Online Asymptotic Geometric Analysis Seminar

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

Tuesday, September 8, 10:30AM (New York, NY time)

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

Saturday, September 19, 11:30AM (New York, NY time)

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).

Slides of the talk

Video of the talk

Tuesday, September 22, 10:30AM (New York, NY time)

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.

Slides of the talk

Video of the talk

Tuesday, September 29, 10:30AM (New York, NY time)

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.

Video of the talk

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

Tuesday, October 20, 10:30AM (New York, NY time)

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.

Video of the talk

Tuesday, October 27, 10:30AM (New York, NY time)

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.

Slides of the talk

Video of the talk

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).

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

Tuesday, November 17, 10:30AM (New York, NY time)

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.

Slides of the talk

Video of the talk

Tuesday, November 24, 10:30AM (New York, NY time)

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

Tuesday, December 8, 10:30AM (New York, NY time)

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.

Slides of the talk

Video of the talk

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.

Video of the talk