## Seminars and Colloquia by Series

Thursday, January 26, 2017 - 11:05 , Location: Skiles 006 , Bill Massey , Princeton , Organizer: Shahaf Nitzan
The multi-server queue with non-homogeneous Poisson arrivals and customer abandonment is a fundamental dynamic rate queueing model for large scale service systems such as call centers and hospitals. Scaling the arrival rates and number of servers gives us fluid and diffusion limits. The diffusion limit suggests a Gaussian approximation to the stochastic behavior. The fluid mean and diffusion variance can form a two-dimensional dynamical system that approximates the actual transient mean and variance for the queueing process. Recent work showed that a better approximation for mean and variance can be computed from a related two-dimensional dynamical system. In this spirit, we introduce a new three-dimensional dynamical system that estimates the mean, variance, and third cumulant moment. This surpasses the previous two approaches by fitting the number in the queue to a quadratic function of a Gaussian random variable. This is based on a paper published in QUESTA and is joint work with Jamol Pender of Cornell University.
Thursday, January 19, 2017 - 11:05 , Location: Skiles 006 , , Univ. of Barcelona , , Organizer: Yao Yao
The long term behavior of dynamical systems can be understood by studying invariant manifolds that act as  landmarks that anchor the orbits. It is important to understand which invariant manifolds persist under modifications of the system. A deep mathematical theory, developed in the 70's shows that invariant manifolds which persist under changes are those that have sharp expansion (in the future or in the past) in the the normal directions. A deep question is what happens in the boundary of these theorems of persistence. This question requires to understand the interplay between the geometric properties and the functional analysis of the functional equations involved.In this talk we present several mechanisms in which properties of normal hyperbolicity degenerate, so leading to the breakdown of the invariant manifold. Numerical studies lead to surprising conjectures relating the breakdown to phenomena in phase transitions. The results have been obtained combining numerical exploration and rigorous reasoning.
Tuesday, January 3, 2017 - 11:05 , Location: Skiles 006 , Barry Simon , California Institute of Technology , Organizer: Shahaf Nitzan
This is not a mathematics talk but it is a talk for mathematicians. Too often, we think of historical mathematicians as only names assigned to theorems. With vignettes and anecdotes, I'll convince you they were also human beings and that, as the Chinese say, "May you live in interesting times" really is a curse.  More tales following up on the talk I gave at GaTech in Nov., 2013.  It is not assumed listeners heard that earlier talk.
Friday, December 9, 2016 - 16:00 , Location: Skiles 006 , Daniel Wise , McGill University , Organizer: John Etnyre
Cube complexes have come to play an increasingly central role within geometric group theory, as their connection to right-angled Artin groups provides a powerful combinatorial bridge between geometry and algebra. This talk will introduce nonpositively curved cube complexes, and then describe the developments that have recently culminated in the resolution of the virtual Haken conjecture for 3-manifolds, and simultaneously dramatically extended our understanding of many infinite groups.
Thursday, December 1, 2016 - 11:05 , Location: Skiles 006 , , Weizmann Institute of Science , , Organizer: Galyna Livshyts
The probabilistic method, pioneered by P. Erdös, has been key in many proofs from asymptotic geometric analysis. This method allows one to take advantage of numerous tools and concepts from probability theory to prove theorems which are not necessarily a-priori related to probability. The objective of this talk is to demonstrate several recent results which take advantage of stochastic calculus to prove results of a geometric nature. We will mainly focus on a specific construction of a moment-generating process, which can be thought of as a stochastic version of the logarithmic Laplace transform. The method we introduce allows us to attain a different viewpoint on the method of semigroup proofs, namely a path-wise point of view. We will first discuss an application of this method to concentration inequalities on high dimensional convex sets. Then, we will briefly discuss an application to two new functional inequalities on Gaussian space; an L1 version of hypercontractivity of the convolution operator related to a conjecture of Talagrand (joint with J. Lee) and a robustness estimate for the Gaussian noise-stability inequality of C.Borell (improving a result of Mossel and Neeman).
Thursday, November 17, 2016 - 11:05 , Location: Skiles 006 , Andrew Suk , University of Illinois at Chicago , Organizer: Shahaf Nitzan
The classic 1935 paper of Erdos and Szekeres entitled A combinatorial problem in geometry" was a starting point of a very rich discipline within combinatorics: Ramsey theory.  In that paper, Erdos and Szekeres studied the following geometric problem.  For every integer n \geq 3, determine the smallest integer ES(n) such that any set of ES(n) points in the plane in general position contains n members in convex position, that is, n points that form the vertex set of a convex polygon.  Their main result showed that ES(n) \leq {2n  - 4\choose n-2} + 1 = 4^{n -o(n)}.  In 1960, they showed that ES(n) \geq 2^{n-2} + 1 and conjectured this to be optimal.  Despite the efforts of many researchers, no improvement in the order of magnitude has been made on the upper bound over the last 81 years. In this talk, we will sketch a proof showing that ES(n) =2^{n +o(n)}.
Thursday, November 10, 2016 - 11:05 , Location: Skiles 006 , , University of Minnesota , Organizer: Anton Leykin
The evolution, through spatially periodic linear dispersion, of rough initial data leads to surprising quantized structures at rational times, and fractal, non-differentiable profiles at irrational times.  The Talbot effect, named after an optical experiment by one of the founders of photography, was first observed in optics and quantum mechanics, and leads to intriguing connections with exponential sums arising in number theory. Ramifications of these phenomena and recent progress on the analysis, numerics, and extensions to nonlinear wave models will be discussed.
Thursday, October 20, 2016 - 11:05 , Location: Skiles 006 , Xingxing Yu , Georgia Tech , Organizer: Shahaf Nitzan
A well-known theorem of Kuratowski  (1930) in graph theory states that a graph is planar if, and only if, it does not contain a subdivision  of $K_5$ or $K_{3,3}$. Wagner (1937) gave a structural characterization of graphs containing no subdivision of $K_{3,3}$.  Seymour in 1977 and, independently, Kelmans in 1979 conjectured that if a graph does not contain a subdivision of $K_5$ then it must be planar or contain a set of at most 4 vertices whose removal results in a disconnected  graph.  In this talk, I will discuss additional background  on this conjecture (including connection to the Four Color Theorem), and outline our recent proof of this conjecture (joint work with Dawei He and Yan Wang).  I will also mention several problems that are related to this conjecture or related to our approach.
Thursday, October 13, 2016 - 11:05 , Location: Skiles 006 , Ernie Croot , Georgia Tech , Organizer: Anton Leykin
In this talk I will discuss some new applications of the polynomial method to some classical problems in combinatorics, in particular the Cap-Set Problem. The Cap-Set Problem is to determine the size of the largest subset A of F_p^n having no three-term arithmetic progressions, which are triples of vectors x,y,z satisfying x+y=2z. I will discuss an analogue of this problem for Z_4^n and the recent progress on it due to myself, Seva Lev and Peter Pach; and will discuss the work of Ellenberg and Gijswijt, and of Tao, on the F_p^n version (the original context of the problem).
Thursday, September 29, 2016 - 11:05 , Location: Skiles 006 , , Technical University of Berlin , , Organizer: Christopher Heil
Modern imaging data are often composed of several geometrically distinct constituents. For instance, neurobiological images could consist of a superposition of spines (pointlike objects) and dendrites (curvelike objects) of a neuron. A neurobiologist might then seek to extract both components to analyze their structure separately for the study of Alzheimer specific characteristics. However, this task seems impossible, since there are two unknowns for every datum. Compressed sensing is a novel research area, which was introduced in 2006, and since then has already become a key concept in various areas of applied mathematics, computer science, and electrical engineering. It surprisingly predicts that high-dimensional signals, which allow a sparse representation by a suitable basis or, more generally, a frame, can be recovered from what was previously considered highly incomplete linear measurements, by using efficient algorithms. Utilizing the methodology of Compressed Sensing, the geometric separation problem can indeed be solved both numerically and theoretically. For the separation of point- and curvelike objects, we choose a deliberately overcomplete representation system made of wavelets (suited to pointlike structures) and shearlets (suited to curvelike structures). The decomposition principle is to minimize the $\ell_1$ norm of the representation coefficients. Our theoretical results, which are based on microlocal analysis considerations, show that at all sufficiently fine scales, nearly-perfect separation is indeed achieved. This project was done in collaboration with David Donoho (Stanford University) and Wang-Q Lim (TU Berlin).