## Seminars and Colloquia by Series

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).
Thursday, September 15, 2016 - 11:05 , Location: Skiles 006 , Saugata Basu , Perdue University , Organizer: Ernie Croot
Effective bounds play a very important role in algebraic geometry with many applications.  In this talk I will survey recent progress and open questions in the quantitative study ofreal varieties and semi-algebraic sets and their connections with other areas of mathematics -- in particular,connections to incidence geometry via the polynomial partitioning method. I will also discuss some results on the topological complexity of symmetric varieties which have a representation-theoretic flavor. Finally, if time permits I will sketch how some of these results extend to the category of constructible sheaves.
Thursday, September 1, 2016 - 11:05 , Location: Skiles 006 , Gérard Ben Arous , Courant Institute, NYU , Organizer: Christian Houdre
This Colloquium will be Part II of the Stelson Lecture.     A function of many variables, when chosen at random, is typically very complex. It has an exponentially large number of local minima or maxima, or critical points. It defines a very complex landscape, the topology of its level lines (for instance their Euler characteristic) is surprisingly complex. This complex picture is valid even in very simple cases, for random homogeneous polynomials of degree p larger than 2. This has important consequences. For instance trying to find the minimum value of such a function may thus be very difficult. The mathematical tool suited to understand this complexity is the spectral theory of large random matrices. The classification of the different types of complexity has been understood for a few decades in the statistical physics of disordered media, and in particular spin-glasses, where the random functions may define the energy landscapes. It is also relevant in many other fields, including computer science and Machine learning. I will review recent work with collaborators in mathematics (A. Auffinger, J. Cerny) , statistical physics (C. Cammarota, G. Biroli, Y. Fyodorov, B. Khoruzenko), and computer science (Y. LeCun and his team at Facebook, A. Choromanska, L. Sagun among others), as well as recent work of E. Subag and E.Subag and O.Zeitouni.
Wednesday, June 8, 2016 - 15:30 , Location: Clary theater , , Brandeis University , , Organizer: Michael Damron
In the early '90s, Gromov introduced a notion of hyperbolicity for geodesic metric spaces.  The study of groups of isometries of such spaces has been an underlying theme in much of the work in geometric group theory since that time.  Many geodesic metric spaces, while not hyperbolic in the sense of Gromov, nonetheless display some hyperbolic-like behavior.  I will discuss a new invariant, the Morse boundary of a space, which captures this behavior.  (Joint work with Harold Sultan and Matt Cordes.)
Thursday, April 14, 2016 - 11:05 , Location: Skiles 006 , , University of Missouri, Columbia , , Organizer: Michael Damron
We consider the following problem. Does there exist an absolute constant C such that for every natural number n, every integer 1 \leq k \leq n, every origin-symmetric convex body L in R^n, and every measure \mu with non-negative even continuous density in R^n, \mu(L) \leq C^k \max_{H \in Gr_{n-k}} \mu(L \cap H}/|L|^{k/n}, where Gr_{n-k} is the Grassmannian of (n-k)-dimensional subspaces of R^n, and |L| stands for volume? This question is an extension to arbitrary measures (in place of volume) and to sections of arbitrary codimension k of the hyperplace conjecture of Bourgain, a major open problem in convex geometry. We show that the above inequality holds for arbitrary origin-symmetric convex bodies, all k and all \mu with C \sim \sqrt{n}, and with an absolute constant C for some special class of bodies, including unconditional bodies, unit balls of subspaces of L_p, and others. We also prove that for every \lambda \in (0,1) there exists a constant C = C(\lambda) so that the above inequality holds for every natural number, every origin-symmetric convex body L in R^n, every measure \mu with continuous density and the codimension of sections k \geq \lambda n. The latter result is new even in the case of volume. The proofs are based on a stability result for generalized intersections bodies and on estimates of the outer volume ratio distance from an arbitrary convex body to the classes of generalized intersection bodies.