Online Asymptotic Geometric Analysis Seminar

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

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.

Slides of the talk

Video of the talk

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.

Slides of the talk

Video of the talk

Tuesday, January 26, 2021, 10:30AM (New York, NY time)

Konstantin Drach, Aix-Marseille Universite, Marseille, France

Topic: Reversing classical inequalities under curvature constraints

Abstract: A convex body $K$ is called uniformly convex if all the principal curvatures at every point along its boundary are bounded by a given constant $\lambda > 0$ either above (\textit{$\lambda$-concave} bodies), or below (\textit{$\lambda$-convex} bodies). We allow the boundary of $K$ to be non-smooth, in which case the bounds on the principal curvatures are defined in the barrier sense. Under uniform convexity assumption, for convex bodies of, say, given volume there are non-trivial upper and lower bounds for various functionals, such as the surface area, in- and outer-radius, diameter, width, etc. The bound in one direction usually constitutes the classical inequality (for example, the lower bound for the surface area is the isoperimetric inequality). The bound in another direction becomes a well-posed and in many cases highly non-trivial \emph{reverse optimization problem}. In the talk, we will give an overview of the results and open questions on the reverse optimization problems under curvature constraints in various ambient spaces.

Video of the talk

Slides of the talk

Tuesday, February 2, 2021, 11:30AM (New York, NY time) - Note the special time an hour later!

Rachel Greenfeld, UCLA, Los-Angeles, CA, USA

Topic: Translational tilings: structure and decidability

Abstract: Let F be a finite subset of Z^d. We say that F is a translational tile of Z^d if it is possible to cover Z^d by translates of F with no overlaps. Given a finite subset F of Z^d, could we determine whether F is a translational tile in finite time? Suppose that F does tile, does it admit a periodic tiling? A well known argument of Wang shows that these two questions are closely related. In the talk, we will discuss the relation between periodicity and decidability; and present some new results, joint with Terence Tao, on the rigidity of tiling structures in Z^2, and their applications to decidability.

Slides of the talk

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).

Video of the talk

Slides of the talk

Tuesday, February 16, 2021, 10:30AM (New York, NY time)

Han Huang, Georgia Institute of Technology, Atlanta, GA, USA

Topic: Rank of Sparse Bernoulli Matrices.

Abstract: Let A be an n by n Bernoulli(p) matrix with p satisfies 1<= pn/ log(n) < +infty. For a fixed positive integer k, the probability that (n-k+1)-th singular value of A equals 0 is (1+o(1)) of the probability that A contains k zero columns or k zero rows.

Video of the talk

Slides of the talk

Tuesday, February 23, 2021, 10:30AM (New York, NY time)

No seminar: this conference intersects with the seminar time.

Tuesday, March 2, 10:30AM (New York, NY time)

Maria de los Angeles Alfonseca-Cubero, North Dakota State University, Fargo, ND, USA

Topic: Solutions to the 5th and 8th Busemann-Petty problems near the ball

Abstract: In this talk we apply classical harmonic analysis tools, such as singular integrals and maximal functions, to two Busemann-Petty problems.

Video of the talk

Whiteboard of the talk

Tuesday, March 9, 2021, 10:30AM (New York, NY time)

Tselil Schramm, Stanford University, Palo Alto, CA, USA

Topic: Computational Barriers to Estimation from Low-Degree Polynomials

One fundamental goal of high-dimensional statistics is to detect and recover structure from noisy data. But even for simple settings (e.g. a planted low-rank matrix perturbed by noise), the computational complexity of estimation is sometimes poorly understood. A growing body of work studies low-degree polynomials as a proxy for computational complexity: it has been demonstrated in various settings that low-degree polynomials of the data can match the statistical performance of the best known polynomial-time algorithms for detection. But prior work has failed to address settings in which there is a "detection-recovery gap" and detection is qualitatively easier than recovery. In this talk, I'll describe a recent result in which we extend the method of low-degree polynomials to address recovery problems. As applications, we resolve (in the low-degree framework) open problems about the computational complexity of recovery for the planted submatrix and planted dense subgraph problems. Based on joint work with Alex Wein.

Video of the talk

Whiteboard of the talk

Tuesday, March 16, 2021, 10:30AM (New York, NY time)

No seminar: this conference intersects with the seminar time.

Tuesday, March 23, 2021, 10:30AM (New York, NY time)

No seminar: this conference intersects with the seminar time.

Tuesday, March 30, 2021, 10:30AM (New York, NY time)

Karoly Boroczky, Central European University, Budapest, Hungary

Topic: Stability of the Prekopa-Leindler inequality and the unconditional Logarithmic Brunn-Minkowski Inequality

Abstract: Recent results about the stability of the Prekopa-Leinder inequality (with Apratim De in the log-concave case, and with Alessio Figalli and Joao Goncalves in general) are discussed. As a consequence, stability of the Logarithmic Brunn-Minkowski Inequality under symmetries of a Coxeter group is obtained.

Video of the talk

Whiteboard of the talk

Tuesday, April 6, 10:30AM (New York, NY time)

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.

Video of the talk

Tuesday, April 13, 2021, 10:30AM (New York, NY time)

Vishesh Jain, Stanford University, Palo Alto, CA, USA

Topic: Singularity of discrete random matrices

Abstract: Let $M_n$ be an $n\times n$ random matrix whose entries are i.i.d copies of a discrete random variable $\xi$. It has been conjectured that the dominant reason for the singularity of $M_n$ is the event that a row or column of $M_n$ is zero, or that two rows or columns of $M_n$ coincide (up to a sign). I will discuss recent work, joint with Ashwin Sah (MIT) and Mehtaab Sawhney (MIT), towards the resolution of this conjecture.

Video of the talk

Slides of the talk

Tuesday, April 20, 2021, 11:30AM (New York, NY time)

Anindya De, University of Pennsilvania

Topic: Convex influences and a quantitative Gaussian correlation inequality.

Abstract: The Gaussian correlation inequality (GCI), proven by Royen in 2014, states that any two centrally symmetric convex sets (say K and L) in the Gaussian space are positively correlated. We will prove a new quantitative version of the GCI which gives a lower bound on this correlation based on the "common influential directions" of K and L. This can be seen as a Gaussian space analogue of Talagrand's well known correlation inequality for monotone functions. To obtain this inequality, we propose a new approach, based on analysis of Littlewood type polynomials, which gives a recipe to transfer qualitative correlation inequalities into quantitative correlation inequalities. En route, we also give a new notion of influences for convex symmetric sets over the Gaussian space which has many of the properties of influences from Boolean functions over the discrete cube. Much remains to be explored, in particular, about this new notion of influences for convex sets. Based on joint work with Shivam Nadimpalli and Rocco Servedio.

Video of the talk

Slides of the talk

Tuesday, April 27, 2021, 10:30AM (New York, NY time)

Almut Burchard, University of Toronto, Canada

Topic: On isodiametric capacitor problems related to aggregation

Abstract: I will describe recent work with Rustum Choksi and Elias Hess-Childs on the strong-attraction limit of a class of non-local shape optimization problems, which serve as toy models for the formation of flocks. In these models, a cloud of particles arranges itself according to forces between pairs of particles that depend on their distance: Neighboring particles repel each other, while at long distance the force is attractive. In the strong-attraction limit, the shape optimization problem amounts to maximizing the capacity of a body, subject to a diameter constraint. Clearly, maximizers are bodies of constant width --- but what is their shape?

Video of the talk

Tuesday, May 4, 2021, 10:30AM (New York, NY time)

Gregory Wyatt, University of Missouri, Columbia, MO, USA (talk 1, 20 minutes)

Topic: Inequalities for the Derivatives of the Radon Transform on Convex Bodies

Abstract: It has been shown that the sup-norm of the Radon transform of a probability density defined on an origin-symmetric convex body of volume 1 is bounded from below by a positive constant that depends only on the dimension. Using Fourier analysis, we extend this estimate to the derivatives of the Radon transform. We also provide a comparison theorem for these derivatives.

Video of the talk

Slides of the talk

Sudan Xing, University of Alberta, Canada (talk 2, 20 minutes)

Topic: The general dual-polar Orlicz-Minkowski problem

Abstract: In this talk, the general dual-polar Orlicz-Minkowski problem will be presented, which is �polar" to the recently initiated general dual Orlicz-Minkowski problem and �dual" to the newly proposed polar Orlicz-Minkowski problem. In particular, the existence, continuity and uniqueness of the solutions for the general dual-polar Orlicz-Minkowski problem will be presented. This talk is based on a joint work with Professors Deping Ye and Baocheng Zhu.

Video of the talk

Slides of the talk

Tuesday, May 11, 2021, 10:30AM (New York, NY time)

Sang Woo Ryoo, Princeton University, NJ, USA

Topic: A sharp form of Assouad's embedding theorem for Carnot groups

Abstract: Assouad's embedding theorem, which embeds snowflakes of doubling metric spaces into Euclidean spaces, has recently been sharpened in many different aspects. Following the work of Tao, which establishes an optimal Assouad embedding theorem for the Heisenberg group, we establish it for general Carnot groups. One main tool is a Nash--Moser type iteration scheme developed by Tao, which we extend into the setting of Carnot groups. The other tool, which is the main novelty of this paper, is a certain orthonormal basis extension theorem in the setting of general doubling metric spaces. We anticipate that this latter tool could be used for other applications.

Tuesday, May 18, 2021, 10:30AM (New York, NY time)

Maria Angeles Hernandez Cifre, Universidad de Murcia, Spain

Topic: On the roots of polynomials with log-convex coefficients

Abstract: In the spirit of the work developed for the Steiner polynomial of convex bodies, we investigate geometric properties of the roots of a general family of n-th degree polynomials closely related to that of dual Steiner polynomials of star bodies, deriving, as a consequence, further properties for the roots of the latter. We study the structure of the set of roots of such polynomials, showing that it is a closed convex cone in the upper half-plane, which covers its interior when n tends to infinity, and giving its precise description for every natural n\geq 2. This is a joint work with J. Yepes-Nicol�s and M. T�rraga.

Video of the talk

Tuesday, May 25, 2021, 11:30AM (New York, NY time)

Alexander Litvak, University of Alberta, AB, Canada

Topic: New bounds on the minimal dispersion

Abstract: We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse. Some of our bounds are sharp up to logarithmic factors. The talk is partially based on a joint work with G. Livshyts.

Video of the talk

Tuesday, June 1, 2021, 10:30AM (New York, NY time)

Ronen Eldan, Weizmann Institute of Science, Rehovot, Israel

Topic: A simple approach to chaos Sherrington-Kirkpatrick model and p-spin models of spin glasses

Abstract: Let G be an n by n matrix of i.i.d standard Gaussians, and consider the maximizer $v(G)$ of the expression $v^T G v$ among all sign vectors $v \in \{-1,1\}^n$. How stable is $v(G)$ under small perturbations of $G$? In 2018, Chen, Handschy and Lerman showed that the corresponding Gaussian field exhibits Chaos in the sense that perturbations of $G$ going to $0$ with the dimension amount to the corresponding maximizers $v(G)$ becoming almost uncorrelated (following Chatterjee '08, this also implies that the corresponding Gaussian field is super-concentrated). Their proof relies heavily on the framework which stems from the cavity method. We will explain an (arguably) much simpler proof which mostly uses classical results in convexity. Our proof also generalizes to mixed p-spin models.

Video of the talk

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

Boaz Slomka, The Open University of Israel, Raanana, Israel

Topic: Discrete variants of Brunn-Minkowski type inequalities

Abstract: I will discuss a family of discrete Brunn-Minkowski type inequalities. As particular cases, this family includes the four functions theorem of Ahlswede and Daykin, a result due to Klartag and Lehec, and other variants, both known and new, Two proofs will be outlined, the first is an elementary short proof and the second is a transport proof which extends a result due to Gozlan, Roberto, Samson and Tetali, and which implies stronger entropic versions of our inequalities. Partly based on joint work with Diana Halikias and Bo'az Klartag

Video of the talk

Schedule Fall 2021:

Tuesday, August 31, 2021, 10:30AM (New York, NY time)

Bernardo Gonzalez Merino, University of Murcia, Spain

Topic: On optimal approximation of functions by log-polynomials

Abstract: Lasserre in 2015 proved that given any $n$-dimensional compact set $K$ there exists a unique $d$-homogeneous polynomial on $n$-variables $g_d$, $d$ even, such that $K\subset G_1(g_d)=\{x\in\R^n:g_d(x)\leq 1\}$ minimizing $|G_1(g)|$ among all polynomials fulfilling such property $K\subset G_1(g)$. In particular, $G_1(g_2)$ coincides with the L�wner ellipsoid of $K$. In this talk, we will explain how to extend (in two different ways) Lasserre's result to some functional settings. Second, we will prove a theorem characterizing the minimizing functions by means of some touching conditions. Finally, we will also comment on some bounds about the corresponding $d$-outer volume (and integral) ratio between the approximating body (or function) and the original set (or function). This is a joint work together with David Alonso-Gutirrez and Rafael Villa.

Video of the talk

Slides of the talk

Tuesday, September 7, 2021, 10:30AM (New York, NY time)

No seminar (conference in Germany at this time)

Tuesday, September 14, 2021, 10:30AM (New York, NY time)

Nir Lev, Bar Ilan University, Ramat Gan, Israel

Topic: Fuglede's tiling-spectrality conjecture for convex domains

Abstract: Which domains in Euclidean space admit an orthogonal basis of exponential functions? For example, the cube is such a domain, but the ball is not. In 1974, Fuglede made a fascinating conjecture that these domains could be characterized geometrically as the domains which can tile the space by translations. While this conjecture was disproved for general sets, in a recent paper with Máté Matolcsi we did prove that Fuglede's conjecture is true for convex domains in all dimensions. I will survey the subject and discuss this result.

Tuesday, September 21, 2021, 10:30AM (New York, NY time)

Tomasz Tkocz, Carnegie Melon University, Pittsburgh, PA, USA

Topic: Rademacher-Gaussian tail comparison for complex coefficients

Abstract: We shall present a generalisation of Pinelis Rademacher-Gaussian tail comparison to complex coefficients. Based on joint work with Chasapis and Liu.

Video of the talk

Slides of the talk

Tuesday, September 28, 2021, 10:30AM (New York, NY time)

Sophie Huberts, CWI, Amsterdam, Netherlands

Topic: Combinatorial Diameter of Random Polyhedra

Abstract: The long-standing polynomial Hirsch conjecture asks if the combinatorial diameter of any polyhedron can be bounded by a polynomial of the dimension and number of facets. Inspired by this question, we study the combinatorial diameter of two classes of random polyhedra. We prove nearly-matching upper and lower bounds, assuming that the number of facets is very large compared to the dimension. One key ingredient is a proof that the complexity of a two-dimensional projection of a random polyhedron is concentrated around the mean. Both analyzing the diameter of random polyhedra and this concentration estimate were 34-year old open research questions first posed by Borgwardt.

Video of the talk

Slides of the talk

Tuesday, October 5, 2021, 10:30AM (New York, NY time)

No seminar because of time conflict with Nirenberg lectures by Klartag and Chen

Tuesday, October 12, 2021, 10:30AM (New York, NY time)

Juan-Carlos Alvarez Paiva, University of Lille, France

Topic: The asymmetric case of Hilbert's fourth problem

Abstract: Hilbert's fourth problem asks to study and construct the class of metrics on open convex subsets of projective $n$-space for which projective lines are geodesics. The problem is often considered solved (by Busemann, Pogorelov, and Szabo), but it is often forgotten that Hilbert also had in mind asymmetric metrics such as those that arise in Minkowski's geometry of numbers (i.e., asymmetric norms) and, indeed, apart from the thesis of Hamel (under Hilbert's supervision), nothing has been done on this problem until recently. In this work I'll explain how classic results in convex geometry together with some easy symplectic geometry can be used to solve this problem in various interesting cases. In particular, we will see how to construct all continuous asymmetric metrics on $n$-dimensional projective space for which projective lines are geodesics.

Video of the talk

Slides of the talk

Tuesday, October 19, 2021, 10:30AM (New York, NY time)

Sinai Robins, Universidade de Sao Paolo, Brazil

Topic: The null set of a of a polytope, and the Pompeiu property for polytopes

Abstract: We study the null set $N(\P)$ of the Fourier transform of a polytope P in $\R^d$, and we find that this null set does not contain (almost all) circles in $\R^d$. As a consequence, the null set does not contain the algebraic varieties $\{ z \in \C^d \mid z_1^2 + \cdots + z_d^2 = \alpha \}$, for each fixed $\alpha \in \C$. In 1929, Pompeiu asked the following question. Suppose we have a convex subset P in R^d, and a function f, defined over R^d, such that the integral of f over P vanishes, and all of the integrals of f, taken over each rigid motion of P, also vanish. Does it necessarily follow that f = 0? If the answer is affirmative, then the convex body P is said to have the Pompeiu property. It is a conjecture that in every dimension, balls are the only convex bodies that do not have the Pompeiu property. Here we get an explicit proof that the Pompeiu property is true for all polytopes, by combining our work with the work of Brown, Schreiber, and Taylor from 1973. Our proof uses the Brion-Barvinok theorem in combinatorial geometry, together with some properties of the Bessel functions. The original proof that polytopes (as well as other bodies) possess the Pompeiu property was given by Brown, Schreiber, and Taylor (1973) for dimension 2. In 1976, Williams observed that the same proof also works for $d>2$ and, using eigenvalues of the Laplacian, gave another proof valid for $d \geq 2$ that polytopes indeed have the Pompeiu property. The null set of the Fourier transform of polytopes has also been used by various people to tackle problems in multi-tiling Euclidean space. Thus, the null set of a polytope is interesting for several applications, including recent discrete versions of this problem. This is joint work with Fabr�cio Caluza Machado.

Video of the talk

Slides of the talk

Tuesday, October 26, 2021, 10:30AM (New York, NY time)

Joe Neeman, UT Austin, TX, USA

Topic: Vector-valued noise stability and quantum max-cut

Abstract: Borell's inequality characterizes the boolean-valued functions on Gaussian space that have extremal Gaussian noise stability. We prove a similar inequality for functions that take values in the sphere. We will also briefly discuss an application to the computational complexity of approximating certain quantum ground states. This is a joint work with Yeongwoo Hwang, Ojas Parekh, Kevin Thompson and John Wright

Slides of the talk

Video of the talk

Tuesday, November 2, 2021, 10:30AM (New York, NY time)

Margalit Glasgow, Stanford University, Palo Alto, USA

Topic: Combinatorial Characterization of Rank of Bernoulli Random Matrices

Abstract: We study the rank of the adjacency matrix A of a random Erdos Reyni graph G~G(n,p). It is well known that when p < log(n)/n, with high probability, A is singular. We prove that when p = omega(1/n), with high probability, the corank of A is equal to the number of isolated vertices remaining in G after the Karp Sipser leaf-removal process, which removes vertices of degree 1 and their unique neighbor. We discuss related results for the asymmetric Bernoulli random matrix, and an application of our techniques to show that the 3-core of G is non-singular with high probability.

Video of the talk

Slides of the talk

Tuesday, November 9, 2021, 10:30AM (New York, NY time)

Eli Putterman, Weizmann Institute of Science, Rehovot, Israel

Topic: Dynamic Gamma-calculus and spectral monotonicity under heat flow

Abstract: We introduce a new, time-inhomogeneous ("dynamic") form of the Bakry-Emery $\Gamma$-calculus, which is closely related to stochastic localization. As an application, we show that the Poincare constant of a log-concave measure in Euclidean space is monotone increasing along the heat flow, and in fact, that the entire spectrum of the associated Laplace operator is monotone decreasing. Time permitting, we will discuss another proof following the approach of Kim and Milman, which derives the spectral monotonicity from the existence of a contracting transport map, and perhaps mention recent extensions of these results to the discrete setting. Joint work with B. Klartag.

Slides of the talk

Video of the talk

Tuesday, November 16, 2021, 10:30AM (New York, NY time)

Andreas Bernig, Frankfurt, Germany

Topic: Intrinsic volumes on Kaehler manifolds

Abstract: The classical Steiner formula for the volume of parallel bodies is the easiest way to define intrinsic volumes of convex bodies. It admits a differential-geometric analogue, Weyl's tube formula, which applies to submanifolds in euclidean spaces. Surprinsingly, the coefficients in Weyl's tube formula only depend on the intrinsic geometry of the submanifold, and not on the embedding. They are also called "intrinsic volumes". In Alesker's modern framework of "valuations on manifolds", both notions of intrinsic volume really are the same. Weyl's theorem can then be rephrased by saying that for every riemannian manifold, there is a canonical algebra of valuations (the Lipschitz-Killing algebra), and this assignment is compatible with isometric embeddings. In a joint work with Joe Fu, Thomas Wannerer, and Gil Solanes, we give a complex version of this theorem. It states that for each Kaehler manifold, there is a canonical algebra of valuations (the Kaehler-Lipschitz-Killing algebra), and this assignment is compatible with holomorphic isometric embeddings. One consequence is that the kinematic formulas on flat hermitian spaces and complex projective spaces are formally the same.

Slides of the talk

Video of the talk

Tuesday, November 23, 2021, 10:30AM (New York, NY time)

Christos Saroglou, University of Ioannina, Greece

Topic: A non-existence result for the $L_p$-Minkowski problem

Abstract: We show that given a real number $p<1$, a positive integer $n$ and a proper subspace $H$ of $\mathbb{R}^n$, the measure on the Euclidean sphere $\mathbb{S}^{n-1}$, which is concentrated in $H$ and whose restriction to the class of Borel subsets of $ \mathbb{S}^{n-1} \cap H$ equals the spherical Lebesgue measure on $ \mathbb{S}^{n-1} \cap H$, is not the $L_p$-surface area measure of any convex body. This, in particular, disproves a conjecture from [Bianchi, B\"or\"oczky, Colesanti, Yang, The $L_p$-Minkowski problem for $-n

Slides of the talk

Video of the talk

Tuesday, November 30, 2021, 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..

Slides of the talk

Video of the talk

Tuesday, December 7, 2021, 10:30AM (New York, NY time)

Stefan Steinerberger, University of Washington, Seattle, WA

Topic: Mean-Value Inequalities for Harmonic Functions

Abstract: The mean-value theorem for harmonic functions says that we can bound the integral of a harmonic function in a ball by the average value on the boundary (and, in fact, there is equality). What happens if we replace the ball by a general convex or even non-convex set? As it turns out, this very simple question has connections to classical potential theory, probability theory, PDEs and even mechanics: one of the arising questions dates back to Saint Venant (1856) and was investigated using a specially built soap bubble machine in the 1920s. There are some fascinating new isoperimetric problems: for example, the worst case convex domain in the plane seems to look a lot like the letter "D" but we cannot prove it. I will discuss some recent results and many open problems.

Slides of the talk

Video of the talk

Schedule Spring 2022:

Tuesday, January 25, 2022, 10:30AM (New York, NY time)

Matthieu Fradelizi, Universite Gustave Eiffel, Paris, France

Topic: Volume of sums of convex sets, mixed volumes, determinants and zonoids.

Abstract: Following, on one hand, analogies between information theory, determinants and mixed volumes and, on the other hand, the relationship between algebraic geometry and convex geometry, many inequalities between mixed volumes of sums of convex sets were conjectured these years like Bezout or Alexandrov-Fenchel's type inequalities. Weak forms of these conjectures imply comparisons of volumes of projections of convex sets. We show these relationships, analyse these conjectures, give counter-examples and partial proofs. In particular we prove these conjectures for zonoids. Work in collaboration with Mokshay Madiman, Mathieu Meyer and Artem Zvavitch.

Slides of the talk

Video of the talk

Tuesday, February 1, 2022, 10:30AM (New York, NY time)

Ahmed El Alaoui, Cornell University, Ithaka, NY

Topic: Algorithmic Stochastic Localization for the Sherrington-Kirkpatrick Model

Abstract: We propose an algorithm which efficiently samples from the SK measure with no external field at all inverse temperatures beta < 1/2. The approach uses a discretized version of the Stochastic Localization (SL) process of Eldan (2013), and the analysis relies on a comparison with a planted model combined with a new information-theoretic interpretation of the SL process. We believe this algorithm should succeed for all beta<1. Finally, we show that due to disorder chaos, 'stable' algorithms cannot approximately sample from the SK measure for beta>1. This result, which pertains to sampling, parallels the use of the overlap gap property to show algorithmic impossibility results for random optimization problems. This is a joint work with Andrea Montanari and Mark Sellke.

Slides of the talk

Video of the talk

Tuesday, February 8, 2022, 10:30AM (New York, NY time)

No seminar because of the clash with this workshop

Tuesday, February 15, 2022, 10:30AM (New York, NY time)

Max Fathi, Etablissement public experimental - Decret No2019-209 du 20 mars 2019, France

Topic: Stability of the spectral gap under a curvature-dimension condition

Abstract: A theorem of Lichnerowicz (1958) states that the spectral gap (or sharp Poincare constant) of a smooth n-dimensional Riemannian manifold with curvature bounded from below by n-1 is bounded by n, which is the spectral gap of the unit n-sphere. This bound has since been extended to metric-measure spaces satisfying a curvature-dimension condition. In this talk, I will present a result on stability of the bound: if a space has almost minimal spectral gap, then the pushforward of the volume measure by a normalized eigenfunction is close to a Beta distribution with parameter n/2, with a sharp estimate on the L1 optimal transport distance. Joint work with Ivan Gentil and Jordan Serres.

Slides of the talk

Video of the talk

Tuesday, February 22, 2022, 10:30AM (New York, NY time)

Zakhar Kabluchko, Institut fur Mathematische Stochastik, Munster, Germany

Topic: Expected face numbers of random beta polytopes

Abstract: Let $X_1,\ldots, X_n$ be independent random points in the $d$-dimensional unit ball with density proportional to $(1-\|x\|^2)^\beta$, where $\beta>-1$ is a parameter. For $\beta=0$ we recover the uniform distribution on the unit ball, the limiting case $\beta \to -1$ corresponds to the uniform distribution on the unit sphere, while the case $\beta\to \infty$ corresponds to the standard Gaussian distribution. The convex hull $[X_1,\ldots,X_n]$ is called the beta polytope (with parameters $n$, $d$, $\beta$). We shall review results on the expected number of $k$-dimensional faces of beta polytopes and two closely related classes of polytopes called beta' and the beta* polytopes. Several objects in stochastic geometry such as the typical cell of the Poisson-Voronoi tessellation or the zero cell of the homogeneous Poisson hyperplane tessellation (in Euclidean space or on the sphere) are related to beta' polytopes, while their analogues in the hyperbolic space are related to beta* polytopes. The expected face numbers of these polytopes can be computed exactly.

Slides of the talk

Video of the talk

Tuesday, March 1, 2022, 10:30AM (New York, NY time)

Chiara Meroni, Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

Topic: Semialgebraicity and constructions with convex bodies

Abstract: Convex geometry has been classically studied from an analytical point of view. In the past two decades, there has been increasing interest in approaching it using tools from real and complex algebraic geometry, with a focus on semialgebraic convex bodies, beyond polytopes. I will introduce some notions and objects that encode this interaction and analyze their behavior in two different constructions: the fiber body of a convex body and the intersection body of a polytope. This is based on two joint works, one with Leo Mathis and one with Katalin Berlow, Marie-Charlotte Brandenburg and Isabelle Shankar.

Slides of the talk

Video of the talk

Tuesday, March 8, 2022, 10:30AM (New York, NY time)

Silouanos Brazitikos, University of Athens, Greece

Topic: On a version of the slicing problem for the surface area of convex bodies

Abstract: We study the slicing inequality for the surface area instead of volume. This is the question whether there exists a constant $\alpha_n$ depending (or not) on the dimension $n$ so that \begin{equation*}S(K)\ls\alpha_n|K|^{\frac{1}{n}}\max_{\xi\in S^{n-1}}S(K\cap\xi^{\perp })\end{equation*} where $S$ denotes surface area and $|\cdot |$ denotes volume. For any fixed dimension we provide a negative answer to this question, as well as to a weaker version in which sections are replaced by projections onto hyperplanes. We also study the same problem for sections and projections of lower dimension and for all the quermassintegrals of a convex body.

Slides of the talk

Video of the talk

Tuesday, March 15, 2022, 10:30AM (New York, NY time)

Stephanie Mui, Courant Institute, New York, NY, USA

Topic: On the Lp Alexandrov problem for negative p

Abstract: Huang, Lutwak, Yang, and Zhang introduced the Lp integral curvature and posed the corresponding Lp Aleksandrov problem, the natural Lp extension of the classical integral curvature and Aleksandrov problem respectively. The problem asks about the existence of a convex body with prescribed Lp integral curvature measure. For the f given even measures, the question will be solved for p ? (-1, 0). Furthermore, a sufficient measure concentration condition will be provided for the case of p = -1, again provided that the given measure is even.

Slides of the talk

Video of the talk

Tuesday, March 22, 2022, 10:30AM (New York, NY time)

Dario Cordero-Erasquin, Sorbonne Universite, Paris, France

Topic: Improved log-concavity for rotationally invariant measures of symmetric convex sets

Abstract: We prove that the (B) conjecture for dilates of symmetric convex sets and the Gardner-Zvavitch conjecture on dimensional Brunn-Minkowski inequalities are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case of log-concave measures, for instance to Cauchy measures as well. For the proof, new sharp spectral inequalities (in particular a sharp weighted Poincar� inequality) are obtained for even probability measures that are log-concave with respect to a rotationally invariant measure. Joint work with Liran Rotem.

Video of the talk

Tuesday, March 29, 2022, 10:30AM (New York, NY time)

Katherina von Dichter, Technische Universitat Munchen, Germany

Topic: Mean inequality for symmetrizations of convex bodies

Abstract: We deal with four symmetrizations of a convex set $C$: the intersection, the harmonic and the arithmetic mean, and the convex hull of $C$ and $-C$. A well-known result of Firey shows that those means build up a subset-chain in the given order. We determine the dilatation factors, depending on the asymmetry of $C$, to reverse the containments between any of those symmetrizations, and tighten the relations proven by Firey and show a stability result concerning those factors near the simplex..

Slides of the talk

Video of the talk

Tuesday, April 5, 2022, 10:30AM (New York, NY time)

Theo McKenzie, University of Berkeley, CA

Topic: Many nodal domains in random regular graphs

Abstract: Discrete graphs have been used as a model for quantum chaos for over 20 years, and it is conjectured that eigenvectors of large regular graphs have Gaussian statistics. If we partition a graph according to the positive and negative components of an eigenvector of the adjacency matrix, the resulting connected subcomponents are called nodal domains, and one consequence of Gaussian behavior would be that there are many nodal domains. Dekel, Lee, and Linial observed that according to simulations, most eigenvectors of the adjacency matrix of random regular graphs have many nodal domains, unlike dense Erdos-R�nyi graphs. In this talk, we show that for the most negative eigenvalues of the adjacency matrix of a random regular graph, there is an almost linear number of nodal domains. Joint work with Shirshendu Ganguly, Sidhanth Mohanty, and Nikhil Srivastava..

Slides of the talk

Video of the talk

Tuesday, April 12, 2022, 10:30AM (New York, NY time)

Gergely Ambrus, Alfred Renyi Institute of Mathematics, and the University of Szeged, Budapest, Hungary

Topic: A generalization of Bang's lemma

Abstract: Plank problems have received a surge of attention recently. In this talk, I will present a generalization of Bang's lemma, which is one of the crucial tools of the subject. This leads to an extension of Kadets' theorem on covering systems, several results related to translative coverings, and further applications to plank problems. .

Slides of the talk

Video of the talk

Tuesday, April 19, 2022, 10:30AM (New York, NY time)

Dan Mikulincer, MIT, Boston, MS

Topic: Causal optimal transport and the Brownian transport map

Abstract: We consider a version of the optimal transport problem, adapted to the filtration of the Wiener space and show that the optimal solution is supported on a graph of a function, dubbed the Brownian transport map. Using tools from Ito's and Malliavin's calculus, we show that the map is Lipschitz when the target measure satisfies appropriate convexity assumptions. This facilitates the proof of several new functional inequalities. In other settings, where a globally Lipschitz transport map cannot exist, we derive Sobolev estimates which are intimately connected to the KLS conjecture. Joint work with Yair Shenfeld.

Slides of the talk

Video of the talk

Tuesday, April 26, 2022, 10:30AM (New York, NY time)

Mokshay Madiman, University of Delaware, US

Topic: Submodularity and fractional subadditivity in geometric functional analysis and information theory

Abstract: We consider two interesting properties of natural set functions that arise in geometric functional analysis and information theory, motivated by the analogies between these fields. Specifically, we consider submodularity and fractional subadditivity properties of functionals such as the volume of Minkowski sums and the entropy of convolutions, sometimes but not always under convexity constraints. The talk will survey results obtained over the last 15 years with a number of collaborators, including Barron, Barthe, Fradelizi, Ghassemi, Marsiglietti, Tetali, and Zvavitch.

Slides of the talk

Video of the talk

Tuesday, May 3, 2022, 10:30AM (New York, NY time)

Joseph Lehec, Paris-Dauphine, France

Topic: Bourgain's slicing problem and KLS isoperimetry up to polylog

Abstract: In a recent joint work with Bo'az Klartag we show that the Kannan, Lovasz, Simonovits conjecture holds true up to a polylog factor in the dimension. This also implies a polylog bound for Bourgain's hyperplane conjecture by a previous result of Eldan and Klartag. In this talk I'll explain the main steps of the proof.

Slides of the talk

Video of the talk

Tuesday, May 10, 2022, 10:30AM (New York, NY time)

Simon Larson, Chalmers University of Techlology, Gothenburg, Sweden

Topic: An inequality for the normal derivative of the Lane-Emden ground state

Abstract: For an open subset of ℝ^d we consider minimizers of the Dirichlet integral among L^q-normalized functions with 1 ≤ q ≤ 2. In this talk I will discuss a sharp lower bound on the L²-norm of the normal derivative in terms of the corresponding energy. The results are valid for arbitrary bounded open Lipschitz sets, and for particular values of q generalize bounds that had previously been obtained under the additional assumption that the set is convex. Based on joint work with Rupert Frank.

Slides of the talk

Video of the talk

Tuesday, May 17, 2022, 10:30AM (New York, NY time)

Jacob Henkel, Friedrich-Schiller-Universitat Jena, Germany

Topic: On some new examples and (non)-existence of Minkowski valuations with certain properties

Abstract: Let $W$ be an irreducible representation of $\mathrm{SL}(n)$ of finite dimension. We ask whether there exists a non trivial continuous and translation invariant Minkowski valuation $\mathcal{K}(\mathbb{R}^n) \to \mathcal{K}(W)$ which is $\mathrm{SL}(n)$ equivariant. Here $\mathcal{K}(\bullet)$ denotes the space of convex bodies in the corresponding vector space. For $W = \mathbb{R}^n$ and $W = (\mathbb{R}^n)^*$ the answer is positive and by a result of Monika Ludwig all such valuations are multiples of the difference body and the projection body respectively. We show that $W = \mathbb{R}$ is the only additional case where such a valuation exists. New examples arise if we omit translation invariance. This is a joint work with Thomas Wannerer.

Video of the talk

Tuesday, May 24, 2022, 10:30AM (New York, NY time)

No seminar because of the clash with this workshop

Schedule Fall 2022:

Tuesday, August 30, 2022, 10:30AM (New York, NY time)

Dmitri Burago, Penn State University

Topic: Something like a Mozaic from Geometry, Dynamics, PDEs, and maybe more.

Abstract: I give very different talks under the same title, this leaves me flexibility. I never know in advance what I want to talk about:) I am going to begin with some unsolved problems, which, in my opinion, deserve more attention that they receive. For some problems, there are partial solutions, which I would try to sketch or at least discuss. I plan then to go to problems where the progress is more substantial, as time permits. What I plan is indeed a mozaic. I am not sure how many topics I would be able to touch upon and which ones would be more of interest, my collaborators include D. Chen, S. Ivanov, B. Kleiner, Ya. Kurylev, M. Lassas, J. Lu, A. Novikov, L. Polterovich. Maybe, I would concentrate оn two or three topics, maybe many more short stories, depending on the reaction.

Slides of the talk

Video of the talk

Tuesday, September 6, 2022, 10:30AM (New York, NY time)

Eduardo Lucas, University of Murcia

Topic: On discrete Brunn-Minkowski type inequalities

Abstract: Finding discrete analogues of well-known inequalities in convex geometry has gained traction in recent decades. In this talk, we will discuss recent discrete results concerning the Brunn-Minkowski inequality, involving both the cardinality measure and, specially, the lattice point enumerator measure, defined as the cardinality of the intersection of a bounded set with the integer lattice. We will also consider extensions to the L_p setting, both for p>1 and the more limited 0\leq p<1. The inequalities will be related to their continuous analogues, and a few additional consequences will be extracted as well. This is a joint work with Maria A. Hernandez Cifre and Jesus Yepes Nicolas.

Slides of the talk

Video of the talk

Tuesday, September 13, 2022, 10:30AM (New York, NY time)

No seminar, intersection with this ICERM workshop

Tuesday, September 20, 2022, 10:30AM (New York, NY time)

Sudan Xing, University of Alberta, Canada

Topic: On Multiple Lp curvilinear-Brunn-Minkowski inequality

Abstract: In this talk, the extension of the curvilinear summation for bounded Borel measurable sets to the Lp space for multiple power parameters is introduced. Based on the multiple Lp-curvilinear summation, we establish the multiple Lp curvilinear-Brunn-Minkowski inequality for bounded Borel measurable sets. We also present the proof of multiple Lp Borell-Brascamp-Lieb inequality as well as its normalized version for functions. This talk is based on the joint work with Dr. Michael Roysdon.

Slides of the talk

Video of the talk

Tuesday, September 27, 2022, 10:30AM (New York, NY time)

No seminar, intersection with this ICERM workshop

THURSDAY, October 6, 2022, 9AM (New York, NY time) -- note the special time!!!

Maud Szusterman, Universite de Paris, France

Topic: Bezout inequality with mixed volumes : is the simplex the only minimizer?

Abstract: A few years ago, C. Saroglou, I. Soprunov, and A. Zvavitch, could rephrase the Bezout inequality (between algebraic varieties) in terms of mixed volumes, yielding a set of inequalities for the simplex. Thanks to Diskant's inequality, these inequalities also hold for an arbitrary convex body K, up to a factor n. The authors conjectured that the simplex is characterized as the only convex body minimizing the ratio, and proved that this characterization holds among polytopes. In a subsequent paper, they introduced a similar, a priori larger ratio, for which the simplex is indeed the unique minimizer. While several classes of convex bodies (decomposable, smooth, ...) have been excluded from the set of possible minimizers of the smaller ratio, the conjecture remains open. I wish to present two conditions, which both exclude a body from being a minimizer : one is a dual condition, the other has geometric flavour, and involves isoperimetric ratios.

Slides of the talk

Video of the talk

Tuesday, October 11, 2022, 10:30AM (New York, NY time)

Huy Tuan Pham, Stanford University

Topic: Talagrand’s selector process conjecture and suprema of positive empirical processes

Abstract: Understanding suprema of stochastic processes is an important subject in probability theory with many applications. While much is known in the case of Gaussian processes thanks to Talagrand’s celebrated majorizing measure theorem, moving beyond the Gaussian case is a much more challenging quest. In this talk, I will discuss recent joint work with Jinyoung Park that resolves a conjecture of Talagrand on suprema of certain stochastic processes driven by sparse Bernoulli random variables (known as selector processes), and a question of Talagrand on general positive empirical processes. Combining with the recent resolution of the (generalized) Bernoulli conjecture, this gives the first steps towards the last missing piece in the study of suprema of general empirical processes. Our proof of Talagrand’s conjecture is combinatorial, and I will discuss how one of the ideas in our proof of Talagrand’s conjecture leads to the proof of the Kahn-Kalai conjecture, an important question in probabilistic combinatorics and random graph theory.

Slides of the talk

Video of the talk

Tuesday, October 18, 2022, 10:30AM (New York, NY time)

No seminar, intersection with this ICERM workshop

Tuesday, October 25, 2022, 10:30AM (New York, NY time)

Barbara Brandolini, University of Palermo, Italy

Topic: Isoperimetric sets for weighted twisted eigenvalues

Abstract: In this talk we discuss a shape optimization problem for the first twisted eigenvalue of the weighted operator $L=-\textrm{div}\left(\gamma(x)\nabla \right)$ among sets with fixed weighted measure. If $\gamma(x)$ satisfies some convenient assumptions, like in the cases $\gamma(x)=e^{-|x|^2}$ or $\gamma(x)=x_N^k \> (k \ge 0)$, we show that this eigenvalue is minimized by the union of two disjoint isoperimetric sets sharing the same measure. The talk is a based on a recent work in collaboration with Antoine Henrot, Anna Mercaldo and Maria Rosaria Posteraro.

Slides of the talk

Tuesday, November 1, 2022, 10:30AM (New York, NY time)

Kasia Wyczesany, Carnegie Melon University, Pittsburgh, PA, USA

Topic: Zoo of Dualities

Abstract: In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We prove that any order reversing quasi involution is of a special form, which arose from the consideration of optimal transport problem with respect to costs that attain infinite values. We will discuss how this unified point of view on order reversing quasi involutions helps to deeper the understanding of the underlying structures and principles. We will provide many examples and ways to construct new order reversing quasi involutions from given ones. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky.

Slides of the talk

Video of the talk

Tuesday, November 8, 2022, 10:30AM (New York, NY time)

Lu Wang, Yale University, New Haven, CT, US

Topic: Density of topologically nontrivial minimal cones

Abstract: In this talk, I will discuss some explicit lower bounds on the density of minimal cones of dimension less than 7 provided that the complements of the cone are topologically nontrivial in certain senses. This is joint with Jacob Bernstein.

Video of the talk

Tuesday, November 15, 2022, 10:30AM (New York, NY time)

Pazit Haim-Kislev, Tel-Aviv University

Topic: An isobarycentric problem

Abstract: In a joint work with Shoni Gilboa and Boaz Slomka we consider an isoperimetric type inequality: Given a finite Borel measure on R^n, which sets have maximal measure among all subsets with prescribed barycenter? As a corollary, we partially answer a question by Henk and Pollehn, which is equivalent to a special case of the Log-Minkowski inequality.

Slides of the talk

Video of the talk

Tuesday, November 22, 2022, 10:30AM (New York, NY time)

Ilya Molchanov, Institute of Mathematical Statistics and Actuarial Science (IMSV), Bern, Switzerland

Topic: Generalised convexity and related limit theorems

Abstract: The standard convex hull of a subset of Euclidean space is defined as the intersection of all images (under the action of a group of rigid motions) of a half-space containing the given set. We propose a generalisation of this classical notion, that we call a (K,H)-hull, and which is obtained from the above construction by replacing a half-space with some other convex closed subset K of the Euclidean space, and a group of rigid motions by a subset H of the group of invertible affine transformations. The main emphasis is on limit theorems for generalised convex hulls of random samples from K. The talk is based on recent works with Alexander Marynych and Zakhar Kabluchko.

Slides of the talk

Video of the talk

Tuesday, November 29, 2022, 10:30AM (New York, NY time)

Thomas Jahn, Catholic University of Eichstatt-Ingolstadt, Germany

Topic: On The Optimal Constants in the Two-Sided Stechkin Inequalities

Abstract: The $\ell_1$-norm of a monotonically decreasing sequence of nonnegative numbers can be sandwiched by a sum involving the best $n$-term approximation errors measured in the $\ell_q$-norm, suitably scaled by positive constants. But what are the optimal constants? In this talk, you will get in touch with the contributions of Copson, Stechkin, Pietsch, Temlyakov, and Bennett, the geometry behind the problem, and variants of the inequality, where the $\ell_1$-norm is replaced by its weak counterpart or where sums are replaced by integrals. This is joint work with Tino Ullrich.

Slides of the talk

Video of the talk

Tuesday, December 6, 2022, 10:30AM (New York, NY time)

Eliza O'Reilly, California Institute of Technology, Pasadena, USA

Topic: Some applications of mixed volumes in data science: random tessellation forests and optimal regularizers

Abstract: Many modern problems in data science aim to efficiently and accurately extract important features and make predictions from complex data sets. Naturally occurring structure in the data underpins the success of many contemporary approaches, but large gaps between theory and practice remain. In this talk I will discuss two different applications where efforts to provide theoretical guarantees and guidance for data analysis tasks has drawn connections to the Brunn-Minkowski theory of convex bodies. First I will present theoretical guarantees for a large class of random forest algorithms based on the theory of stationary random tessellations where the bounds depend on certain mixed volumes related to the geometry of the induced partitions. Second, I will discuss recent progress on finding optimal convex and non-convex regularizers for a given probability distribution modeling a data source. In particular, we are able to characterize an optimal star body regularizer by interpreting a relevant functional as a dual mixed volume. This talk is based on joint works with Ngoc Mai Tran, Oscar Leong, Yong Sheng Soh, and Venkat Chandrasekaran. .

Slides of the talk

Video of the talk

Tuesday, December 13, 2022, 10:30AM (New York, NY time)

Dongbin Li, University of Delaware, Newark, DE, USA

Topic: An information-theoretic approach to the Kneser-Poulsen conjecture in discrete geometry

Abstract: The Kneser-Poulsen conjecture in discrete geometry asserts that the volume of a union of balls in Euclidean space decreases if their centers are brought closer. In this talk, we will introduce an information-theoretic approach to tackle this problem. Our approach revolves around a broad question regarding whether Rényi entropies of independent sums decrease when one of the summands is contracted by a 1-Lipschitz map. We answer this broad question affirmatively in various cases. Specifically, we show that when $W$ is a radially symmetric log-concave random vector in $\R^d$, then for any random vector $X$ in $\R^d$ and any 1-Lipschitz map $T$, we have $h_2(T(X)+W) \leq h_2(X+W),$ which can be viewed as an entropic analogue of the Kneser-Poulsen conjecture. The talk is based on a joint work with Gautam Aishwarya, Irfan Alam, Sergii Myroshnychenko, and Oscar Zatarain-Vera.

Slides of the talk

Video of the talk

Schedule Spring 2023:

Thursday, January 19, 2023, 9:30AM (New York, NY time)

Francesco Chiacchio, University of Naples, Italy

Topic: Talenti type results for linear and nonlinear Robin problems

Abstract: We consider the p-Poisson equation with Robin boundary conditions, where the Robin parameter is a function. By means of some weighted isoperimetric inequalities, we provide various sharp bounds for the solutions to the problems under consideration. We also derive some Faber-Krahn type inequalities.

Slides of the talk

Thursday, January 26, 2023, 9:30AM (New York, NY time)

Steven Hoehner, Longwood University

Topic: An intrinsic volume metric for the class of convex bodies in $\mathbb{R}^n$

Abstract: A new intrinsic volume metric is introduced for the class of convex bodies in $\mathbb{R}^n$. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.

Slides of the talk

Thursday, February 2, 2023, 9:30AM (New York, NY time)

Victor Reis, University of Washington, Seattle, WA

Topic: The Vector Balancing Constant for Zonotopes

Abstract: The vector balancing constant vb(K,Q) of two symmetric convex bodies K,Q is the minimum r≥0 so that any number of vectors from K can be balanced into an r-scaling of Q. A question raised by Schechtman is whether for any d-dimensional zonotope one has vb(K,K) < O(sqrt(d)). Intuitively, this asks whether a natural geometric generalization of Spencer's Theorem (for which K is a cube) holds. We prove that for any d-dimensional zonotope one has vb(K,K) <= O(sqrt(d) * log log log d). Our main technical contribution is a tight lower bound on the Gaussian measure of any section of a normalized zonotope, generalizing Vaaler's Theorem for cubes.

Slides of the talk

Video of the talk

Thursday, February 9, 2023, 10:00AM (New York, NY time) -- NOTE THE SPECIAL TIME

Paul Simanjuntak, University of Missouri, Columbia, USA

Topic: A probabilistic approach to intersection bodies

Abstract: The Busemann intersection inequality is a fundamental isoperimetric inequality for intersection bodies. A distinguishing feature is that it holds even when the intersection body is not convex, but merely star-shaped. I will discuss two probabilistic approaches to approximating intersection bodies. The first involves radial sums of random ellipsoids, building on work of Goodey-Weil and Kalton-Koldobsky-Yaskin-Yaskina. The second is inspired by a construction of Anttilla, Ball, and Perissinaki, involving random approximation of slabs. Unifying both approaches is a representation of star bodies as special averages of convex sets. In turn, our methods suggest a new approach to proving isoperimetric inequalities for star-shaped set. Based on joint works with R. Adamczak, G. Paouris, and P. Pivovarov.

Slides of the talk

Video of the talk

Thursday, February 16, 2023, 9:30AM (New York, NY time)

Rotem Assouline, Weizmann Institute of Science, Israel

Topic: Brunn-Minkowski for path spaces on Riemannian surfaces

Abstract: the Minkowski average of two sets on a Riemannian manifold can be defined by replacing straight lines with geodesics. The Brunn Minkowski inequality is then equivalent to nonnegative Ricci curvature. We propose a generalization of this operation in which geodesics are replaced by an arbitrary family of curves. We show that horocycles in the hyperbolic plane satisfy the Brunn Minkowski inequality, in stark contrast to geodesics. Our main tool is needle decomposition. Joint work with Bo'az Klartag.

Slides of the talk

Video of the talk

Thursday, February 23, 2023, 10:00AM (New York, NY time) -- NOTE THE SPECIAL TIME

Petros Valettas, University of Missouri, Columbia

Topic: Conditional concentration for functions of high-dimensional random arrays

Abstract: The use of martingale methods for establishing concentration properties occupy central role in algorithmic discrete mathematics, in particular in situations that the functionals of study lack smoothness assumptions (e.g., Lipschitz conditions, bounds for the L_2 norm of the gradient, etc.) A prototypical example which highlights this utility is the so-called conditional concentration. In this talk we will discuss how we can extend this more combinatorial (rather than standard) form of concentration to functions of high-dimensional random arrays, which do not have necessarily independent entries but enjoy some symmetries. Based on a joint work with P. Dodos and K. Tyros (NKUA).

Slides of the talk

Video of the talk

Thursday, March 2, 2023, 9:30AM (New York, NY time)

Michael Roysdon, Brown University, Providence RI

Topic: Comparison Problems for the Radon Transform

Abstract: The classical Busemann-Petty Problem from the 1950s asked the following tomographic question: Assuming you have two origin-symmetric convex bodies $K$ and $L$ in the $n$-dimensional Euclidean space satisfying the volume inequality \[ |K \cap \theta^{\perp}| \leq |L \cap \theta^{\perp}| for all \theta \in S^{n-1}, \] does it follow that $|K| \leq |L|$? The answer is affirmative for $n \leq 4$ and negative whenever $n \geq 5$. However, if $K$ belongs to a certain class of convex bodies introduced by E. Lutwak in the 1980s, the intersection bodies, then the answer to the Busemann-Petty problem is affirmative in all dimensions. Several extensions of this result have been shown in the case of measures on convex bodies, and isomorphic results of the same type have been established. Moreover, the isomorphic Busemann-Petty problem is actually equivalent to the isomorphic slicing problem of Bourgain from the 1980s, which remains open to this day. In this talk, we will address the following comparison problem for the Radon transform: Given a pair of non-negative, even, continuous functions $\varphi, \psi$ defined on $\R^n$ and belonging to $L^p(\R^n)$, for some $p \geq 1$, satisfying the inequality $R\varphi \leq R \psi$, where $\R$ denotes the Radon transform, does it follows that $\|\varphi|_{L^p} \leq \|\psi\|_{L^p}$. As with the Busemann-Petty problem, to study this comparison problem in a meaningful way, one must introduce a special class of functions which emulate the class of intersection bodies. This is based on a joint work with Alexander Koldobsky and Artem Zvavitch.

Thursday, March 9, 2023, 9:30AM (New York, NY time)

Purvi Gupta, Indian Institute of Science, Bangalore.

Topic: Polyhedral-like approximations in complex analysis

Abstract: In convex geometry, there is an abundance of results on the approximation of convex bodies. The approximants are sourced from different families of special convex bodies such as inscribed or circumscribed (and possibly random) polyhedra, convex floating bodies, illumination bodies, etc. The asymptotics of such approximations are sensitive to the geometric and combinatorial features of the underlying convex bodies. Many invariants, such as the Blaschke surface area measure, can be completely recovered from such asymptotic analysis. In multi-variate complex analysis, while there are many natural notions of convexity, approximations of the kind described above have not been studied extensively. In this talk, we will present multiple volume approximation results in the complex setting. We will particularly focus on strongly $\mathbb C$-convex domains, which are complex-projective analogues of strongly convex domains. We will describe what the appropriate analogues of polyhedra are, and how the Heisenberg group plays a role in determining the constants of asymptotics. Familiarity with several complex variables will not be assumed.

Slides of the talk

Video of the talk

Thursday, March 16, 2023, 9:30AM (New York, NY time)

No seminar

Thursday, March 23, 2023, 9:30AM (New York, NY time)

No seminar

Thursday, March 30, 2023, 9:30AM (New York, NY time)

Grigoris Paouris, Texas A&M, College Station, TX, USA

Topic: On Subgaussian behavior of marginals of the uniform measure on Schatten balls.

Abstract: Consider the Uniform measure on the unit ball of the $p$-Schatten norms on matrices over real or complex numbers. We investigate the tail estimates of the marginals of these measures. In particular, we show that for $ p\geq 2$, these measures are "subgaussian," i.e., the tails of all marginals are dominated by the tails of Gaussian random variables. Based on joint work with Kavita Ramanan.

Video of the talk

Thursday, April 6, 2023, 9:30AM (New York, NY time)

Elisabeth Werner, Case Western Reserve University, Cleveland, OH

Topic: Spherical convex hull of random points on a wedge

Abstract: Consider two half-spaces $H_1^+$ and $H_2^+$ in $\mathbb{R}^{d+1}$ whose bounding hyperplanes $H_1$ and $H_2$ are orthogonal and pass through the origin. The intersection $\mathbb{S}_{2,+}^d:=\mathbb{S}^d\cap H_1^+\cap H_2^+$ is a spherical convex subset of the $d$-dimensional unit sphere $\mathbb{S}^d$ and is called a spherical wedge. Choose $n$ independent random points uniformly at random on $\mathbb{S}_{2,+}^d$ and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of $\log n$. The result is compared to the corresponding behavior of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere. Based on joint work with Florian Besau, Anna Gusakova, Matthias Reitzner, Carsten Schuett and Christoph Thaele.

Slides of the talk

Video of the talk

Thursday, April 13, 2023, 9:30AM (New York, NY time)

Dylan Langharst, Kent State University, Kent, OH

Topic: Higher-Order Affine Isoperimetric Inequalities

Abstract: Schneider generalized the difference body of a convex body to higher-order, and also established the higher-order analogue of the Rogers-Shephard inequality. In this talk, we extend this idea to the projection body, centroid body, LYZ Body, and radial mean bodies, as well as prove the associated inequalities (analogues of Zhang's projection inequality, Petty's projection inequality, the Busemann-Petty centroid inequality, Busemann's random simplex inequality and the affine Sobolev inequality). Joint work with J. Haddad, E. Putterman, M. Roysdon and D. Ye

Slides of the talk

Video of the talk

Thursday, April 20, 2023, 9:30AM (New York, NY time)

Apostolos Giannopoulos, National Technical University of Athens, Greece

Topic: Threshold for the expected measure of random polytopes

Abstract: We discuss the question how to obtain a threshold for the expected measure of a random polytope defined as the convex hull of independent random points with a log-concave distribution. We describe a general approach to the problem, which is based on the Cramer transform and works for an arbitrary log-concave distribution. We state a number of concrete questions that might lead to a complete affirmative answer in full generality. The talk is based on joint works with S. Brazitikos and M. Pafis.

Slides of the talk

Video of the talk

Thursday, April 27, 2023, 9:30AM (New York, NY time)

Yanir Rubinstein, University of Maryland, MD

Topic: L^p-polarity, Mahler functionals, and the isotropic constant

Abstract: The polar $K^\circ$ and the support function $h_K$ of a convex body $K$ are fundamental objects in Functional and Convex Analysis. The Mahler and Bourgain Conjectures have motivated an enormous amount of research in those fields over the past 85 years. Motivated by Bergman kernels, in joint work with B. Berndtsson and V. Mastrantonis we point out that $K^\circ$ and $h_K$ are $L^\infty$-versions of a more general one-parameter family of objects $$ K^{\circ, p} $$ and $$ h_{p,K}, $$ introduce the associated one-parameter generalization of the Mahler volume and Conjectures, establish some of their fundamental properties, and, finally, explore a new connection between these objects and Bourgain's slicing conjecture motivated by complex geometry.

Video of the talk

Schedule Fall 2023:

Thursday, September 7, 2023, 9:30AM (New York, NY time)

Tsuji Hiroshi, Osaka University, Japan

Topic: A new connection between the volume product and regularization of heat flow

Abstract: This talk is based on a joint work with Shohei Nakamura (Osaka). We discuss the Blaschke--Santal\'{o} inequality and Mahler's conjecture regarding the volume product of a convex body via heat flow. A key ingredient in our talk is the smoothing property of Ornstein--Uhlenbeck semigroup called hypercontractivity. We reveal a new link between the volume product and the smoothing property. This point of view is also extended to the Brascamp—Lieb inequality for multilinear functionals.

Slides of the talk

Video of the talk

Thursday, September 14, 2023, 9:30AM (New York, NY time)

Orli Herscovici, St. John's University, New York, NY

Topic: Stability and the equality case in the B-theorem

Abstract: The B-theorem of Cordero-Erausquin, Fradelizi, and Maurey states that for every symmetric convex set $K\subset\mathbb{R}^n$, and every $a,b>0$ the Gaussian measure $\gamma$ satisfies $\gamma(\sqrt{ab}K)\geq\sqrt{\gamma(aK)\gamma(bK)}$. In this talk we consider the stability and equality case of this inequality. This talk is based on a joint work with Galyna Livshyts, Liran Rotem, and Alexander Volberg.

Board of the talk

Video of the talk

Thursday, September 21, 2023, 10 AM (New York, NY time) -- note the special time!

Andriy Prymak, University of Manitoba

Topic: Convex bodies of constant width with exponential illumination number

Abstract: Borsuk's number $b(n)$ is the smallest integer such that any set of diameter 1 in the $n$-dimensional Euclidean space can be covered by $b(n)$ sets of smaller diameter. Currently best known asymptotic upper bound $b(n)\le (\sqrt{3/2}+o(1))^n$ was obtained by Shramm (1988) and by Bourgain and Lindenstrauss (1989) using different approaches. Bourgain and Lindenstrauss estimated the minimal number $g(n)$ of open balls of diameter 1 needed to cover a set of diameter 1 and showed $1.0645^n\le g(n)\le (\sqrt{3/2}+o(1))^n$. On the other hand, Schramm used the connection $b(n)\le h(n)$, where $h(n)$ is the illumination number of $n$-dimensional convex bodies of constant width, and showed $h(n)\le (\sqrt{3/2}+o(1))^n$. The best known asymptotic lower bound on $h(n)$ is subexponential and is the same as for $b(n)$, namely $h(n)\ge b(n)\ge c^{\sqrt{n}}$ for large $n$ established by Kahn and Kalai with $c\approx 1.203$ (1993) and by Raigorodskii with $c\approx 1.2255$ (1999). In 2015 Kalai asked if an exponential lower bound on $h(n)$ can be proved. We show $h(n)\ge (\cos(\pi/14)+o(1))^{-n}$ by constructing the corresponding $n$-dimensional bodies of constant width, which answers Kalai's question in the affirmative. The construction is based on a geometric argument combined with a probabilistic result establishing the existence of ``thinly spread'' discrete subsets of the sphere which possess certain separation (are a spherical code) and uniformly low density properties. The result allows to improve the lower bound of Bourgain and Lindenstrauss to $g(n)\ge (2/\sqrt{3}+o(1))^{n} \approx 1.1547^n$. We will also discuss applications of thinly spread subsets of the sphere in the context of constructing examples of convex bodies which are close to the ball but are hard to illuminate/cover. The talk is based on a joint work with Andrii Arman and Andriy Bondarenko.

Slides of the talk

Video of the talk

Thursday, September 28, 2023, 9:30AM (New York, NY time)

Esther Bou Dagher, Imperial College London, UK

Topic: Coercive Inequalities and U-Bounds

Abstract: In the setting of step-two Carnot groups, we prove Poincaré and $\beta$-Logarithmic Sobolev inequalities for probability measures as a function of various homogeneous norms. To do that, the key idea is to obtain an intermediate inequality called the U-Bound inequality (based on joint work with B. Zegarlinski). Using this U-Bound inequality, we show that certain infinite dimensional Gibbs measures- with unbounded interaction potentials as a function of homogeneous norms- on an infinite product of Carnot groups satisfy the Poincaré inequality (based on joint work with Y. Qiu, B. Zegarlinski, and M. Zhang). We also enlarge the class of measures as a function of the Carnot-Carathéodory distance that gives us the q−Logarithmic Sobolev inequality in the setting of Carnot groups. As an application, we use the Hamilton-Jacobi equation in that setting to prove the p−Talagrand inequality and hypercontractivity.

Slides of the talk

Video of the talk

Thursday, October 5, 2023, 9:30AM (New York, NY time)

Federico Glaudo, IAS, Princeton, NJ

Topic: On the isoperimetric profile of the hypercube

Abstract: The isoperimetric problem in the hypercube is a strikingly simple topic which is not yet completely understood. We will review the known facts on the problem, focusing on how the gaussian isoperimetric inequality provides a lower bound for the isoperimetric profile of the hypercube. Then, we will exploit this observation to prove that if the volume is close to 1/2, then the isoperimetric set within the cube is a half-space (intersected with the cube). Furthermore, we will show that the lower bound provided by the gaussian isoperimetric inequality is not sharp when the dimension of the hypercube goes to infinity (contrasting with the case of the sphere).

Slides of the talk

Video of the talk

Thursday, October 12, 2023, 9:30AM (New York, NY time)

Marta Strzelecka, University of Warsaw, Poland

Topic: Operator $\ell_p\to\ell_q$ norms of random matrices with iid entries

Abstract: During the talk we focus on the expectation of operator norms from $\ell_p^n$ to $\ell_q^m$ of random matrices $(X_{i,j})_{i\le m, j\le n}$, where the entries $X_{i,j}$ are iid random variables satisfying a mild moment growth assumption. It turns out that in most ranges of $p$ and $q$ this expectation behaves like a certain nice quantity depending only on $p,q, n, m$, and the distribution of $X_{i,j}$'s. Moreover, if the $X_{i,j}$'s are additionally assumed to be subgaussian, then the same holds true in the whole range $p,q\ge 1$. We shall also see how this problem is related to the two-sided Chevet-type inequalities for iid Weibull random variables with shape parameter $r\in[1,2]$. Based on joint work with Rafal Latala.

Slides of the talk

Video of the talk

Thursday, October 19, 2023, 9:30AM (New York, NY time)

Gautam Aishwarya, Technion University, Israel

Topic: Dimensional Brunn--Minkowski inequalities for the relative entropy

Abstract: In this talk, we will discuss a further development of the connection between the convexity properties of a measure $\nu$ and the convexity properties of the associated relative entropy $D(\cdot \Vert \nu)$ on Wasserstein space, to include the study of dimensional Brunn-Minkowski inequalities for log-concave measures when the admissible sets are assumed to have some symmetry. In particular, the results presented will imply dimensional inequalities for centred star-shaped bodies when the measure $\nu$ is log-concave with a p-homogeneous potential (such as the Gaussian measure), thereby going beyond the convexity assumption on admissible sets that appears essential for the prominent differential-geometric technique in this area. Several fundamental questions about optimal transport that naturally arise upon a finer look at the Gaussian case will also be discussed. This talk is based on joint work with Liran Rotem.

Slides of the talk

Video of the talk

Thursday, October 26, 2023, 9:30AM (New York, NY time)

The seminar is postponed until January and we don't have any energy to search for a replacement.

Thursday, November 2, 2023, 9:30AM (New York, NY time)

Manuel Fernandez, Georgia Tech, Atlanta, GA

Topic: On the $ell_0$ isoperimetry of measurable sets

Abstract: Gibbs-sampling, also known as coordinate hit-and-run (CHAR), is a random walk used to sample points uniformly from convex bodies. Its transition rule is simple: Given the current point p, pick a random coordinate i and resample the ith coordinate of p according to the distribution induced by fixing all other coordinates. Despite its use in practice, strong theoretical guarantees regarding the mixing time of CHAR for sampling from convex bodies were only recently shown in works of Narayanan et al. and Laddha et al. 
 In the work of Laddha et al., as part of their proof strategy, the authors introduced the notion of the $\ell_0$ isoperimetric coefficient of a measurable set and provided a lower bound for the quantity in the case of axis-aligned cubes. In this talk we will present some new results regarding the $\ell_0$ isoperimetric coefficient of measurable sets. In particular we pin down the exact order of magnitude of the $\ell_0$ isoperimetric coefficient of axis-aligned cubes and present a general upper bound of the $\ell_0$ isoperimetric coefficient for any measurable set. As an application, we will mention how the results give a moderate improvement in the mixing time of CHAR.

Slides of the talk

Video of the talk

Thursday, November 9, 2023, 9:30AM (New York, NY time)

Shohei Nakamura, Osaka University

Topic: Functional volume product, regularizing effect of heat flow, and Brascamp—Lieb inequality

Abstract: This talk is based on the joint work with Hiroshi Tsuji (Osaka). I am going to report three results that are related from each other: (1) A monotonicity of the functional volume product along the Fokker—Planck flow for even functions. (2) An improvement of Borell’s reverse hypercontractivity in terms of Nelson’s time condition. (3) Some stability result of the functional Blaschke—Santal\’{o} inequality initiated by Barthe—B\”{o}r\”{o}czky—Fradelizi (which is a wealth of the monotonicity). All these results are based on a simple idea, that is, the functional volume product can be regarded as a limiting case of the regularizing effect of Ornstein—Uhlenbeck heat flow: for some explicit $C_s$, $$ \lim_{s\to0} C_s \big(\int f dx\big)^{-\frac{q_s}{p_s}} \big\| P_s \big[ \big( \frac{f}{\gamma} \big)^\frac1{p_s} \big] \big\|_{L^{q_s}(\gamma)}^{q_s} = \int f dx \int f^\circ dx, $$ where $p_s \sim 2s$ and $q_s\sim -2s$. This identity is motivated from the work of Bobkov—Gentil—Ledoux. This can be further understood in terms of the Brascamp—Lieb theory. From this view point, our result can be read as an example of the inverse Brascamp—Lieb inequality initiated by Barthe—Wolff but without their non-degenerate condition. I am also going to explain an importance to go beyond their non-degenerate condition for the purpose of the link to an open problem in convex geometry.

Slides of the talk

Video of the talk

Thursday, November 16, 2023, 9:30AM (New York, NY time)

Tatiana Brailovskaya, Princeton University, NJ, USA

Topic: Universality and matrix concentration inequalities

Abstract: Random matrices are ubiquitous across many fields — physics, computer science, applied and pure mathematics. Oftentimes the random matrix of interest will have non-trivial structure — entries that are dependent and have potentially different means and variances (e.g. sparse Wigner matrices, matrices corresponding to adjacencies of random graphs, sample covariance matrices). However, current understanding of such complex random matrices remains lacking. In this talk, I will discuss recent results concerning the spectrum of sums of independent random matrices with some boundedness constraints on their operator norms. In particular, I will demonstrate that under some fairly general conditions, such sums will exhibit the following universality phenomenon — their spectrum will lie close to that of a Gaussian random matrix with the same mean and covariance. (joint with Ramon van Handel)

Thursday, November 23, 2023, 9:30AM (New York, NY time)

No seminar (Thanksgiving, Banff,...)

Schedule Spring 2024:

Thursday, January 11, 2023, 9:30AM (New York, NY time)

Martin Rapaport, Universite Gustave Eiffel, Paris, France

Topic: On the monotonicity of discrete entropy for log-concave random variables on $\mathbb{Z}^{d}$.

Abstract: In this talk, the focus is on the discrete entropy of log-concave random vectors in $\mathbb{Z}^{d}$. Initially, we will provide motivation, context, and a brief overview as well as discuss the definition of log-concavity on $\mathbb{Z}^{d}$ that we are using. Subsequently, we will prove a discrete generalised Entropy Power Inequality (EPI) for isotropic log-concave sums of independent identically distributed random vectors. To achieve this, we will explain our proof strategy, which primarily involves two important stages: an approximation result between discrete entropy and differential entropy as well as a discrete analogue of the upper bound on the isotropic of a log-concave function, which is of independent interest. To accomplish these goals, tools from convex geometry are needed since the isotropic position is required. Thus, the talk fits into the interactions between information theory and convex geometry. Finally, several open questions will be discussed. This talk is based on joint work with Matthieu Fradelizi and Lampros Gavalakis.

Slides of the talk

Video of the talk

Thursday, January 18, 2024, 9:30AM (New York, NY time)

Emma Pollard, Boise State University, US

Topic: Symmetrization Resistance

Abstract: An asymmetric random variable X is said to be symmetrization resistant if every independent random variable Y that produces a symmetric sum X+Y has a greater variance than that of X. Asymmetric Bernoulli random variables were shown to be symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. Proving symmetrization resistance appears to be difficult: little is known about other asymmetric distributions. We introduce the notion of entropic symmetrization resistance which is the same as symmetrization resistance except that the entropy (rather than variance) of Y must exceed that of X. We show that Bernoulli random variables exhibit entropic symmetrization resistance exactly when they exhibit symmetrization resistance. We also extend the underlying entropy and variance inequalities to the hypercube. Finally, we explore the possibility of extensions to non-Bernoulli random variables. This talk is based on joint work with Mokshay Madiman.

Slides of the talk

Video of the talk

Thursday, January 25, 2024, 9:30AM (New York, NY time)

Jnyaneshwar Baslingker, Indian Institute of Science

Topic: Log-concavity in 1D Coulomb Gas ensumbles

Abstract: The ordered elements in several one-dimensional Coulomb gas ensembles arising in probability and mathematical physics are shown to have log- concave distributions. Examples include the beta ensembles with convex potentials (in the continuous setting) and the orthogonal polynomial ensembles (in the discrete setting). In particular, we prove the log-concavity of the Tracy-Widom β distributions, Airy distribution, Airy-2 process. Log-concavity of last passage times in percolation is proven using their connection to Meixner ensembles. As a result we prove the log-concavity of top rows of Young diagrams under Poissonized Plancherel measure, which is Poissonized version of a conjecture of Chen. This is ongoing joint work with Manjunath Krishnapur and Mokshay Madiman.

Slides of the talk

Video of the talk

Thursday, February 1, 2024, 9:30AM (New York, NY time)

Devraj Duggal, University of Minnesota, MN

Topic: On Spherical Covariance Representations

Abstract: We first motive the study of covariance representations by surveying preceding results in the Gauss space. Their spherical counterparts are then derived thereby allowing applications to the spherical concentration phenomenon. The applications include second order concentration inequalities. This talk is based on joint work with Sergey Bobkov.

Slides of the talk

Video of the talk

Thursday, February 8, 2024, 9:30AM (New York, NY time)

Marius Tiba, University of Oxford, UK

Topic: Sharp stability for the Brunn-Minkowski inequality for arbitrary sets

Abstract: The Brunn-Minkowski inequality states that for (open) sets A and B in R^d, we have |A+B|^{1/d} \geq |A|^{1/d}+|B|^{1/d}. Equality holds if and only if A and B are convex and homothetic sets in R^d. In this talk, we present a sharp stability result for the Brunn-Minkowski inequality, concluding a long line of research on this problem. We show that if we are close to equality in the Brunn-Minkowski inequality, then A and B are close to being homothetic and convex, establishing the exact dependency between the three notions of closeness. This is based on joint work with Alessio Figalli and Peter van Hintum.

Slides of the talk

Video of the talk

Thursday, February 22, 2024, 9:30AM (New York, NY time)

No seminar, intersection with a workshop

TUESDAY, February 27, 2024, 9:30AM (New York, NY time) -- NOTE THE TALK IS ON TUESDAY NOT THURSDAY

Steven Hoehner, Longwood University

Topic: A New Geometric Definition of Euler's Number with an Application to Random Polytopes

Abstract: Place $N$ geodesic caps on the sphere $\mathbb{S}^{d-1}=\{x\in\R^d:\,\|x\|_2=1\}$, each of measure $1/N$. How much of the sphere can we cover? When $d\geq 3$ and $N\geq 3$, it is not obvious from a geometric standpoint that such a partial covering can capture a substantial proportion of the measure of $\mathbb{S}^{d-1}$. Using a probabilistic approach, in which the caps are chosen uniformly and independently from the sphere, shows that the expected proportion covered is $1-e^{-1}+o(1)$. Thus a natural question arises: is the random covering optimal, as the number of caps and dimension tend to infinity? We show that the answer to this question is affirmative. Our main result is an asymptotic estimate for the maximum volume of a partial covering with caps of different sizes. As a corollary, we obtain a new geometric definition of Euler's number in terms of the volume of the sphere in high dimensions. We will discuss the proof, which uses concentration of measure along with tools from probability and statistics. Finally, we discuss possible applications related to the optimality of random polytopes in high dimensions. This talk is based on joint work with Gil Kur.

Slides of the talk

Video of the talk

Thursday, March 7, 2024, 9:30AM (New York, NY time)

Hannah Alpert, Auburn University

Topic: Unintuitive properties of Urysohn 1-width

Abstract: A metric space has small Urysohn 1-width if it admits a continuous map to a 1-dimensional complex where the preimage of each point has small diameter. An open problem is, if a space's universal cover has small Urysohn 1-width, must the original space also have small Urysohn 1-width? Naively we would guess yes, but various strange examples suggest maybe not. Joint work with Panos Papasoglu, Arka Banerjee, Alexey Balitskiy, and Larry Guth.

Slides of the talk

Video of the talk

Thursday, March 14, 2024, 9:30AM (New York, NY time)

No seminar, intersection with a workshop

Thursday, March 21, 2024, 9:30AM (New York, NY time)

Beatrice Helen Vritsiou, University of Alberta, Edmonton, Canada

Topic: On the Illumination Conjecture for convex bodies with many symmetries

Abstract: We will present a unified treatment of the Hadwiger-Boltyanski Illumination Conjecture for 1-symmetric convex bodies in ALL dimensions, and we will also show how to settle the conjecture (along with its equality cases) for some instances of 1-unconditional convex bodies. This is joint work with Wen Rui Sun.

Slides of the talk

Thursday, April 11, 2024, 9:30AM (New York, NY time)

Colin Tang, Carnegie Melon University, Pittsburgh, PA

Topic: Simplex slicing: an asymptotically-sharp lower bound

Abstract: We show that for the regular n-simplex, the 1-codimensional central slice that's parallel to a facet will achieve the minimum area (up to a $1-o(1)$ factor) among all 1-codimensional central slices, thus improving the previous best known lower bound (Brzezinski 2013) by a factor of $\frac{2\sqrt{3}}{e} \approx 1.27$. In addition to the standard technique of interpreting geometric problems as problems about probability distributions and standard Fourier-analytic techniques, we rely on a new idea, mainly changing the contour of integration of a meromorphic function.

Slides of the talk

Video of the talk

Thursday, May 2, 2024, 9:30AM (New York, NY time)

Shay Sadovsky, Tel Aviv University, Israel

Topic: Godbersen's conjecture for locally anti-blocking polytopes

Abstract: Godbersen's conjecture states that for any convex body $K$, the mixed volume $V(K[j],-K[n-j])$ is tightly bounded by $\binom{n}{j} Vol(K)$. The main object of this talk will be the class locally anti-blocking polytopes, which will be introduced in detail. The special properties of this class allow us to prove Godbersen's conjecture for it. We will also discuss other results that have been proved for locally anti-blocking bodies such as Kalai's 3d conjecture, recently proved by Sanyal and Winter, and various geometric inequalities proved together with Artstein-Avidan and Sanyal.

Slides of the talk

Video of the talk

Schedule Fall 2024:

Wednesday, September 18, 2024, 10AM (New York, NY time)

Dylan Langharst, Institut de Mathematiques de Jussieu, Sorbonne Universite, Paris, France

Topic: Functional Volume Product Along the Fokker-Planck Heat Flow

Abstract: The now-classical Santal\'o inequality for an integrable, nonnegative function $f$ on $\mathbb R^n$ states $M(f):=\left(\int_{\mathbb R^n}f\right)\inf_z\left(\int_{\mathbb R^n}(\tau_zf)^\circ\right) \leq M(e^{-|x|^2}),$ where $\tau_z f(x)= f(x-z)$ and $f^\circ$ is the polar of a function. We approximate polarity by defining the $L^p$ Laplace transform of a function $f$, $p\in (0,1)$ as, $$\mathcal{L}_p(f)(x) = \left(\int_{\mathbb R^n}f(y)^\frac{1}{p}e^{x\cdot y}dy\right)^\frac{p}{p-1}.$$ Under mild regularity assumptions on $f$, such as log-concavity or continuity, $\lim_{p\to 0^{+}}\mathcal{L}_p(f)(x/p)=f^{\circ}(x)$. In this work, which is joint with Cordero-Erausquin and Fradelizi, we define the $L^p$ volume product of a nonnegative function $f$ as \[ M_p(f):=\Big(\int_{\mathbb R^n}f \Big)\, \inf_z \Big(\int_{\mathbb R^n}{\mathcal L}_p (\tau_z f)(x/p)\, dx \Big)^{1-p}. \] Our main thoerem is that $M_p(f_t)$ is increasing along the Fokker-Planck heat-semi group. This extends a recent result by Nakamura-Tsuji, who obtained the same monotonicity when $f$ is even and integrable. An immediate corollary is $M_p(f)\leq M_p(e^{-|x|^2})$. Sending $p\to 0^+$, we also obtain that $M(f_t)$ is increasing in $t$. Perhaps even more interesting is the analysis for the infimum: the infimum may be zero if $p$ is not sufficiently close to $0$. We characterize exactly when this occurs by studying the Laplace transform of log-concave functions, using and elaborating on some ideas by Klartag. If the infimum is not zero, then it is obtained at a unique point, which we call the $p$th Laplace-Santal\'o point of $f$.

Slides of the talk

Video of the talk

Wednesday, September 25, 2024, 10AM (New York, NY time)

Rotem Assouline, Weizmann Istitute of Science, Rehovot, Israel

Topic: Magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on Riemannian manifolds

Abstract: I will present a magnetic version of the Riemannian Brunn-Minkowski and Borell-Brascamp-Lieb inequalities of Cordero-Erausquin-McCann-Schmuckenschläger and Sturm, replacing geodesics by minimizers of a magnetic action functional. Both results involve a notion of magnetic Ricci curvature.

Slides of the talk

Video of the talk

Wednesday, October 9, 2024, 10AM (New York, NY time)

Daniela Di Donato, Universita di Pavia, Italy

Topic: Rectifiability in Carnot groups

Abstract: Intrinsic regular surfaces in Carnot groups play the same role as C^1 surfaces in Euclidean spaces. As in Euclidean spaces, intrinsic regular surfaces can be locally defined in different ways: e.g. as non critical level sets or as continuously intrinsic differentiable graphs. The equivalence of these natural definitions is the problem that we are studying. Precisely our aim is to generalize some results proved by Ambrosio, Serra Cassano, Vittone valid in Heisenberg groups to the more general setting of Carnot groups. This is joint work with Antonelli, Don and Le Donne.

Slides of the talk

Video of the talk

Wednesday, October 30, 2024, 10AM (New York, NY time)

Manuel Fernandez, Georgia Institute of Technology, Atlanta, GA, USA

Topic: Distance theorems and the smallest singular of inhomogeneous random rectangular matrices

Abstract: In recent years, significant progress has been made in our understanding of the quantitative behavior of random matrices. Such results include delocalization properties of eigenvectors and tail estimates for the smallest singular value. A key ingredient in their proofs is a 'distance theorem', which is a small ball estimate for the distance between a random vector and subspace. Building on work of Livshyts and Livshyts, Tikhomirov and Vershynin, we introduce a new distance theorem for inhomogeneous vectors and subspaces spanned by the columns of an inhomogeneous matrix. Such a result has a number of applications for generalizing results about the quantitative behavior of i.i.d. matrices to matrices without any identical distribution assumptions. To highlight this, we show that the smallest singular value estimate of Rudelson and Vershynin, proven for i.i.d. subgaussian rectangular matrices, holds true for inhomogeneous and heavy-tailed matrices. This talk is partially based on joint work with Max Dabagia.

Slides of the talk

Video of the talk

Wednesday, November 6, 2024, 10AM (New York, NY time)

Mira Gordin, Princeton University, NJ, USA

Topic: Vector-Valued Concentration Inequalities on Discrete Spaces

Abstract: Existing vector-valued concentration inequalities with no assumption on the target Banach space are known only in very special settings, the Gaussian measure on R^n and the uniform measure on the discrete hypercube {-1,1}^n. We present a novel vector-valued concentration inequality on the symmetric group which goes beyond the product setting of the prior known results. Furthermore, we discuss the implications of this result for the nonembeddability of the symmetric group into Banach spaces of nontrivial Rademacher type, a question of interest in the metric geometry of Banach spaces. In addition, we will discuss similar results for the k-slice of the hypercube and the biased measure on the hypercube. This talk is based on joint work with Ramon van Handel.

Slides of the talk

Video of the talk

Wednesday, November 13, 2024, 10AM (New York, NY time)

Pazit Haim-Kislev, Princeton University, NJ, USA

Topic: Symplectic capacities of convex bodies

Abstract: Although convexity is not a symplectic invariant, convex domains appear to exhibit particular symplectic features. One example of this phenomenon is Viterbo's volume-capacity conjecture, which asserts that among all convex bodies of the same volume, the ball has the largest capacity. Since its introduction in 2000, this conjecture has been highly influential in the study of symplectic capacities, sparking extensive research. In particular, Artstein-Avidan, Karasev, and Ostrover showed that a special case of Viterbo's conjecture is equivalent to the famous Mahler conjecture regarding volume products of centrally symmetric convex domains. In this talk, I will present a recent counterexample to Viterbo's conjecture in every dimension, demonstrating that not all capacities coincide on the class of convex domains. This is joint work with Yaron Ostrover.

Slides of the talk

Video of the talk

Wednesday, November 20, 2024, 10AM (New York, NY time)

Eli Putterman, Tel Aviv University, Israel

Topic: On Schneider's higher-order difference body

Abstract: The volume of the difference body K - K is bounded above and below via of the most fundamental inequalities in convex geometry: the Brunn-Minkowski inequality, which implies |K - K| \ge 2^n |K| with equality iff K is centrally symmetric, and the Rogers-Shephard inequality, which states that |K - K| \le \binom{2n}{n} |K|, with equality iff K is a simplex. In the 1970s, Schneider introduced a generalization of the difference body, the higher-order difference body D^m(K), and showed that its volume is maximized (for |K| fixed) precisely when K is a simplex; moreover, if K is planar then |D^m(K)| is minimized iff K is centrally symmetric. He also conjectured that if K is at least three-dimensional and |K| = 1, then |D^m(K)| is minimized when K is an ellipsoid. In our work, we reinterpret |D^m(K)| as a sum of mixed volumes of different linear images of K, and use this to give a simple proof of Schneider's result in the plane, as well as to solve a related optimization problem: if K is a zonoid with isotropic generating measure in dimension at least 3, then the volume of D^m(K) is maximized precisely when K is a cube. If time permits, we will also discuss the connection between Schneider's conjecture and Petty's conjecture on the minimal volume of the projection body.

Slides of the talk

Video of the talk