# Workshop: High dimensional measures: geometric and probabilistic aspects

**Dates:** March 22-26, 2021

**Venue:** Online

**Organizers:** Ronen Eldan (Rehovot), Alexander Litvak (Alberta), Assaf Naor (New York), Elisabeth M. Werner (Cleveland)

**Description: **concentration phenomena, $L^p$-surface area, $L^p$-Brunn-Minkowsky theory, log-concavity, isoperimetric inequalities, high-dimensional convex bodies, approximation of convex bodies, random matrices, isotropic convex bodies

Speakers:

Andreas Bernig, Yuansi Chen, Matthieu Fradelizi, Olivier Guedon, Han Huang, Bo'az Klartag, Galyna Livshyts, Mark Meckes, Emanuel Milman, Stanislav Nagy, Grigorios Paouris, Liran Rotem, Dmitry Ryabogin, Franz Schuster, Alina Stancu, Nike Sun, Stanislaw Szarek, Kateryna Tatarko, Konstantin Tikhomirov, Santosh Vempala, Dongmeng Xi, Vladyslav Yaskin, Pierre Youssef, Dmitry Zaporozhets

Talks are given online and access data will be sent to the registered participants in due course. Recordings of the talks will be posted online.

If you are interested in attending the workshop, please click here for online registration.

Click here for the abstracts.

Click here for the schedule.

## Personalized Zoom sessions for follow-up questions

**Olivier Guédon**

Zoom Link

Meeting-ID: 994 7365 7133

Thursday, 6:30-7:45 p.m. (CET)

## Video recordings and slides

**Emanuel Milman: Functional Inequalities on sub-Riemannian manifolds via QCD**

We are interested in obtaining Poincar ́e and log-Sobolev inequalities on domains in sub-Riemannian manifolds (equipped with their natural sub-Riemannian metric and volume measure).

It is well-known that strictly sub-Riemannian manifolds do not satisfy any type of Curvature-Dimension condition CD(K,N), introduced by Lott–Sturm–Villani some 15 years ago, so we must follow a dif- ferent path. We show that while ideal (strictly) sub-Riemannian manifolds do not satisfy any type of CD condition, they do satisfy a quasi-convex relaxation thereof, which we name QCD(Q, K, N ), where Q > 1 is a slack parameter. As a consequence, these spaces satisfy numerous functional inequalities with exactly the same quantitative dependence (up to a factor of Q) as their CD counterparts. We achieve this by extending the localization paradigm to completely general interpolation inequalities, and a one-dimensional comparison of QCD densities with their “CD upper envelope”. We thus ob- tain the best known quantitative estimates for (say) the Lp-Poincar ́e and log-Sobolev inequalities on domains in ideal sub-Riemannian manifolds and in general corank 1 Carnot groups, which in partic- ular are independent of the topological dimension. For instance, the classical Li–Yau / Zhong–Yang spectral-gap estimate holds on all Heisenberg groups of arbitrary dimension up to a factor of 4.

**Olivier Guédon: On the asymptotic geometry of the unit ball of Schatten classes
**

I will discuss about asymptotic results of geometric parameters associated to the unit balls of Schatten classes and the relations with various geometric conjectures. Joint work with B. Dadoun, M. Fradelizi, P.A. Zitt.

**Han Huang: Rank of Sparse Bernoulli Matrices**

**Dongmeng Xi: On the Gaussian Minkowski Problem**

We would like to talk about the Minkowski problem for Gaussian surface area measure. Both the uniqueness and existence results are investigated. This is a joint work with Yong Huang and Yiming Zhao.

**Kateryna Tatarko: Unique determination of ellipsoids by their dual volumes**

Gusakova and Zaporozhets conjectured that ellipsoids in $R^n$ are uniquely determined up to an isometry by their intrinsic volumes. In this talk, we will present a solution to the dual problem in all dimensions. We show that an ellipsoid is uniquely determined up to an isometry by its dual Steiner polynomial. We also discuss an alternative proof of the analogous known result of Petrov and Tarasov for classical Steiner polynomials in $R^3$. This is joint work with S. Myroshnychenko and V. Yaskin.

**Franz Schuster: Blaschke–Santaló Inequalities for Minkowski and Asplund Endomorphisms**

The Blaschke–Santaló inequality is one of the best known and most powerful affine isoperimetric inequalities in convex geometric analysis. In particular, it is significantly stronger than the classical Euclidean Urysohn inequality. In this talk, we present new isoperimetric inequalities for monotone Minkowski endomorphisms of convex bodies, each one stronger than the Urysohn inequality. Among this large family of new inequalities, the only affine invariant one – the Blaschke–Santaló inequality – turns out to be the strongest one. A further extension of these inequalities to merely weakly monotone Minkowski endomorphisms is proven to be impossible which, in turn, uncovers an unexpected phenomenon. Moreover, for Asplund endomorphisms of log-concave functions, which are functional analogues of Minkowski endomorphisms, a family of analytic inequalities is presented which generalizes the functional Blaschke–Santaló inequality. This is joint work with Georg Hofstätter.

**Andreas Bernig: Intrinsic volumes on pseudo-Riemannian manifolds**

The intrinsic volumes in Euclidean space can be defined via Steiner’s tube formula and were characterized by Hadwiger as the unique continuous, translation and rotation invariant valuations. By the Weyl principle, their extension to Riemannian manifolds behaves naturally under isometric embeddings. In a series of papers with Dmitry Faifman and Gil Solanes, we developed a theory of intrinsic volumes in pseudo-Euclidean spaces and on pseudo-Riemannian manifolds. Fundamental results like Hadwiger’s theorem, Weyl’s principle and Crofton formulas on spheres have their natural analogues in the pseudo-Riemannian setting.

**Bo’az Klartag: One more proof of the Alexandrov-Fenchel inequality**

We present a short proof of the Alexandrov-Fenchel inequalities for mixed volumes of convex bodies. Joint work with Cordero-Erausquin, Merigot and Santambrogio.

**Yuansi Chen: Recent progress on the KLS conjecture**

Kannan, Lovász and Simonovits (KLS) conjectured in 1995 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.

**Stanislaw Szarek: The projective/injective ratio and GPTs**

Among natural tensor products of normed spaces, the projective and the injective are the extreme ones. The question : How much do they differ? was considered by Grothendieck and Pisier (in the 1950s and 1980s), but - surprisingly - no systematic quantitative analysis of the finite- dimensional case seems to have been ever made. Recently, a renewed interest in the problem came up in foundations of physics, in the context of generalized probabilistic theories (GPTs) and XOR games, where it can be restated as: How powerful are global strategies compared to local ones? We show that, in a nutshell, the discrepancy between the projective and injective norms on tensor products of normed spaces is always lower-bounded by the power of the (smaller) dimension, with the exponent depending on the generality of the setup. Some of the results are essentially optimal, but other can be likely improved. The methods involve a wide range of techniques from geometry of Banach spaces and random matrices. Joint work with G. Aubrun, L. Lami, C. Palazuelos, A. Winter.

**Stanislav Nagy: Quantiles, depth, and symmetries: Geometry in multivariate statistics**

There are tools of multivariate statistics with natural counterparts in geometry. We examine these connections and outline the amount of research that has been conducted in parallel in the two fields. Advances from geometry allow us to approach problems in multivariate statistics that were considered open.

**Matthieu Fradelizi: The functional form of Mahler conjecture for even log-concave functions in dimension 2**

We prove the functional form of Mahler conjecture concerning the functional volume product of an even log-concave function in dimension 2: we give the sharp lower bound of this quantity and characterize the equality case. Joint work with Elie Nakhle.

**Liran Rotem: Riesz Representation Theorem for Log-Concave Functions**

**Dmitry Ryabogin: On bodies floating in equilibrium in every direction**

We discuss old and new results related to Ulam’s Problem 19 from the Scottish Book asking, is it a solid of uniform density which will float in water in every position a sphere?

**Konstantin Tikhomirov: Random Graph Matching with Improved Noise Robustness**

Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields such as computer vision and biology. In this work we will discuss a new algorithm for exact matching of correlated Erdos-Renyi graphs. Based on joint work with Cheng Mao and Mark Rudelson.

**Pierre Youssef: Outliers in sparse Wigner matrices**

Given a Wigner matrix with centered bounded entries, we study the effect of sparsity on the extreme eigenvalues. More precisely, multiplying the entries by independent Bernoulli variables with parameter pn, we show that as $p_n$ decreases, outliers start emerging in the semi-circular law which is the limiting spectral distribution. We illustrate this in the case of Erdos-Renyi graph, where we capture a phase transition and a particular value of sparsity (i.e. of $p_n$) above which there are no outliers and below which there are. This is a joint work with Konstantin Tikhomirov.

**Galyna Livshyts: On some tight convexity inequalities for symmetric convex sets**

We conjecture an inequality which strengthens the Ehrhard inequality for symmetric convex sets, in the case of the standard Gaussian measure. We explain its relation to other questions, such as the isoperimetric problem, and (if time permits), to the tight bound in a version of the Dirichlet- Poincare inequality. All of the aforementioned questions have round k-cylinders as optimizers. We make progress towards this question, using the L2 methods; we shall also discuss energy minimization and its relation to this question. We show a related estimate which is tight only for round cylinders; the equality case characterization is based on the quantitative stability in the energy minimization estimates, as well as the quantitative stability in the Brascamp-Lieb inequality (both of which are obtained using the L2 methods together with some tools from trace theory), and an approximation argument. If time permits, we will also discuss some new inequalities in the case of other measures.

**Santosh Vempala: Reducing Isotropy to KLS: An Almost Cubic Volume Algorithm**

Computing the volume of a convex body is an ancient problem whose study has led to many interesting mathematical developments. In the most general setting, the convex body is given only by a membership oracle. In this talk, we present a faster algorithm for isotropic transformation of an arbitrary convex body in n-dimensional space, with complexity $n^3\psi^2$, where $\psi$ bounds the KLS constant for isotropic convex bodies. Together with the known bound of $\psi = n^{o(1)}$ [Chen 2020] and the Cousins-Vempala n3 volume algorithm for well-rounded convex bodies [2015], this gives an $n^{3+o(1)}$ volume algorithm for general convex bodies, improving on the $n^4$ algorithm of Lovász-Vempala [2003]. A positive resolution of the KLS conjecture ($\psi = O(1)$) would imply an $n^3$ volume algorithm (up to log factors). No background on algorithms, KLS or ABC will be assumed for the talk. This is joint work with He Jia, Aditi Laddha and Yin Tat Lee.

**Alina Stancu: Some comments on the fundamental gap of the Dirichlet Laplacian in hyperbolic space**

I will present some results on the fundamental gap of convex domains in hyperbolic space for different types of convexity. The results are in contrast with the behaviour of the fundamental gap in Euclidean space and I will make some comments on the aspects of the problem that are different between the two spaces.

**Vladyslav Yaskin: A solution to the 5th and 8th Busemann-Petty problems near the Euclidean ball**

We show that the 5th and the 8th Busemann-Petty problems have positive solutions for bodies that are sufficiently close to the Euclidean ball in the Banach-Mazur distance. Joint work with M. Angeles Alfonseca, Fedor Nazarov, and Dmitry Ryabogin.

**Dmitry Zaporozhets: Gaussian from its maximum**

In this talk, we will discuss the following question. Suppose that $X = (X_1,...,X_n)$ is a centered Gaussian vector in $R^n$. Is it true that the distribution of the $\max_{k=1,\ldots,n}(X_1,\ldots,X_n)$ uniquely (up to rotations) defines the distribution of $X$?

**Mark Meckes: Magnitude and intrinsic volumes in subspaces of $L_1$**

Magnitude is an isometric invariant of metric spaces, with origins in category theory, which turns out to be related to a wide variety of classical geometric invariants, including Minkowski dimension, volume, and surface measure. For convex bodies in $l_1^n$ , magnitude turns out to be an $l_1$ analogue of both the classical Steiner polynomial and the Wills functional. I will discuss this result and some of its consequences for magnitude in subspaces of $L_1$, including a new proof of Sudakov’s minoration inequality.

**Grigorios Paouris: Non-Asymptotic results for singular values of Gaussian matrix products**

I will discuss non-asymptotic results for the singular values of products of Gaussian matrices. In particular, I will discuss the rate of convergence of the empirical measure to the triangular law and discuss quantitive results on asymptotic normality of Lyapunov exponents. The talk is based on joint work with Boris Hanin.

**Nike Sun: On the Ising perceptron model**

The perceptron is a toy model of a simple neural network that stores a collection of given patterns. Its analysis reduces to a simple problem in high-dimensional geometry, namely, the intersection of the cube (or sphere) with a collection of random half-spaces. Despite the simplicity of this model, its high-dimensional asymptotics are not very well understood, particularly in the case of the cube. I will describe what is known and present recent results. This lecture is based on joint work with Jian Ding.