Online Asymptotic Geometric Analysis Seminar
Welcome to the Online AGA seminar webpage! If you are interested in giving a talk, please let us know. Also, please suggest speakers which you would like to hear speak. Most talks are 50 minutes, but some 20-minute talks will be paired up as well. The talks will be video recorded conditioned upon the speakers' agreement. PLEASE SHARE THE SEMINAR INFO WITH YOUR DEPRARTMENT AND ANYONE WHO MAY BE INTERESTED! Please let the organizers know if you would like to be added to the mailing list.
The Zoom link to join the seminar
The seminar "sea-side" social via gather.town for after the talk
Note that on Tuesdays, the lectures start at:
7:30am in Los-Angeles, CA
8:30am in Edmonton, AB
9:30am in Columbia MO; College Station, TX; Chicago, IL
10:30am in Kent, OH; Atlanta, GA; Montreal; New York, NY
11:30am in Rio de Janeiro, Buenos Aires
3:30pm (15:30) in London
4:30pm (16:30) in Paris, Milan, Budapest, Vienna
5:30pm (17:30) in Tel Aviv.
Spring 2020: abstracts, slides, videos of the talks
Tuesday, April 7, 10:30AM (New York, NY time)
Alexander Koldobsky, University of Missouri, Columbia, MO, USA
Topic: A new version of the isomorphic Busemann-Petty problem for arbitrary functions
Abstract (click this link to view pdf)
Slides of the talk
Video of the talk
Saturday, April 11, 11:30AM (New York, NY time)
Sergey Bobkov, University of Minnesota, Minneapolis, MN, USA
Topic: A Fourier-analytic approach to transport inequalities
Abstract: We will be discussing a Fourier-analytic approach to optimal matching between independent samples, with an elementary proof of the Ajtai-Komlos-Tusnady theorem. The talk is based on a joint work with Michel Ledoux.
Tuesday, April 14, 10:30AM (New York, NY time)
Elisabeth Werner, Case Western Reserve University, Cleveland, OH, USA
Topic: Constrained convex bodies with maximal affine surface area
Abstract: Given a convex body K in R^n, we study the maximal affine surface area of K, i.e., the quantity AS(K) = sup_{C} as(C) where as(C) denotes the affine surface area of C, and the supremum is taken over all convex subsets of K. In particular, we give asymptotic estimates on the size of AS(K).
Károly Böröczky, Central European University, Budapest, Hungary
Topic: Symmetry and Structure within the Log-Brunn-Minkowski Conjecture
Abstract: After reviewing some formulations of the Log-Brunn-Minkowski Conjecture in R^n in terms of Monge-Ampere equations, of Hilbert Operator and of Brunn-Minkowski Theory, I will report on some recent advances, like Livshyts' and Kolesnikov's improvement on the fundamental approach of Milman and Kolesnikov, and the verification of the conjecture for bodies with n hyperplane symmetries by Kalantzopoulos and myself using an idea due to Bathe and Fradelizi.
Uri Grupel, University of Innsbruk, Austria
Topic: Metric distortion of random spaces
Abstract: We consider a random set in the unit circle. Is the induced discrete metric of the set closer to that of another independent random set or to the evenly spaced set of the same cardinality? We measure the distortion by looking at the smallest bi-Lipschitz norm of all the bijections between the two sets. Since the distortion between two random sets has infinite expectation, the talk will focus on the median. We show that two random sets have "typically" smaller distortion than a random set and an evenly spaced set.
Grigoris Paouris, Texas A&M University, College Station, TX, USA
Topic: Quantitative Triangle law and Joint Normality of Lyapunov exponents for products of Gaussian matrices
Abstract: We will discuss spectral properties of products of independent Gaussian square matrices with independent entries. Non-asymptotic results for the statistics of the singular values will be presented as well as the rate of convergence to the triangle law. We will also show quantitative estimates on the asymptotic joint normality of the Lyapunov exponents. The talk is based on a joint work with Boris Hanin.
Ilaria Fragalà, Politecnico di Milano, Italy
Topic: Symmetry problems for variational functionals: from continuous to discrete.
Abstract: I will discuss some symmetry problems for variational energies on the class of convex polygons with a prescribed number of sides, in which the regular n-gon can be proved or is expected to be optimal. Such symmetry results can be viewed as the "discrete" analogue of well-known "continuous" isoperimetric inequalities with balls as optimal domains. I will focus in particular on the following topics
(i) Discrete isoperimetric type inequalities
(ii) Discrete Faber-Krahn type inequalities
(iii) Overdetermined boundary value problems on polygons.
Saturday, May 2, 11:30AM (New York, NY time)
Alina Stancu, Concordia University, Montreal, Canada
Topic: On the fundamental gap and convex sets in hyperbolic space
Abstract: The lower bound on the fundamental gap of the Laplacian on convex domains in R^n, with Dirichlet boundary conditions, has a long history and has been finally settled a few years ago with parabolic methods by Andrews and Clutterbuck. More recently, the same lower bound, which depends on the diameter of the domain, has been proved for convex sets on the standard sphere in several stages with several groups of authors, 2016-2018. Over the past year, together with collaborators, we have found that the gap on the hyperbolic space behaves strikingly different and we aim to explain it, particularly for this audience, as a difference in the nature of convex sets in H^n versus R^n or S^n.
Tuesday, May 5, 10:30AM (New York, NY time)
Andrea Colesanti, University of Florence, Italy
Topic: Brunn-Minkowski type inequalities and affine surface area
Abstract: Does the affine surface area verify a concavity inequality of Brunn-Minkowski type? We will try to provide an answer to this question, and we will see that the answer depends on the dimension, and on the type of addition that we choose. The results presented in this talk were obtained in collaboration with Karoly Boroczky, Monika Ludwig and Thomas Wannerer.
Monika Ludwig, Vienna University of Technology, Austria
Topic: Valuations on Convex Functions
Tuesday, May 12, 10:30AM (New York, NY time)
Bo'az Klartag, Weizmann Institute of Science, Rehovot, Israel
Topic: Rigidity of Riemannian embeddings of discrete metric spaces
Abstract: Let M be a complete, connected Riemannian surface and suppose that S is a discrete subset of M. What can we learn about M from the knowledge of all distances in the surface between pairs of points of S? We prove that if the distances in S correspond to the distances in a 2-dimensional lattice, or more generally in an arbitrary net in R^2, then M is isometric to the Euclidean plane. We thus find that Riemannian embeddings of certain discrete metric spaces are rather rigid. A corollary is that a subset of Z^3 that strictly contains a two-dimensional lattice cannot be isometrically embedded in any complete Riemannian surface. This is a joint work with M. Eilat.
Notebook of the talk
Chat file from the talk
Saturday, May 16, 11:30AM (New York, NY time)
Galyna Livshyts, Georgia Tech, Atlanta, GA, USA
Topic: On the Log-Brunn-Minkowski conjecture and related questions
Abstract: We shall discuss the Log-Brunn-Minkowski conjecture, a conjectured strengthening of the Brunn-Minkowski inequality proposed by Boroczky, Lutwak, Yang and Zhang, focusing on the local versions of this and related questions. The discussion will involve introduction and explanation of how the local version of the conjecture arises naturally, a collection of ''hands on'' examples and elementary geometric tricks leading to various related partial results, statements of related questions as well as a discussion of more technically involved approaches and results. Based on a variety of joint results with several authors, namely, Colesanti, Hosle, Kolesnikov, Marsiglietti, Nayar, Zvavitch. REMARK: THIS TALK IS A LAST MINUTE REPLACEMENT OF THE EARLIER ANNOUNCED TALK BY TIKHOMIROV; TIKOMIROV'S TALK IS NOW SCHEDULED FOR THE FALL.
Olivier Guedon, Université Gustave Eiffel, Paris, France
Topic: Floating bodies and random polytopes
Abstract: I will present some results about the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\R^n$. Under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector---namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body. This is joint work with F. Krahmer, C. Kummerle, S. Mendelson and H. Rauhut.
Vitaly Milman, Tel Aviv University and Liran Rotem, Technion, Haifa, Israel
Topic: Novel view on classical convexity theory
Abstract: In this talk we will introduce and study the class of flowers. A flower in R^n is an arbitrary union of balls which contain the origin. While flowers are not necessarily convex, they are in one to one correspond with the class of convex bodies containing the origin, so by studying flowers we are also studying convex bodies from a new viewpoint. We will give several equivalent definitions of flowers and describe some of their basic properties. We will also discuss how to apply an arbitrary (real) function to a flower, and the corresponding construction for convex bodies. In particular, we will explain how to raise a flower to a given power. Finally, we will discuss some elements of the asymptotic theory of flowers. In particular we will present a Dvoretzky-type theorem for flowers which actually gives better estimates than the corresponding estimates for convex bodies. Based on two papers by the speakers, the first of which is joint with E. Milman.
Alexandros Eskenazis, Institut de Mathématiques de Jussieu, Sorbonne Université, Paris, France
Topic: The dimensional Brunn-Minkowski inequality in Gauss space
Abstract: We will present a complete proof of the dimensional Brunn-Minkowski inequality for origin symmetric convex sets in Gauss space. This settles a problem raised by Gardner and Zvavitch (2010). The talk is based on joint work with G. Moschidis.
Timothy Gowers, Cambridge University, UK
Topic: High-dimensional tennis balls
Abstract: In this talk, it will be explained what a high-dimensional tennis ball is, how one can construct it and its connection to V. Milman's question about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\ell_2^n$.
Tuesday, June 2, 10:30AM (New York, NY time)
Yair Shenfeld, Princeton University, NJ, USA
Topic: The extremal structures of the Alexandrov-Fenchel inequality
Abstract: The Alexandrov-Fenchel inequality is one of the fundamental results in the theory of convex bodies. Yet its equality cases, which are solutions to isoperimetric-type problems, have been open for more than 80 years. I will discuss recent progress on this problem where we confirm some conjectures by R. Schneider. Joint work with Ramon van Handel.
Saturday, June 6, 11:30AM (New York, NY time)
Mark Meckes, Case Western Reserve University, Cleveland, USA
Topic: Magnitude and intrinsic volumes of convex bodies
Abstract: Magnitude is an isometric invariant of metric spaces with origins in category theory. Although it is very difficult to exactly compute the magnitude of interesting subsets of Euclidean space, it can be shown that magnitude, or more precisely its behavior with respect to scaling, recovers many classical geometric invariants, such as volume, surface area, and Minkowski dimension. I will survey what is known about this, including results of Barcelo--Carbery, Gimperlein--Goffeng, Leinster, Willerton, and myself, and sketch the proof of an upper bound for the magnitude of a convex body in Euclidean space in terms of intrinsic volumes.
Matthieu Fradelizi, Marne-la-Vallée, Paris, France
Topic: Volume product, polytopes and finite dimensional Lipschitz-free spaces.
Abstract: We shall present some results on the volume product of polytopes, including the question of its maximum among polytopes with a fixed number of vertices. Then we shall focus on the polytopes that are unit balls of Lipschitz-free Banach spaces associated to finite metric spaces. We characterize when these polytopes are Hanner polytopes and when two such polytopes are isometric to each others. We also also study the maximum of the volume product in this class. Based on joint works with Matthew Alexander, Luis C. Garcia-Lirola and Artem Zvavitch.
Saturday, June 13, 11:30AM (New York, NY time)
Julian Haddad, Federal University of Minas Gerais, Belo Horizonte, Brasil
Topic: From affine Poincaré inequalities to affine spectral inequalities
Abstract: We develop the basic theory of $p$-Rayleigh quotients in bounded domains, in the affine case, for $p \geq 1$. We establish p-affine versions of the affine Poincaré inequality and introduce the affine invariant $p$-Laplace operator $\Delta_p^{\mathcal A}$ defining the Euler-Lagrange equation of the minimization problem. For $p=1$ we obtain the existence of affine Cheeger sets and study preliminary results towards a possible spectral characterization of John's position.
Tuesday, June 16, 10:30AM (New York, NY time)
Semyon Alesker, Tel Aviv University, Israel
Topic: Multiplicative structure on valuations and its analogues over local fields.
Abstract: Valuation on convex sets is a classical notion of convex geometry. Multiplicative structure on translation invariant smooth valuations was introduced by the speaker years ago. Since then several non-trivial properties of it have been discovered as well as a few applications to integral geometry. In the first part of the talk we will review some of these properties. Then we discuss analogues of the algebra of even translation invariant valuations over other locally compact (e.g. complex, p-adic) fields. While any interpretation of these new algebras is missing at the moment, their properties seem (to the speaker) to be non-trivial and having some intrinsic beauty.
Orli Herscovici, University of Haifa, Israel
Topic: The best constant in the Khinchine inequality for slightly dependent random variables
Abstract: We solve the open problem of determining the best constant in the Khintchine inequality under condition that the Rademacher random variables are slightly dependent. We also mention some applications in statistics of the above result. The talk is based on a joint work with Susanna Spektor.
Elizabeth Meckes, Case Western Reserve University, Cleveland, USA
Topic: On the eigenvalues of Brownian motion on \mathbb{U}(n)
Abstract: Much recent work in the study of random matrices has focused on the non-asymptotic theory; that is, the study of random matrices of fixed, large size. I will discuss one such example: the eigenvalues of unitary Brownian motion. I will describe an approach which gives uniform quantitative almost-sure estimates over fixed time intervals of the distance between the random spectral measures of this parametrized family of random matrices and the corresponding measures in a deterministic parametrized family \{\nu_t\}_{t\ge 0} of large-n limiting measures. I will also discuss larger time scales. This is joint work with Tai Melcher.
A cool video of the spectral measure evolution
Elizaveta Rebrova, UCLA, Los-Angeles, CA, USA
Topic: Modewise methods for tensor dimension reduction
Abstract: Although tensors are a natural multi-modal extension of matrices, going beyond two modes (that is, rows and columns) presents many interesting non-trivialities. For example, the notion of singular values is no longer well-defined, and there are various versions of the rank. One of the most natural (and mathematically challenging) definitions of the tensor rank is so-called CP-rank: for a tensor X, it is a minimal number of rank one tensors whose linear combination constitutes X. Main focus of my talk will be an extension of the celebrated Johnson-Lindenstrauss lemma to low CP-rank tensors. Namely, I will discuss how modewise randomized projections can preserve tensor geometry in the subspace oblivious way (that is, a projection model is not adapted for a particular tensor subspace). Modewise methods are especially interesting for the tensors as they preserve the multi-modal structure of the data, acting on a tensor directly, without initial conversion of tensors to matrices or vectors. I will also discuss an application for the least squares fitting CP model for tensors. Based on our joint work with Mark Iwen, Deanna Needell, and Ali Zare.
Peter Van Hintum, Cambridge University, UK
Topic: Sharp stability of the Brunn-Minkowski inequality
Abstract: We consider recent results concerning the stability of the classic Brunn-Minkowski inequality. In particular we shall focus on the linear stability for homothetic sets. Resolving a conjecture of Figalli and Jerison, we show there are constants C,d>0 depending only on n such that for every subset A of R^n of positive measure, if |(A+A)/2 - A| <= d |A|, then |co(A) - A| <= C |(A+A)/2 - A| where co(A) is the convex hull of A. The talk is based on joint work with Hunter Spink and Marius Tiba.
Fall 2020: abstracts, slides, videos of the talks
Schedule Fall 2020:
Tuesday, August 25, 10:30AM (New York, NY time)
Bo Berndtsson, Chalmers University of Technology and the University of Goteborg, Sweden
Topic: Complex integrals and Kuperberg's proof of the Bourgain-Milman theorem
Abstract: We give a proof of the Bourgain-Milman theorem using complex methods. The proof is inspired by Kuperberg's, but considerably shorter. Time permitting, we will also comment on Nazarov's proof and estimates of Bergman kernels.
Saturday, August 29, 11:30AM (New York, NY time)
Marton Naszodi, Alfred Renyi Inst. of Math. and Eotvos Univ., Budapest, Hungary
Topic: Some new quantitative Helly-type theorems
Abstract: Quantitative Helly-type theorems were introduced by Bárány, Katchalski and Pach in 1982, who, among other results, showed the following. There is a constant C_d depending on the dimension d only, such that if the intersection of a finite family of convex bodies in R^d is of volume at most one, then the intersection of some subfamily of 2d members is of volume at most C_d. We consider colorful and fractional versions of this result.
Tuesday, September 1, 10:30AM (New York, NY time)
Kateryna Tatarko, Texas A&M University, College Station, TX, USA and the University of Alberta, Canada
Topic: On the unique determination of ellipsoids by dual intrinsic volumes
Abstract: In this talk, we show that an ellipsoid is uniquely determined up to an isometry by its dual Steiner polynomial. We reduce this result to the moment problem, and as a by-product obtain an alternative proof of the analogous known result for classical Steiner polynomials in $R^3$. This is joint work with S. Myroshnychenko and V. Yaskin.
Saturday, September 5, 11:30AM (New York, NY time)
Adam Kashlak, University of Alberta, Edmonton, Canada
Topic: Analytic Permutation Testing via Kahane--Khintchine Inequalities
Abstract: The permutation test is a versatile type of exact nonparametric significance test that requires drastically fewer assumptions than similar parametric tests by considering the distribution of a test statistic over a discrete group of distributionally invariant transformations. The main downfall of the permutation test is the high computational cost of running such a test making this approach laborious for complex data and experimental designs and completely infeasible in any application requiring speedy results. We rectify this problem through application of Kahane--Khintchine-type inequalities under a weak dependence condition and thus propose a computation free permutation test---i.e. a permutation-less permutation test. This general framework is studied within both commutative and non-commutative Banach spaces. We further improve these Kahane-Khintchine-type bounds via a transformation based on the incomplete beta function and Talagrand's concentration inequality. For k-sample testing, we extend the theory presented for Rademacher sums to weakly dependent Rademacher chaoses making use of modified decoupling inequalities. We test this methodology on classic functional data sets including the Berkeley growth curves and the phoneme dataset. We also consider hypothesis testing on speech samples under two experimental designs: the Latin square and the complete randomized block design.
Grigory Ivanov, IST Austria, Austria, and MIPT, Moscow, Russia.
Topic: Functional John-Lowner ellipsoids of a log-concave function
Abstract: We extend the notion of the John ellipsoid (the largest volume ellipsoid contained within a convex body) to the setting of log-concave functions. For every s > 0, we define a class of log-concave functions derived from ellipsoids. For any log-concave function f, and any fixed s > 0, we consider functions belonging to this class and find the one with the largest integral under the condition that it is pointwise less than or equal to f. We show that it exists and is unique, and call it the John s-function of f. We give a characterization of this function similar to the one provided by John in his fundamental theorem. As an application, we obtain a quantitative Helly-type result about the integral of the pointwise minimum of a family of log-concave functions. Next, we will discuss the concept of duality for log-concave functions and extend the notion of the Löwner ellipsoid (the smallest volume ellipsoid containing a convex body) to the setting of log-concave functions. Time permitting, we will discuss the difference between the behavior of convex sets and log-concave functions concerning our problems. Based on joint works with Márton Naszódi and Igor Tsiutsiurupa.
Saturday, September 12, 11:30AM (New York, NY time)
Masha Gordina, University of Connecticut, Storrk, CT, USA
Topic: Uniform doubling on SU(2) and beyond
Abstract: Suppose G is a compact Lie group equipped with a left-invariant Riemannian metric. Such metrics usually form a finite-dimensional cone. The Riemannian volume measure corresponding to such a metric is the Haar measure of the group (up to a multiplicative constant). Because of compactness, each left-invariant metric g has the doubling property, that is, there exists a doubling constant D=D(G, g) such that the volume of the ball of radius 2r is at most D times the volume of the ball of radius r. We are concerned with the following question: does there exist a constant D(G) such that, for all left-invariant metrics g on G, the constant D(G, g) is bounded above by D(G)? This is what we call uniformly doubling. The conjecture is that any compact Lie group is uniformly doubling. The only cases for which the conjecture is known are Riemannian tori and the group SU(2). The talk will describe a number of analytic consequences of uniform doubling (in absence of curvature bounds) and our approach to proving uniform doubling on SU(2). The work in progress for U(2) might be mentioned as well. This is joint work with Nathaniel Eldredge (University Northern Colorado) and Laurent Saloff-Coste (Cornell University). Reference: Left-invariant geometries on SU(2) are uniformly doubling, GAFA 2018.
Tuesday, September 15, 10:30AM (New York, NY time)
Santosh Vempala, Georgia Institute of Technology, Atlanta, GA, USA
Topic: Reducing Isotropy to KLS: An n^3\psi^2 Volume Algorithm
Abstract: Computing the volume of a convex body is an ancient problem whose study has led to many interesting mathematical developments. In the most general setting, the convex body is given only via a membership oracle. In this talk, we present a faster algorithm for isotropic transformation of an arbitrary convex body in R^n, with complexity n^3\psi^2, where \psi bounds the KLS constant for isotropic convex bodies. Together with the known bound of \psi = O(n^{1/4}) [2017] and the Cousins-Vempala n^3 volume algorithm for well-rounded convex bodies [2015], this gives an n^{3.5} volume algorithm for general convex bodies, the first improvement on the n^4 algorithm of Lovász-Vempala [2003]. A positive resolution of the KLS conjecture (\psi = O(1)) would imply an n^3 volume algorithm. No background on algorithms, KLS or ABC will be assumed for the talk. Joint work with He Jia, Aditi Laddha and Yin Tat Lee.
Remark: a follow up talk with more details will take place at the Georgia Tech High Dimensional Seminar on Wednesday, September 16, at 3:15pm (NYC time). Zoom link here.
Ferenc Fodor, University of Szeged, Hungary
Topic: Strengthened inequalities for the mean width and the $\ell$-norm
Abstract: According to a result of Barthe the regular simplex maximizes the mean width of convex bodies whose John ellipsoid is the Euclidean unit ball. This is equivalent to the fact that the regular simplex maximizes the $\ell$-norm of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball. The reverse statement that the regular simplex minimizes the mean width of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball is also true as proved by Schmuckenschl\"ager. In this talk we prove strengthened stability versions of these results and some related stability statements for the convex hull of the support of centered isotropic measures on the unit sphere. This is joint work with K\'aroly J. B\"or\"oczky (Budapest, Hungary) and Daniel Hug (Karlsruhe, Germany).
Dan Mikulincer, Weizmann Institute of Science, Rehovot, Israel
Topic: Stability of Stein kernels, moment maps and invariant measures
Abstract: Suppose that \mu is some nice measure on a Euclidean space. We can associate it with several different constructions of interest: Stein kernels, arising from Stein's theory, the moment map, which is of a more geometric flavour and a particular choice of a stochastic process for which \mu is the invariant measure. We will discuss the connections between these different objects and show that they are stable with respect to the original measure. That is, a small perturbation to either construction will yield a new measure which is close to \mu. Joint work with Max Fathi.
Konstantin Tikhomirov, Georgia Tech, Atlanta, GA, USA
Topic: Non-asymptotic bound for the smallest singular value of powers of random matrices
Abstract: I will discuss a joint work with H.Huang on the smallest singular value of powers of Gaussian matrices and challenges in extending the obtained bound to non-Gaussian setting.
Tuesday, October 6, 10:30AM (New York, NY time)
Paata Ivanisvili, North Carolina State University, NC, USA
Topic: Enflo's problem
Abstract: A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. I will speak about the joint work with Ramon van Handel and Sasha Volberg where we prove that Rademacher and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the Hamming cube.
Tuesday, October 13, 10:30AM (New York, NY time)
Keith Ball, University of Warwick, UK
Topic: Rational approximations to the zeta function
Abstract: I will describe the construction of a sequence of rational functions with rational coefficients that converge to the zeta function. These approximations extend and make precise the spectral interpretations of the Riemann zeros found by Connes and by Berry and Keating.
Naomi Feldheim, Bar Ilan University, Israel
Topic: Persistence of Gaussian stationary processes
Abstract: Let f:R->R be a Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution. What is the probability that f remains above a certain fixed line for a long period of time? This simple question, which was posed by mathematicians and engineers more than 60 years ago (e.g. Rice, Slepian), has some surprising answers which were discovered only recently. I will describe how a spectral point of view leads to those results. Based on joint works with O. Feldheim, F. Nazarov, S. Nitzan, B. Jaye and S. Mukherjee.
Rafal Latala, University of Warsaw, Poland
Topic: Order Statistics of Log-Concave Vectors
Abstract: I will discuss two-sided bounds for expectations of order statistics (k-th maxima) of moduli of coordinates of centered log-concave random vectors with uncorrelated coordinates. Our bounds are exact up to multiplicative universal constants in the unconditional case for all k and in the isotropic case for k = n-cn^{5/6}. We also present two-sided estimates for expectations of sums of k largest moduli of coordinates for some classes of random vectors. Joint work with Marta Strzelecka.
Tuesday, November 3, 10:30AM (New York, NY time) (two talks 20 min each)
Michael Roysdon, Tel Aviv University, Israel
Topic:$L_p$-Brunn-Minkoswki type inequalities and an $L_p$-Borell-Brascamp-Lieb inequality, 10:30-10:50
Abstract: the classical Brunn-Minkowski inequality asserts that the volume of convex Minkowski combination exhibits (1/n)-concavity when applied for any pair of convex bodies (or more generally, Borel sets). Many advancements of this inequality have been studied throughout the year, famous examples of such mathematicians who pursued these studies are Prekopa, Leindler, and Brascamp and Lieb. The goal of this talk is to introduce the "L_p" versions of such inequalities following the L_p-Minkowski sum introduced by Firey (and later more generally by Lutwak, Yang, and Zhang), as well as it's associated L_p_ Brunn-Minkowksi inequality. In particular, we show that such inequalities hold in the class of s-concave measures, and discuss the related isoperimetric inequality (joint with S. Xing).
Jesús Yepes Nicolás, Universidad de Murcia, Spain
Topic: Further inequalities for the Wills functional of convex bodies.
Abstract: The Wills functional of a convex body, defined as the sum of its intrinsic volumes, turned out to have many interesting applications and properties. In this talk, making profit of the fact that it can be represented as the integral of a log-concave function, which is furthermore the Asplund product of other two log-concave functions, we will show new properties of the Wills functional. Among others, we get Brunn-Minkowski and Rogers-Shephard type inequalities for this functional and show that the cube of edge-length 2 maximizes it among all 0-symmetric convex bodies in John position. Joint work with David Alonso-Guti�rrez and Mar�a A. Hern�ndez Cifre.
Tuesday, November 10, 10:30AM (New York, NY time)
Dima Faifman, Tel Aviv University, Israel
Topic: Crofton formulas in isotropic pseudo-Riemannian spaces.
Abstract: The length of a curve in the plane can be computed by counting the intersection points with a line, and integrating over all lines. More generally, the intrinsic volumes (quermassintegrals) of a subset of Euclidean space can be computed by Crofton integrals, bringing forth their fundamental role in integral geometry. In spherical and hyperbolic geometry, such formulas are also known and classical. In pseudo-Riemannian isotropic spaces, such as de Sitter or anti-de Sitter space, one can similarly ask for an integral-geometric formula for the volume of a submanifold, or more generally for the intrinsic volumes of a subset, which were introduced only recently. I will explain how to obtain and apply such formulas, and how in fact there is a universal Crofton formula depending on a complex parameter extending the Riemannian Crofton formulas, for which all indefinite signatures are distributional boundary values. This is a joint work in progress with Andreas Bernig and Gil Solanes.
Mark Sellke, Stanford University, Palo Alto, CA, USA
Topic: Chasing Convex Bodies
Abstract: I will explain the chasing convex bodies problem posed by Friedman and Linial in 1991. In this problem, an online player receives a request sequence K_1,...,K_T of convex sets in d dimensional space and moves his position online into each requested set. The player's movement cost is the length of the resulting path. Chasing convex bodies asks if there is an online algorithm with cost competitive against the offline optimal path. This is both an interesting metrical task system and (equivalent to) a competitive analysis view on online convex optimization. This problem has recently been solved twice. The first solution gives a 2^{O(d)} competitive algorithm while the second gives a nearly optimal min(d,sqrt(d*log(T))) competitive algorithm for T requests. The latter result is based on the Steiner point, which is the exact optimal solution to a related geometric problem called Lipschitz selection and dates from 1840. In the talk, I will briefly outline the first solution and fully explain the second. Partially based on joint works with Sébastien Bubeck, Bo'az Klartag, Yin Tat Lee, and Yuanzhi Li.
Kasia Wyczesany, Tel Aviv University, Israel
Topic: Existence of potentials for non-traditional cost functions
Abstract: In this talk, we will present a new approach to the problem of existence of a potential for the optimal transport problem and apply it to non-traditional cost functions (i.e. costs that may attain infinite values). As a by-product, we give a new transparent proof of Rockafellar-Ruschendorf theorem. As an example of a non-traditional cost, we discuss the polar cost, which is particularly interesting as it induces the polarity transform and the class of geometric convex functions. This is joint work with S. Artstein-Avidan and S. Sadovsky.
Tuesday, December 1, 10:30AM (New York, NY time)
Shay Sadovsky, Tel Aviv University, Israel
Topic: Existence of potentials for non-traditional cost functions (part 2)
Abstract: In this talk, we present a constructive method for finding solutions to Monge's problem of mass-transport between two measures with respect to the polar cost. Our costruction, generalizing an idea of Keith Ball, utilizes a new notion of 'Hall polytopes', which we introduce. Our method applies to non-traditional transport problems, i.e. those with costs which can attain the value infinity, as well as the classical case. Based on joint work with Shiri Artstein-Avidan and Kasia Wyczesany.
Emanuel Milman, Technion, Haifa, Israel
Topic: Sharp Isoperimetric Inequalities for Affine Quermassintegrals
Abstract: The affine quermassintegrals associated to a convex body in $\R^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn--Minkowski theory, and thus constitute a central pillar of affine convex geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke--Santal\'o and Petty projection inequalities, respectively. In this work we confirm Lutwak's conjecture, including characterization of the equality cases, for all values of $k=1,\ldots,n-1$, in a single unified framework. In fact, it turns out that ellipsoids are the only \emph{local} minimizers with respect to the Hausdorff topology. In addition, we address a related conjecture of Lutwak on the validity of certain Alexandrov--Fenchel-type inequalities for affine (and more generally $L^p$-moment) quermassintegrals. The case $p=0$ corresponds to a sharp averaged Loomis--Whitney isoperimetric inequality. Based on joint work with Amir Yehudayoff.
Tuesday, December 15, 10:30AM (New York, NY time)
Pierre Youssef, NYU Abu Dhabi, United Arab Emirates
Topic: Mixing time of the switch chain on regular bipartite graphs.
Abstract: Given a fixed integer d, we consider the switch chain on the set of d-regular bipartite graphs on n vertices equipped with the uniform measure. We prove a sharp Poincar� and log-Sobolev inequality implying that the mixing time of the switch chain is at most O(n log^2n) which is optimal up to a logarithmic term. This improves on earlier results of Kannan, Tetali, Vempala and Dyer et al. who obtained the bounds O(n^13 log n) and O(n^7 log n) respectively. This is a joint work with Konstantin Tikhomirov.
Schedule Spring 2021:
Tuesday, January 5, 2021, 10:30AM (New York, NY time)
Yuansi Chen, Duke University, Durham, NC, USA
Topic: Recent progress on the KLS conjecture and the stochastic localization scheme of Eldan
Abstract: Kannan, Lovasz and Simonovits (KLS) conjectured in 1993 that the Cheeger isoperimetric coefficient of any log-concave density is achieved by half-spaces up to a universal constant factor. This conjecture also implies other important conjectures such as Bourgain's slicing conjecture (1986) and the thin-shell conjecture (2003). In this talk, first we briefly survey the origin and the main consequences of these conjectures. Then we present the development and the refinement of the main proof technique, namely Eldan's stochastic localization scheme, which results in the current best bounds of the Cheeger isoperimetric coefficient in the KLS conjecture.
Tuesday, January 12, 2021:
No seminar. This winter school is happening at the time of the seminar.
Tuesday, January 19, 2021, 10:30AM (New York, NY time)
Mark Agranovsky, Bar Ilan University, Ramat Gan, Israel
Topic: On integrable domains and surfaces
Abstract: Integrability of domains or surfaces in R^n is defined in terms of sectional or solid volume functions, evaluating the volumes of the intersections with affine planes or half-spaces. Study of relations between the geometry of domains and types of their volume functions is motivated by a problem of V.I. Arnold about algebraically integrable bodies, which in turn goes back to celebrated Newton's Lemma about ovals. The talk will be devoted to a survey of some recent works in this area.
Konstantin Drach, Aix-Marseille Universite, Marseille, France
Topic: TBA
Abstract: TBA
Tuesday, February 2, 2021, 11:30AM (New York, NY time) - Note the special time an hour later!
Rachel Greenfeld, UCLA, Los-Angeles, CA, USA
Tuesday, February 9, 2021, 10:30AM (New York, NY time)
Alexey Garber, The University of Texas Rio Grande Valley, Brownsville, TX, USA
Topic: Convex polytopes that tile space with translations: Voronoi domains and spectral sets
Abstract: In this talk I am going to discuss convex d-dimensional polytopes that tile R^d with translations and their properties related to two conjectures. The first conjecture, the Fuglede conjecture, claims that every spectral set in R^d tiles the space with translations; this conjecture was recently settled for convex domains by Lev and Matolcsi. The second conjecture, the Voronoi conjecture, claims that every convex polytope that tiles R^d with translations is the Voronoi domain for some d-dimensional lattice. The conjecture originates from the Voronoi�s geometric theory of positive definite quadratic forms and is related to many questions in mathematical crystallography including Hilbert�s 18th problem. I mostly plan to discuss recent progress in the Voronoi conjecture and the proof of the conjecture for five-dimensional parallelohedra; in the general setting the Voronoi conjecture is still open. The talk is based on a joint work with Alexander Magazinov (Skoltech).
Tuesday, February 16, 2021, 10:30AM (New York, NY time)
Han Huang, Georgia Institute of Technology, Atlanta, GA, USA
Boaz Slomka, The Open University of Israel, Raanana, Israel
Tuesday, March 2, 10:30AM (New York, NY time)
Maria de los Angeles Alfonseca-Cubero, North Dakota State University, Fargo, ND, USA
Tuesday, March 9, 2021, 10:30AM (New York, NY time)
Tselil Schramm, Stanford University, Palo Alto, CA, USA
Tuesday, March 16, 2021, 10:30AM (New York, NY time)
No seminar: this conference intersects with the seminar time.
Tuesday, March 30, 10:30AM (New York, NY time)
Susanna Spektor, Sheridan College, Toronto, ON, Canada
Topic: On the applications of the Khinchine type inequality for Independent and Dependent Poisson random variables.
Abstract: We will obtain the Khinchine type inequality for Poisson random variables in two settings-when random variables are independent and when the sum of them is equal to a fixed number. We will look at the applications of these inequalities in Statistics..
Semyon Alesker, Tel Aviv University, Tel Aviv, Israel
Topic: New inequalities for mixed volumes of convex bodies and valuations theory
Abstract: I will present a few new inequalities for mixed volumes of general convex bodies. In a special case they can be considered as a new isoperimetric property of Euclidean ball in R^n. The inequalities are consequences of a recent result of J. Kotrbaty on Hodge-Riemann type inequalities on the space of translation invariant valuations on convex sets.
Tuesday, April 13, 2021, 10:30AM (New York, NY time)
Vishesh Jain, Stanford University, Palo Alto, CA, USA
Anindya De, University of Pennsilvania
Ronen Eldan, Weizmann Institute of Science, Rehovot, Israel
Tuesday, May 4, 2021, 10:30AM (New York, NY time)
Gregory Wyatt, University of Missouri, Columbia, MO, USA (talk 1, 20 minutes)
Sudan Xing, University of Alberta, Canada (talk 2, 20 minutes)
Tuesday, May 11, 2021, 10:30AM (New York, NY time)
Sang Woo Ryoo, Princeton University, NJ, USA
Maria Angeles Hernandez Cifre, Universidad de Murcia, Spain
Alexander Litvak, University of Alberta, AB, Canada
Tuesday, June 1, 2021, 10:30AM (New York, NY time)
Almut Burchard, University of Toronto, Canada
TBA, TBA
Galyna Livshyts, Georgia Tech glivshyts6@math.gatech.edu
Liran Rotem, Technion lrotem@technion.ac.il
Dmitry Ryabogin, Kent State University ryabogin@math.kent.edu
Konstantin Tikhomirov, Georgia Tech ktikhomirov6@gatech.edu