quasi-alternating knots, and two new concordance invariants of knots, one of which (unlike other invariants arising from Heegaard Floer homology) detects non-sliceness of the figure-eight knot. We also give a formula for how this theory behaves under connected sum, and use it to give examples not homology cobordant to anything computable via our surgery formula. This is joint work with C. Manolescu; the last part of is also joint with I. Zemke.
Monday, March 13, 2017 - 15:00 , Location: Skiles 006 , Hwangrae Lee , Auburn University , Organizer: Greg Blekherman
For a given generic form, the problem of finding the nearest rank-one form with respect to the Bombieri norm is well-studied and completely solved for binary forms. Nonetheless, higher-rank approximation is quite mysterious except in the quadratic case. In this talk we will discuss such problems in the binary case.
Monday, March 13, 2017 - 15:30 , Location: Skiles 006 , David Gay , UGA , Organizer: Caitlin Leverson
This is joint work with Jeff Meier. The Gluck twist operation removes an S^2XB^2 neighborhood of a knotted S^2 in S^4 and glues it back with a twist, producing a homotopy S^4 (i.e. potential counterexamples to the smooth Poincare conjecture, although for many classes of 2-knots theresults are in fact known to be smooth S^4's). By representing knotted S^2's in S^4 as doubly pointed Heegaard triples and understanding relative trisection diagrams of S^2XB^2 carefully, I'll show how to produce trisection diagrams (a.k.a. Heegaard triples) for these homotopy S^4's.(And for those not up on trisections I'll review the foundations.) The resulting recipe is surprisingly simple, but the fun, as always, is in the process.
Series: Math Physics Seminar
We study regularity properties of weak solutions of the homogeneous Boltzmann equation. While under the so called Grad cutoff assumption the homogeneous Boltzmann equation is known to propagate smoothness and singularities, it has long been suspected that the non-cutoff Boltzmann operator has similar coercivity properties as a fractional Laplace operator. This has led to the hope that the homogenous Boltzmann equation enjoys similar smoothing properties as the heat equation with a fractional Laplacian. We prove that any weak solution of the fully nonlinear non-cutoff homogenous Boltzmann equation (for Maxwellian molecules) with initial datum $f_0$ with ﬁnite mass, energy and entropy, that is, $f_0 \in L^1_2(\R^d) \cap L \log L(\R^d)$, immediately becomes Gevrey regular for strictly positive times, i.e. it gains infinitely many derivatives and even (partial) analyticity.This is achieved by an inductive procedure based on very precise estimates of nonlinear, nonlocal commutators of the Boltzmann operator with suitable test functions involving exponentially growing Fourier multipliers.(Joint work with Jean-Marie Barbaroux, Dirk Hundertmark, and Semjon Vugalter)
Wednesday, March 15, 2017 - 11:05 , Location: Skiles 006 , Max Alekseyev , George Washington University , firstname.lastname@example.org , Organizer: Torin Greenwood
Genome median and genome halving are combinatorial optimization problems that aim at reconstruction of ancestral genomes by minimizing the number of possible evolutionary events between the reconstructed genomes and the genomes of extant species. While these problems have been widely studied in past decades, their known algorithmic solutions are either not efficient or produce biologically inadequate results. These shortcomings have been recently addressed by restricting the problems solution space. We show that the restricted variants of genome median and halving problems are, in fact, closely related and have a neat topological interpretation in terms of embedded graphs and polygon gluings. Hence we establish a somewhat unexpected link between comparative genomics and topology, and further demonstrate its advantages for solving genome median and halving problems in some particular cases. As a by-product, we also determine the cardinality of the genome halving solution space.
Series: Research Horizons Seminar
Every graph G has canonically associated to a finite abelian group called the Jacobian group. The cardinality of this group is the number of spanning trees in G. If G is planar, the Jacobian group admits a natural simply transitive action on the set of spanning trees. More generally, for any graph G one can define a whole family of (non-canonical) simply transitive group actions. The analysis of such group actions involves ideas from tropical geometry. Part of this talk is based on joint work with Yao Wang, and part is based on joint work with Spencer Backman and Chi Ho Yuen.
Wednesday, March 15, 2017 - 14:05 , Location: Skiles 006 , Shane Scott , Georgia Tech , Organizer: Justin Lanier
Much of what is known about automorphisms of free groups is given by analogy to results on mapping class groups. One desirable result is the celebrated Nielson-Thurston classification of the mapping class group into reducible, periodic, or pseudo Anosov homeomorphisms. We will discuss attempts at analogous results for free group automorphisms.
Series: Analysis Seminar
In this talk we will discuss several ways to construct new convex bodies out of old ones. We will start by defining various methods of "averaging" convex bodies, both old and new. We will explain the relationships between the various definitions and their connections to basic conjectures in convex geometry. We will then discuss the power operation, and explain for example why every convex body has a square root, but not every convex body has a square. If time permits, we will briefly discuss more complicated constructions such as logarithms. The talk is based on joint work with Vitali Milman.
Thursday, March 16, 2017 - 11:00 , Location: Skiles 005 , Rick Durrett , Duke University , email@example.com , Organizer: Megan Bernstein
The use of evolutionary game theory biology dates to work of Maynard-Smith who used it to explain why most fights between animals were of the limited war type. Nowak and collaborators have shown that a spatial distribution of players can explain the existence of altruism, which would die out in a homogeneously mixing population. For the last twenty years, evolutionary games have been used to model cancer. In applications to ecology and cancer, the system is not homogeneously mixing so it is important to understand how space changes the outcome of these games. Over the last several years we have developed a theory for understanding the behavior of evolutionary games in the weak selection limit. We will illustrate this theory by discussing a number of examples. The most recent work was done in collaboration with a high school student so the talk should be accessible to a broad audience.
Series: Graph Theory Seminar
For a graph G, the Colin de Verdière graph parameter mu(G) is the maximum corank of any matrix in a certain family of generalized adjacency matrices of G. Given a non-negative integer t, the family of graphs with mu(G) <= t is minor-closed and therefore has some nice properties. For example, a graph G is planar if and only if mu(G) <= 3. Colin de Verdière conjectured that the chromatic number chi(G) of a graph satisfies chi(G) <= mu(G)+1. For graphs with mu(G) <= 3 this is the Four Color Theorem. We conjecture that if G has at least t vertices and mu(G) <= t, then |E(G)| <= t|V(G)| - (t+1 choose 2). For planar graphs this says |E(G)| <= 3|V(G)|-6. If this conjecture is true, then chi(G) <= 2mu(G). We prove the conjectured edge upper bound for certain classes of graphs: graphs with mu(G) small, graphs with mu(G) close to |V(G)|, chordal graphs, and the complements of chordal graphs.
Series: Stochastics Seminar
Estimation of the covariance matrix has attracted significant attention of the statistical research community over the years, partially due to important applications such as Principal Component Analysis. However, frequently used empirical covariance estimator (and its modifications) is very sensitive to outliers, or ``atypical’’ points in the sample. As P. Huber wrote in 1964, “...This raises a question which could have been asked already by Gauss, but which was, as far as I know, only raised a few years ago (notably by Tukey): what happens if the true distribution deviates slightly from the assumed normal one? As is now well known, the sample mean then may have a catastrophically bad performance…” Motivated by Tukey's question, we develop a new estimator of the (element-wise) mean of a random matrix, which includes covariance estimation problem as a special case. Assuming that the entries of a matrix possess only finite second moment, this new estimator admits sub-Gaussian or sub-exponential concentration around the unknown mean in the operator norm. We will present extensions of our approach to matrix-valued U-statistics, as well as applications such as the matrix completion problem. Part of the talk will be based on a joint work with Xiaohan Wei.
Series: School of Mathematics Colloquium
We present algorithms for performing sparse univariate polynomial interpolation with errors in the evaluations of the polynomial. Our interpolation algorithms use as a substep an algorithm that originally is by R. Prony from the French Revolution (Year III, 1795) for interpolating exponential sums and which is rediscovered to decode digital error correcting BCH codes over finite fields (1960). Since Prony's algorithm is quite simple, we will give a complete description, as an alternative for Lagrange/Newton interpolation for sparse polynomials. When very few errors in the evaluations are permitted, multiple sparse interpolants are possible over finite fields or the complex numbers, but not over the real numbers. The problem is then a simple example of list-decoding in the sense of Guruswami-Sudan. Finally, we present a connection to the Erdoes-Turan Conjecture (Szemeredi's Theorem). This is joint work with Clement Pernet, Univ. Grenoble.
Friday, March 17, 2017 - 11:05 , Location: Skiles 006 , Erich Kaltofen , North Carolina State University , Organizer: Anton Leykin
Error-correcting decoding is generalized to multivariate sparse polynomial and rational function interpolation from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (``outliers''). Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers (``cleans up data'') by techniques from the Welch/Berlekamp decoder for Reed-Solomon codes. Our algorithms can build a sparse function model from a number of evaluations that is linear in the sparsity of the model, assuming that there are a constant number of ouliers and that the function probes can be randomly chosen.
Series: ACO Student Seminar
Optimization problems arising in decentralized multi-agent systems have gained significant attention in the context of cyber-physical, communication, power, and robotic networks combined with privacy preservation, distributed data mining and processing issues. The distributed nature of the problems is inherent due to partial knowledge of the problem data (i.e., a portion of the cost function or a subset of the constraints is known to different entities in the system), which necessitates costly communications among neighboring agents. In this talk, we present a new class of decentralized first-order methods for nonsmooth and stochastic optimization problems which can significantly reduce the number of inter-node communications. Our major contribution is the development of decentralized communication sliding methods, which can skip inter-node communications while agents solve the primal subproblems iteratively through linearizations of their local objective functions.This is a joint work with Guanghui (George) Lan and Yi Zhou.
Friday, March 17, 2017 - 14:00 , Location: Skiles 006 , John Etnyre , Georgia Tech , Organizer: John Etnyre
This will be a 1.5 hour (maybe slightly longer) seminar.
Following up on the previous series of talks we will show how to construct Lagrangian Floer homology and discuss it properties.
Friday, March 17, 2017 - 15:00 , Location: Skiles 006 , Rick Durett , Duke University , firstname.lastname@example.org , Organizer: Megan Bernstein
In the latent voter model, which models the spread of a technology through a social network, individuals who have just changed their choice have a latent period, which is exponential with rate λ during which they will not buy a new device. We study site and edge versions of this model on random graphs generated by a configuration model in which the degrees d(x) have 3 ≤ d(x) ≤ M. We show that if the number of vertices n → ∞ and log n << λn << n then the latent voter model has a quasi-stationary state in which each opinion has probability ≈ 1/2 and persists in this state for a time that is ≥ nm for any m <∞. Thus, even a very small latent period drastically changes the behavior of the voter model.