Nov 13, 2017
Monday

09:15 AM  09:30 AM


Welcome

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09:30 AM  10:30 AM


On Gaussianwidth gradient complexity and meanfield behavior of interacting particle systems and random graphs.
Ronen Eldan (Weizmann Institute of Science)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The motivating question for this talk is: What does a sparse Erd\"osR\'enyi random graph, conditioned to have twice the number of triangles than the expected number, typically look like? Motivated by this question, In 2014, Chatterjee and Dembo introduced a framework for obtaining Large Deviation Principles (LDP) for nonlinear functions of Bernoulli random variables (this followed an earlier work of ChatterjeeVaradhan which used limit graph theory to answer this question in the dense regime). The aforementioned framework relies on a notion of "low complexity" functions on the discrete cube, defined in terms of the covering numbers of their gradient. The central lemma used in their proof provides a method of estimating the lognormalizing constant $\log \sum_{x \in \{1,1\}^n} e^{f(x)}$ by a corresponding meanfield functional. In this talk, we will introduce a new notion of complexity for measures on the discrete cube, namely the meanwidth of the gradient of the logdensity. We prove a general structure theorem for such measures which goes beyond the discrete cube. In particular, we show that a measure $\nu$ attaining low complexity (with no extra smoothness assumptions needed) are close to a product measure in the following sense: there exists a measure $\tilde \nu$ a small "tilt" of $\nu$ in the sense that their logdensities differ by a linear function with small slope, such that $\tilde \nu$ is close to a product measure in transportation distance. An easy corollary of our result is a strengthening of the framework of ChatterjeeDembo, which in particular simplifies the derivation of LDPs for subgraph counts, and improves the attained bounds. We will demonstrate how our framework can be used to study the behavior of lowcomplexity measures beyond the approximation of the partition function. As an example application, we prove that exponential random graphs behave roughly like mixtures of stochastic block models.
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10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
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11:00 AM  12:00 PM


On optimal matching of Gaussian samples
Michel Ledoux (Institut de Mathématiques de Toulouse)

 Location
 MSRI: Simons Auditorium
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Optimal matching problems are random variational problems widely investigated in the mathematics and physics literature. We discuss here the optimal matching problem of an empirical measure on a sample of iid random variables to the common law in KantorovichWasserstein distances, which is a classical topic in probability and statistics. Twodimensional matching of uniform samples gave rise to deep results investigated from various view points (combinatorial, generic chaining). We study here the case of Gaussian samples, first in dimension one on the basis of explicit representations of Kantorovich metrics and a sharp analysis of more general logconcave distributions in terms of their isoperimetric profile (joint work with S. Bobkov), and then in dimension two (and higher) following the PDE and transportation approach recently put forward by L. Ambrosia, F. Stra and D. Trevisan.
 Supplements



12:00 PM  01:30 PM


Lunch

 Location
 MSRI: Atrium
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01:30 PM  02:30 PM


Renyi divergence and the central limit theorem
Sergey Bobkov (University of Minnesota, Twin Cities)

 Location
 MSRI: Simons Auditorium
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We will be discussing strong distances between probability distributions given by the Renyi divergences.
The main focus is put on the study of the asymptotic behavior along normalized convolutions and the associated central limit theorem.
 Supplements



02:45 PM  03:45 PM


Asymptotics in discrete Betaensembles
Alice Guionnet (Massachusetts Institute of Technology)

 Location
 MSRI: Simons Auditorium
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We will discuss the fluctuations in discrete Betaensembles. Such distributions describe for instance the positions of lozenge tilings and ressemble the continuous Betaensembles which appear in random matrix ensembles. We will highlight the role of equations introduced by Nekrasov in this context, which play the same crucial role than DysonSchwinger equations in the continuous case, but are not obtained by a straightforward integration by part.
 Supplements



03:45 PM  04:15 PM


Tea

 Location
 MSRI: Atrium
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04:15 PM  05:15 PM


Rigidity of the 3D hierarchical Coulomb gas
Sourav Chatterjee (Stanford University)

 Location
 MSRI: Simons Auditorium
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The mathematical analysis of Coulomb gases, especially in dimensions higher than one, has been the focus of much recent activity. For the 3D Coulomb, there is a famous prediction of Jancovici, Lebowitz and Manificat that if N is the number of particles falling in a given region, then N has fluctuations of order cuberoot of E(N). I will talk about the recent proof of this conjecture for a closely related model, known as the 3D hierarchical Coulomb gas. I will also try to explain, through some toy examples, why such unusually small fluctuations may be expected to appear in interacting gases
 Supplements




Nov 14, 2017
Tuesday

09:30 AM  10:30 AM


Refinements of Gaussian concentration for norms
Grigoris Paouris (Texas A & M University)

 Location
 MSRI: Simons Auditorium
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10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
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11:00 AM  12:00 PM


Subgaussian and lacunary uniformly bounded orthonormal systems
Gilles Pisier (Texas A & M University)

 Location
 MSRI: Simons Auditorium
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12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
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02:00 PM  03:00 PM


Billiards and Caustics
Shiri Artstein (Tel Aviv University)

 Location
 MSRI: Simons Auditorium
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We shall survey some known, some open, and some new results and questions related to billiard dynamics in convex domains.
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03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
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03:30 PM  04:30 PM


On matrixvalued logconcave functions
Dario CorderoErausquin (University of Paris VI, France)

 Location
 MSRI: Simons Auditorium
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In a 2013 preprint, H. Raufi obtained, using complex geometry, a matrixvalued extension of Prekopa’s theorem. I will discuss a new approach to this result, and in particular present a BrascampLieb variance inequality for matrixvalued logconcave potentials.
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04:30 PM  06:20 PM


Reception

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 MSRI: Atrium
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Nov 15, 2017
Wednesday

09:30 AM  10:30 AM


Bezout Inequality for Mixed volumes
Artem Zvavitch (Kent State University )

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 MSRI:
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In this talk we will discuss the following analog of Bezout inequality for mixed volumes: $$ V(P_1,\dots,P_r,\Delta^{nr})V_n(\Delta)^{r1}\leq \prod_{i=1}^r V(P_i,\Delta^{n1})\ \text{ for }2\leq r\leq n. $$ We will show that the above inequality is true when $\Delta$ is an $n$dimensional simplex and $P_1, \dots, P_r$ are convex bodies in ${\mathbb R}^n$. We present a conjecture that if the above inequality is true for all convex bodies $P_1, \dots, P_r$, then $\Delta$ must be an $n$dimensional simplex. We will show that the conjecture is true for many special cases, for example, in ${\mathbb R}^2$ or if we assume that $\Delta$ is a convex polytope. Next we will discuss an isomorphic version of the Bezout inequality: what is the best constant $c(n,r)>0$ such that $$ V(P_1,\dots,P_r,\Delta^{nr})V_n(D)^{r1}\leq c(n,r)\prod_{i=1}^r V(P_i,D^{n1})\ \text{ for }2\leq r\leq n, $$ where $P_1, \dots, P_r, D$ are convex bodies in ${\mathbb R}^n.$ Finally, we will present a connection of the above inequality to inequalities on the volume of orthogonal projections of convex bodies as well as inequalities for zonoids.
 Supplements



10:30 AM  11:00 AM


Tea

 Location
 MSRI: Atrium
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11:00 AM  12:00 PM


Hessian Valuations
Monika Ludwig (Technische Universität Wien)

 Location
 MSRI: Simons Auditorium
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Different approaches to introduce intrinsic volumes and more generally mixed volumes for convex and logconcave functions were proposed by Bobkov, Colesanti and Fragal\` a, by Rotem and Milman and by Alesker. They all turn out to be valuations on the corresponding spaces. Here a new class of continuous valuations on the space of convex functions on ${\mathbb R}^n$ is introduced. On smooth convex functions, they are defined for $i=0,\dots,n$ by \begin{equation*} u\mapsto \int_{{\mathbb R}^n} \zeta(u(x),x,\nabla u(x))\,[{{D}^2} u(x)]_i\,d x \end{equation*} where $\zeta\in C({\mathbb R}\times{\mathbb R}^n\times{\mathbb R}^n)$ and $[{{D}^2} u]_i$ is the $i$th elementary symmetric function of the eigenvalues of the Hessian matrix, ${{D}^2} u$, of $u$. Under suitable assumptions on $\zeta$, these valuations are shown to be invariant under translations and rotations on convex and coercive functions. Ultimately, a complete classification of continuous and rigid motion invariant valuations on this space of functions is the aim of this approach. The connection to Hadwiger's theorem will be discussed. The results presented in this talk are joint with Andrea Colesanti (University of Florence) and Fabian Mussnig (Technische Universit\"at Wien).
 Supplements




Nov 16, 2017
Thursday

09:30 AM  10:30 AM


A Reverse Minkowski Theorem
Oded Regev (New York University, Courant Institute)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
Informally, Minkowski's first theorem states that lattices that are globally dense (have small determinant) are also locally dense (have lots of points in a small ball around the origin). This fundamental result dates back to 1891 and has a very wide range of applications.
I will present a proof of a reverse form of Minkowski's theorem, conjectured by Daniel Dadush in 2012, showing that locally dense lattices are also globally dense (in the appropriate sense).
The talk will be self contained and I will not assume any familiarity with lattices.
Based on joint papers with Daniel Dadush and Noah StephensDavidowitz.
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10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
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11:00 AM  12:00 PM


Optimality of the JohnsonLindenstrauss lemma
Jelani Nelson (Harvard University)

 Location
 MSRI: Simons Auditorium
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Dimensionality reduction in Euclidean space, as attainable by the JohnsonLindenstrauss lemma, has been a fundamental tool in algorithm design and machine learning. The JL lemma states that any n point subset of l_2 can be mapped to l_2^m with distortion 1+epsilon, where m = O(epsilon^{2} log n). In this talk, I discuss our recent proof that the JL lemma is optimal, in the sense that for any n, d, epsilon, where epsilon is not too small, there is a point set X in l_2^d such that any f:X>l_2^m with 1+epsilon must have m = Omega(epsilon^{2} log n). I will also discuss some subsequent work and future directions. Joint work with Kasper Green Larhus (Aarhus University).
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12:00 PM  01:30 PM


Lunch

 Location
 MSRI: Atrium
 Video


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01:30 PM  02:30 PM


Progress on the KLS Conjecture
Santosh Vempala (Georgia Tech)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We show that the Cheeger constant of any logconcave density is at least Tr(A^2)^{1/4} where A is its covariance matrix, i.e., n^{1/4} for isotropic logconcave densities. This improves on known bounds for the KLS, thinshell, concentration and Poincare constants, and gives an alternative proof of the current best bound for the slicing constant. We then show how our proof can be used to derive a nearly tight bound for the logSobolev constant of isotropic logconcave distributions.
The talk is joint work with Yin Tat Lee (UW and MSR).
 Supplements



02:45 PM  03:45 PM


Convergence of Hamiltonian Monte Carlo and Faster Polytope Volume Computation
Yin Tat Lee (University of Washington)

 Location
 MSRI: Simons Auditorium
 Video

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We explain the Hamiltonian Monte Carlo method and apply it to the problem of 1) generating uniform random points from polytopes, 2) computing the volume of polytopes. For polytopes in R^n specified by O(n) inequalities, the resulting algorithm for both problems takes merely O*(n^1.667) steps. For volume computation, this is a huge improvement over the previous best algorithm that requires O(n^4) steps. The key idea is to prove certain isoperimetric inequalities on manifolds defined by log barrier functions. Joint work with Santosh Vempala
 Supplements



03:45 PM  04:15 PM


Tea

 Location
 MSRI: Atrium
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04:15 PM  05:15 PM


The dimensionfree structure of nonhomogeneous random matrices
Ramon Van Handel (Princeton University)

 Location
 MSRI: Simons Auditorium
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What does the spectrum of a random matrix look like when the entries can have an arbitrary variance pattern? Such questions, which are of interest in several areas of pure and applied mathematics, are largely orthogonal to problems of classical random matrix theory. For example, one might ask the following basic question: when does an infinite matrix with independent Gaussian entries define a bounded operator on l_2? In this talk, I will describe recent work with Rafal Latala and Pierre Youssef in which we completely answer this question, settling an old conjecture of Latala. More generally, we provide optimal estimates on the Schatten norms of random matrices with independent Gaussian entries. These results not only answer some basic questions in this area, but also provide significant insight on what such matrices look like and how they behave.
 Supplements




Nov 17, 2017
Friday

09:30 AM  10:30 AM


The smallest singular value of a $d$regular random square matrix
Alexander Litvak (University of Alberta)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
We derive a lower bound on the smallest singular value of a random $d$regular matrix, that is, the adjacency matrix of a random $d$regular directed graph. More precisely, let $C_1<d< c_1 n/\log^2 n$ and let $M$ be uniformly distributed on the set of all $0/1$valued $n\times n$ matrices such that each row and each column of a matrix has exactly $d$ ones. Then the smallest singular value $s_{n} (M)$ of $M$ is greater than $c_2 n^{6}$ with probability at least $1C_2\log^2 d/\sqrt{d}$, where $c_1$, $c_2$, $C_1$, and $C_2$ are absolute positive constants. This is a joint work with A. Lytova, K. Tikhomirov, N. TomczakJaegermann, and P. Youssef
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


On the convex Poincaré inequality and weak transportation inequalities
Radoslaw Adamczak (University of Warsaw; Polish Academy of Sciences)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
I will discuss the connections between weak transportation inequalities with quadraticlinear cost functions, BobkovLedoux type modified logSobolev inequalities for convex/concave functions and the convex Poincare inequality. If time permits I will also mention some self normalized moment inequalities for general convex functions, which can be easily obtained from concentration estimates for Lipschitz convex functions. Based on joint work with Michal Strzelecki.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
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02:00 PM  03:00 PM


Invertibility via distance for random matrices with continuous distributions
Konstantin Tikhomirov (Princeton University)

 Location
 MSRI: Simons Auditorium
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 Abstract
Let A be an n by n random matrix with independent centered rows, so that each row has realvalued components, is isotropic and logconcave. Further, let M be any fixed n by n real matrix. We derive small ball probability estimates for the smallest singular value of the noncentered random matrix A+M. Our method is free from any use of covering arguments, and is principally different from a standard (by now) approach involving a decomposition of the unit sphere and coverings, as well as from an approach of SankarSpielmanTeng for noncentered Gaussian matrices. Our method gives an estimate for the condition number of A+M which essentially matches the known bounds for a noncentered Gaussian matrix.
 Supplements



03:00 PM  03:30 PM


Tea

 Location
 MSRI: Atrium
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03:30 PM  04:30 PM


Uniform cover inequalities for the volume of coordinate sections and projections of convex bodies
Apostolos Giannopoulos (National and Kapodistrian University of Athens)

 Location
 MSRI: Simons Auditorium
 Video

 Abstract
The classical LoomisWhitney inequality and the uniform cover inequality of Bollobás and Thomason provide upper bounds for the volume of a compact set in terms of its lower dimensional coordinate projections. We provide further extensions of these inequalities in the setting of convex bodies. We also establish the corresponding dual inequalities for coordinate sections; these uniform cover inequalities for sections may be viewed as extensions of Meyer's dual LoomisWhitney inequality. Joint work with S. Brazitikos and DM. Liakopoulos. We also discuss applications of these inequalities to questions regarding the surface area of lower dimensional projections and the average section functional of lower dimensional sections of a convex body, which were the motivation for them (joint works with A. Koldobsky, S. Dann, S. Brazitikos and P. Valettas).
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