# Online Asymptotic Geometric Analysis Seminar

** Welcome to the Online AGA seminar webpage!
If you are interested in giving a talk, please let us know. Also, please suggest speakers which you would like to hear speak. Most talks are 50 minutes, but some 20-minute talks will be paired up as well. The talks will be video recorded conditioned upon the speakers' agreement. PLEASE SHARE THE SEMINAR INFO WITH YOUR DEPRARTMENT AND ANYONE WHO MAY BE INTERESTED! Please let the organizers know if you would like to be added to the mailing list.
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**

Note that on Tuesdays, the lectures start at:

7:30am in Los-Angeles, CA

8:30am in Edmonton, AB

9:30am in Columbia MO; College Station, TX; Chicago, IL

10:30am in Kent, OH; Atlanta, GA; Montreal; New York, NY

11:30am in Rio de Janeiro, Buenos Aires

3:30pm (15:30) in London

4:30pm (16:30) in Paris, Milan, Budapest, Vienna

5:30pm (17:30) in Tel Aviv.

### Tuesday, April 7, 10:30AM (New York, NY time)

### Alexander Koldobsky, University of Missouri, Columbia, MO, USA

#### Topic: A new version of the isomorphic Busemann-Petty problem for arbitrary functions

### Saturday, April 11, 11:30AM (New York, NY time)

### Sergey Bobkov, University of Minnesota, Minneapolis, MN, USA

#### Topic: A Fourier-analytic approach to transport inequalities

#### Abstract: We will be discussing a Fourier-analytic approach to optimal matching between independent samples, with an elementary proof of the Ajtai-Komlos-Tusnady theorem. The talk is based on a joint work with Michel Ledoux.

### Tuesday, April 14, 10:30AM (New York, NY time)

### Elisabeth Werner, Case Western Reserve University, Cleveland, OH, USA

#### Topic: Constrained convex bodies with maximal affine surface area

#### Abstract: Given a convex body K in R^n, we study the maximal affine surface area of K, i.e., the quantity
AS(K) = sup_{C} as(C)
where as(C) denotes the affine surface area of C, and the supremum is taken over all convex
subsets of K. In particular, we give asymptotic estimates on the size of AS(K).

### Saturday, April 18, 11:30AM (New York, NY time)

### Károly Böröczky, Central European University, Budapest, Hungary

#### Topic: Symmetry and Structure within the Log-Brunn-Minkowski Conjecture

#### Abstract: After reviewing some formulations of the Log-Brunn-Minkowski Conjecture in R^n in terms of Monge-Ampere equations, of Hilbert Operator and of Brunn-Minkowski Theory, I will report on some recent advances, like Livshyts' and Kolesnikov's improvement on the fundamental approach of Milman and Kolesnikov, and the verification of the conjecture for bodies with n hyperplane symmetries by Kalantzopoulos and myself using an idea due to Bathe and Fradelizi.

### Tuesday, April 21, 10:30AM (New York, NY time)

### Uri Grupel, University of Innsbruk, Austria

#### Topic: Metric distortion of random spaces

#### Abstract: We consider a random set in the unit circle. Is the induced
discrete metric of the set closer to that of another independent random
set or to the evenly spaced set of the same cardinality? We measure the
distortion by looking at the smallest bi-Lipschitz norm of all the
bijections between the two sets. Since the distortion between two random
sets has infinite expectation, the talk will focus on the median. We
show that two random sets have "typically" smaller distortion than a
random set and an evenly spaced set.

### Saturday, April 25, 11:30AM (New York, NY time)

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

#### Topic: Quantitative Triangle law and Joint Normality of Lyapunov exponents for products of Gaussian matrices

#### Abstract: We will discuss spectral properties of products of independent Gaussian square matrices with independent entries. Non-asymptotic results for the statistics of the singular values will be presented as well as the rate of convergence to the triangle law. We will also show quantitative estimates on the asymptotic joint normality of the Lyapunov exponents. The talk is based on a joint work with Boris Hanin.

### Tuesday, April 28, 10:30AM (New York, NY time)

### Ilaria Fragalà, Politecnico di Milano, Italy

#### Topic: Symmetry problems for variational functionals: from continuous to discrete.

#### Abstract: I will discuss some symmetry problems for variational energies on the class of convex polygons
with a prescribed number of sides, in which the regular n-gon can be proved or is expected to be optimal.
Such symmetry results can be viewed as the “discrete” analogue of well-known “continuous” isoperimetric inequalities with balls as optimal domains.
I will focus in particular on the following topics

(i) Discrete isoperimetric type inequalities

(ii) Discrete Faber-Krahn type inequalities

(iii) Overdetermined boundary value problems on polygons.

### Saturday, May 2, 11:30AM (New York, NY time)

### Alina Stancu, Concordia University, Montreal, Canada

#### Topic: On the fundamental gap and convex sets in hyperbolic space

#### Abstract: The lower bound on the fundamental gap of the Laplacian on convex domains in R^n, with Dirichlet boundary conditions, has a long history and has been finally settled a few years ago with parabolic methods by Andrews and Clutterbuck. More recently, the same lower bound, which depends on the diameter of the domain, has been proved for convex sets on the standard sphere in several stages with several groups of authors, 2016-2018. Over the past year, together with collaborators, we have found that the gap on the hyperbolic space behaves strikingly different and we aim to explain it, particularly for this audience, as a difference in the nature of convex sets in H^n versus R^n or S^n.

### Tuesday, May 5, 10:30AM (New York, NY time)

#### Topic: Brunn-Minkowski type inequalities and affine surface area

#### Abstract: Does the affine surface area verify a concavity inequality of Brunn-Minkowski type? We will try to provide an answer to this question, and we will see that the answer depends on the dimension, and on the type of addition that we choose. The results presented in this talk were obtained in collaboration with Karoly Boroczky, Monika Ludwig and Thomas Wannerer.

### Saturday, May 9, 11:30AM (New York, NY time)

### Monika Ludwig, Vienna University of Technology, Austria

#### Topic: Valuations on Convex Functions

### Tuesday, May 12, 10:30AM (New York, NY time)

### Bo'az Klartag, Weizmann Institute of Science, Rehovot, Israel

#### Topic: Rigidity of Riemannian embeddings of discrete metric spaces

#### Abstract: Let M be a complete, connected Riemannian surface and
suppose that S is a discrete subset of M. What can we learn about M
from the knowledge of all distances in the surface between pairs of
points of S? We prove that if the distances in S correspond to the
distances in a 2-dimensional lattice, or more generally in an
arbitrary net in R^2, then M is isometric to the Euclidean plane. We
thus find that Riemannian embeddings of certain discrete metric spaces
are rather rigid. A corollary is that a subset of Z^3 that strictly
contains a two-dimensional lattice cannot be isometrically embedded in
any complete Riemannian surface. This is a joint work with M. Eilat.

### Saturday, May 16, 11:30AM (New York, NY time)

### Galyna Livshyts, Georgia Tech, Atlanta, GA, USA

#### Topic: On the Log-Brunn-Minkowski conjecture and related questions

#### Abstract: We shall discuss the Log-Brunn-Minkowski conjecture, a conjectured strengthening of the Brunn-Minkowski inequality proposed by Boroczky, Lutwak, Yang and Zhang, focusing on the local versions of this and related questions. The discussion will involve introduction and explanation of how the local version of the conjecture arises naturally, a collection of ‘’hands on’’ examples and elementary geometric tricks leading to various related partial results, statements of related questions as well as a discussion of more technically involved approaches and results. Based on a variety of joint results with several authors, namely, Colesanti, Hosle, Kolesnikov, Marsiglietti, Nayar, Zvavitch. REMARK: THIS TALK IS A LAST MINUTE REPLACEMENT OF THE EARLIER ANNOUNCED TALK BY TIKHOMIROV; TIKOMIROV'S TALK IS NOW SCHEDULED FOR THE FALL.

### Tuesday, May 19, 10:30AM (New York, NY time)

### Olivier Guedon, Université Gustave Eiffel, Paris, France

#### Topic: Floating bodies and random polytopes

#### Abstract: I will present some results about the geometry of centrally-symmetric random polytopes, generated by $N$ independent copies of a random vector $X$ taking values in $\R^n$. Under minimal assumptions on $X$, for $N \gtrsim n$ and with high probability, the polytope contains a deterministic set that is naturally associated with the random vector---namely, the polar of a certain floating body. This solves the long-standing question on whether such a random polytope contains a canonical body.
This is joint work with F. Krahmer, C. Kummerle, S. Mendelson and H. Rauhut.

### Saturday, May 23, 11:30AM (New York, NY time)

### Vitaly Milman, Tel Aviv University and Liran Rotem, Technion, Haifa, Israel

#### Topic: Novel view on classical convexity theory

#### Abstract: In this talk we will introduce and study the class of flowers. A flower in R^n is an arbitrary union of balls which contain the origin. While flowers are not necessarily convex, they are in one to one correspond with the class of convex bodies containing the origin, so by studying flowers we are also studying convex bodies from a new viewpoint. We will give several equivalent definitions of flowers and describe some of their basic properties. We will also discuss how to apply an arbitrary (real) function to a flower, and the corresponding construction for convex bodies. In particular, we will explain how to raise a flower to a given power. Finally, we will discuss some elements of the asymptotic theory of flowers. In particular we will present a Dvoretzky-type theorem for flowers which actually gives better estimates than the corresponding estimates for convex bodies.
Based on two papers by the speakers, the first of which is joint with E. Milman.

### Tuesday, May 26, 10:30AM (New York, NY time)

### Alexandros Eskenazis, Institut de Mathématiques de Jussieu, Sorbonne Université, Paris, France

#### Topic: The dimensional Brunn-Minkowski inequality in Gauss space

#### Abstract: We will present a complete proof of the dimensional Brunn-Minkowski inequality for origin symmetric convex sets in Gauss space. This settles a problem raised by Gardner and Zvavitch (2010). The talk is based on joint work with G. Moschidis.

### Saturday, May 30, 11:30AM (New York, NY time)

#### Topic: High-dimensional tennis balls

#### Abstract: In this talk, it will be explained what a high-dimensional tennis ball is, how one can construct it and its connection to V. Milman's question about well-complemented almost Euclidean subspaces of spaces uniformly isomorphic to $\ell_2^n$.

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

### Yair Shenfeld, Princeton University, NJ, USA

#### Topic: The extremal structures of the Alexandrov-Fenchel inequality

#### Abstract: The Alexandrov-Fenchel inequality is one of the fundamental results in the theory of convex bodies. Yet its equality cases, which are solutions to isoperimetric-type problems, have been open for more than 80 years. I will discuss recent progress on this problem where we confirm some conjectures by R. Schneider. Joint work with Ramon van Handel.

### Saturday, June 6, 11:30AM (New York, NY time)

### Mark Meckes, Case Western Reserve University, Cleveland, USA

#### Topic: Magnitude and intrinsic volumes of convex bodies

#### Abstract: Magnitude is an isometric invariant of metric spaces with origins in category theory. Although it is very difficult to exactly compute the magnitude of interesting subsets of Euclidean space, it can be shown that magnitude, or more precisely its behavior with respect to scaling, recovers many classical geometric invariants, such as volume, surface area, and Minkowski dimension. I will survey what is known about this, including results of Barcelo--Carbery, Gimperlein--Goffeng, Leinster, Willerton, and myself, and sketch the proof of an upper bound for the magnitude of a convex body in Euclidean space in terms of intrinsic volumes.

### Tuesday, June 9, 10:30AM (New York, NY time)

#### Topic: Volume product, polytopes and finite dimensional Lipschitz-free spaces.

#### Abstract: We shall present some results on the volume product of polytopes, including the question of its maximum among polytopes with a fixed number of vertices. Then we shall focus on the polytopes that are unit balls of Lipschitz-free Banach spaces associated to finite metric spaces. We characterize when these polytopes are Hanner polytopes and when two such polytopes are isometric to each others. We also also study the maximum of the volume product in this class.
Based on joint works with Matthew Alexander, Luis C. Garcia-Lirola and Artem Zvavitch.

### Saturday, June 13, 11:30AM (New York, NY time)

### Julian Haddad, Federal University of Minas Gerais, Belo Horizonte, Brasil

#### Topic: From affine Poincaré inequalities to affine spectral inequalities

#### Abstract: We develop the basic theory of $p$-Rayleigh quotients in
bounded domains, in the affine case, for $p \geq 1$. We establish
p-affine versions of the affine Poincaré inequality and introduce the
affine invariant $p$-Laplace operator $\Delta_p^{\mathcal A}$ defining
the Euler-Lagrange equation of the minimization problem. For $p=1$ we
obtain the existence of affine Cheeger sets and study preliminary
results towards a possible spectral characterization of John's
position.

### Tuesday, June 16, 10:30AM (New York, NY time)

### Semyon Alesker, Tel Aviv University, Israel

#### Topic: Multiplicative structure on valuations and its analogues over local fields.

#### Abstract: Valuation on convex sets is a classical notion of convex geometry. Multiplicative structure on translation invariant smooth valuations was introduced by the speaker
years ago. Since then several non-trivial properties of it have been discovered as well as a few applications to integral geometry. In the first part of the talk we will review some of these properties.
Then we discuss analogues of the algebra of even translation invariant valuations over other locally compact (e.g. complex, p-adic) fields. While any interpretation of these new algebras is missing at the moment,
their properties seem (to the speaker) to be non-trivial and having some intrinsic beauty.

### Saturday, June 20, 11:30AM (New York, NY time)

#### Topic: The best constant in the Khinchine inequality for slightly dependent random variables

#### Abstract: We solve the open problem of determining the best constant in the Khintchine inequality under condition that the Rademacher random variables are slightly dependent. We also mention some applications in statistics of the above result. The talk is based on a joint work with Susanna Spektor.

### Tuesday, June 23, 10:30AM (New York, NY time)

### Elizabeth Meckes, Case Western Reserve University, Cleveland, USA

#### Topic: On the eigenvalues of Brownian motion on \mathbb{U}(n)

#### Abstract: Much recent work in the study of random matrices has focused on the non-asymptotic theory; that is, the study of random matrices of fixed, large size. I will discuss one such example: the eigenvalues of unitary Brownian motion. I will describe an approach which gives uniform quantitative almost-sure estimates over fixed time intervals of the distance between the random spectral measures of this parametrized family of random matrices and the corresponding measures in a deterministic parametrized family \{\nu_t\}_{t\ge 0} of large-n limiting measures. I will also discuss larger time scales. This is joint work with Tai Melcher.

### Saturday, June 27, 11:30AM (New York, NY time)

#### Topic: Modewise methods for tensor dimension reduction

#### Abstract: Although tensors are a natural multi-modal extension of matrices, going beyond two modes (that is, rows and columns) presents many interesting non-trivialities. For example, the notion of singular values is no longer well-defined, and there are various versions of the rank. One of the most natural (and mathematically challenging) definitions of the tensor rank is so-called CP-rank: for a tensor X, it is a minimal number of rank one tensors whose linear combination constitutes X. Main focus of my talk will be an extension of the celebrated Johnson-Lindenstrauss lemma to low CP-rank tensors. Namely, I will discuss how modewise randomized projections can preserve tensor geometry in the subspace oblivious way (that is, a projection model is not adapted for a particular tensor subspace). Modewise methods are especially interesting for the tensors as they preserve the multi-modal structure of the data, acting on a tensor directly, without initial conversion of tensors to matrices or vectors. I will also discuss an application for the least squares fitting CP model for tensors. Based on our joint work with Mark Iwen, Deanna Needell, and Ali Zare.

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

#### Topic: Sharp stability of the Brunn-Minkowski inequality

#### Abstract: We consider recent results concerning the stability of the classic Brunn-Minkowski inequality. In particular we shall focus on the linear stability for homothetic sets. Resolving a conjecture of Figalli and Jerison, we show there are constants C,d>0 depending only on n such that for every subset A of R^n of positive measure, if |(A+A)/2 - A| <= d |A|, then |co(A) - A| <= C |(A+A)/2 - A| where co(A) is the convex hull of A. The talk is based on joint work with Hunter Spink and Marius Tiba.

## Schedule Fall 2020:

### Tuesday, August 25, 10:30AM (New York, NY time)

### Bo Berndtsson, Chalmers University of Technology and the University of Goteborg, Sweden

#### Topic: Complex integrals and Kuperberg's proof of the Bourgain-Milman theorem

#### Abstract: We give a proof of the Bourgain-Milman theorem using complex methods. The proof is inspired by Kuperberg's, but considerably shorter. Time permitting, we will also comment on Nazarov's proof and estimates of Bergman kernels.

### Saturday, August 29, 11:30AM (New York, NY time)

### Marton Naszodi, Alfred Renyi Inst. of Math. and Eotvos Univ., Budapest, Hungary

#### Topic: Some new quantitative Helly-type theorems

#### Abstract:
Quantitative Helly-type theorems were introduced by Bárány, Katchalski and Pach in 1982,
who, among other results, showed the following. There is a constant C_d depending on
the dimension d only, such that if the intersection of a finite family of convex bodies in R^d is
of volume at most one, then the intersection of some subfamily of 2d members is of volume
at most C_d. We consider colorful and fractional versions of this result.

### Tuesday, September 1, 10:30AM (New York, NY time)

### Kateryna Tatarko, Texas A&M University, College Station, TX, USA and the University of Alberta, Canada

#### Topic: On the unique determination of ellipsoids by dual intrinsic volumes

#### Abstract: In this talk, we show that an ellipsoid is uniquely determined up to an isometry by its dual Steiner polynomial. We reduce this result to the moment problem, and as a by-product obtain an alternative proof of the analogous known result for classical Steiner polynomials in $R^3$. This is joint work with S. Myroshnychenko and V. Yaskin.

### Saturday, September 5, 11:30AM (New York, NY time)

### Adam Kashlak, University of Alberta, Edmonton, Canada

#### Topic: Analytic Permutation Testing via Kahane--Khintchine Inequalities

#### Abstract: The permutation test is a versatile type of exact nonparametric significance test that requires drastically fewer assumptions than similar parametric tests by considering the distribution of a test statistic over a discrete group of distributionally invariant transformations. The main downfall of the permutation test is the high computational cost of running such a test making this approach laborious for complex data and experimental designs and completely infeasible in any application requiring speedy results. We rectify this problem through application of Kahane--Khintchine-type inequalities under a weak dependence condition and thus propose a computation free permutation test---i.e. a permutation-less permutation test. This general framework is studied within both commutative and non-commutative Banach spaces. We further improve these Kahane-Khintchine-type bounds via a transformation based on the incomplete beta function and Talagrand's concentration inequality. For k-sample testing, we extend the theory presented for Rademacher sums to weakly dependent Rademacher chaoses making use of modified decoupling inequalities. We test this methodology on classic functional data sets including the Berkeley growth curves and the phoneme dataset. We also consider hypothesis testing on speech samples under two experimental designs: the Latin square and the complete randomized block design.

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

### Grigory
Ivanov, IST Austria, Austria, and MIPT, Moscow, Russia.

#### Topic: Functional John-Lowner ellipsoids of a log-concave function

#### Abstract: We extend the notion of the John ellipsoid (the largest volume ellipsoid contained within a convex body) to the setting of log-concave functions. For every s > 0, we define a class of log-concave functions derived from ellipsoids. For any log-concave function f, and any fixed s > 0, we consider functions belonging to this class and find the one with the largest integral under the condition that it is pointwise less than or equal to f. We show that it exists and is unique, and call it the John s-function of f. We give a characterization of this function similar to the one provided by John in his fundamental
theorem. As an application, we obtain a quantitative Helly-type result about the integral of the pointwise minimum of a family of log-concave functions.
Next, we will discuss the concept of duality for log-concave functions and extend the notion of the Löwner ellipsoid (the smallest volume ellipsoid containing a convex body) to the setting of log-concave functions. Time permitting, we will discuss the difference between the behavior of convex sets and log-concave functions concerning our problems.
Based on joint works with Márton Naszódi and Igor Tsiutsiurupa.

### Saturday, September 12, 11:30AM (New York, NY time)

### Masha Gordina, University of Connecticut, Storrk, CT, USA

#### Topic: Uniform doubling on SU(2) and beyond

#### Abstract: Suppose G is a compact Lie group equipped with a left-invariant Riemannian metric. Such metrics usually form a finite-dimensional cone. The Riemannian volume measure corresponding to such a metric is the Haar measure of the group (up to a multiplicative constant). Because of compactness, each left-invariant metric g has the doubling property, that is, there exists a doubling constant D=D(G, g) such that the volume of the ball of radius 2r is at most D times the volume of the ball of radius r. We are concerned with the following question: does there exist a constant D(G) such that, for all left-invariant metrics g on G, the constant D(G, g) is bounded above by D(G)? This is what we call uniformly doubling. The conjecture is that any compact Lie group is uniformly doubling. The only cases for which the conjecture is known are Riemannian tori and the group SU(2). The talk will describe a number of analytic consequences of uniform doubling (in absence of curvature bounds) and our approach to proving uniform doubling on SU(2). The work in progress for U(2) might be mentioned as well. This is joint work with Nathaniel Eldredge (University Northern Colorado) and Laurent Saloff-Coste (Cornell University). Reference: Left-invariant geometries on SU(2) are uniformly doubling, GAFA 2018.

### Tuesday, September 15, 10:30AM (New York, NY time)

### Santosh Vempala, Georgia Institute of Technology, Atlanta, GA, USA

#### Topic: Reducing Isotropy to KLS: An n^3\psi^2 Volume Algorithm

#### Abstract: Computing the volume of a convex body is an ancient problem whose study has led to many interesting mathematical developments. In the most general setting, the convex body is given only via a membership oracle. In this talk, we present a faster algorithm for isotropic transformation of an arbitrary convex body in R^n, with complexity n^3\psi^2, where \psi bounds the KLS constant for isotropic convex bodies. Together with the known bound of \psi = O(n^{1/4}) [2017] and the Cousins-Vempala n^3 volume algorithm for well-rounded convex bodies [2015], this gives an n^{3.5} volume algorithm for general convex bodies, the first improvement on the n^4 algorithm of Lovász-Vempala [2003]. A positive resolution of the KLS conjecture (\psi = O(1)) would imply an n^3 volume algorithm.
No background on algorithms, KLS or ABC will be assumed for the talk.
Joint work with He Jia, Aditi Laddha and Yin Tat Lee.

Remark: a follow up talk with more details will take place at the Georgia Tech High Dimensional Seminar on Wednesday, September 16, at 3:15pm (NYC time).
Zoom link here.

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

### Ferenc Fodor, University of Szeged, Hungary

#### Topic: Strengthened inequalities for the mean width and the $\ell$-norm

#### Abstract: According to a result of Barthe the regular simplex maximizes the mean width of convex bodies whose John ellipsoid is the Euclidean unit ball. This is equivalent to the fact that the regular simplex maximizes the $\ell$-norm of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball.
The reverse statement that the regular simplex minimizes the mean width of convex bodies whose L\"owner ellipsoid is the Euclidean unit ball is also true as proved by Schmuckenschl\"ager. In this talk we prove strengthened stability versions of these results and some related stability statements for the convex hull of the support of centered isotropic measures on the unit sphere. This is joint work with K\'aroly J. B\"or\"oczky (Budapest, Hungary) and Daniel Hug (Karlsruhe, Germany).

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

### Dan Mikulincer, Weizmann Institute of Science, Rehovot, Israel

#### Topic: Stability of Stein kernels, moment maps and invariant measures

#### Abstract: Suppose that \mu is some nice measure on a Euclidean space. We can associate it with several different constructions of interest: Stein kernels, arising from Stein's theory, the moment map, which is of a more geometric flavour and a particular choice of a stochastic process for which \mu is the invariant measure. We will discuss the connections between these different objects and show that they are stable with respect to the original measure. That is, a small perturbation to either construction will yield a new measure which is close to \mu. Joint work with Max Fathi.

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

#### Topic: Non-asymptotic bound for the smallest singular value of powers of random matrices

#### Abstract: I will discuss a joint work with H.Huang on the smallest singular value of powers of Gaussian matrices and challenges in extending the obtained bound to non-Gaussian setting.

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

### Paata Ivanisvili, North Carolina State University, NC, USA

#### Topic: Enflo's problem

#### Abstract: A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of Enflo. The key feature of Enflo type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. I will speak about the joint work with Ramon van Handel and Sasha Volberg where we prove that Rademacher and Enflo type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the Hamming cube.

### Tuesday, October 13, 10:30AM (New York, NY time)

### Keith Ball, University of Warwick, UK

#### Topic: Rational approximations to the zeta function

#### Abstract: I will describe the construction of a sequence of rational functions with rational coefficients that converge to the zeta function. These approximations extend and make precise the spectral interpretations of the Riemann zeros found by Connes and by Berry and Keating.

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

### Naomi Feldheim, Bar Ilan University, Israel

#### Topic: Persistence of Gaussian stationary processes

#### Abstract: Let f:R->R be a Gaussian stationary process, that is, a random function which is invariant to real shifts and whose marginals have multi-normal distribution.
What is the probability that f remains above a certain fixed line for a long period of time?
This simple question, which was posed by mathematicians and engineers more than 60 years ago (e.g. Rice, Slepian), has some surprising answers which were discovered only recently. I will describe how a spectral point of view leads to those results.
Based on joint works with O. Feldheim, F. Nazarov, S. Nitzan, B. Jaye and S. Mukherjee.

### 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.

### Tuesday, November 3, 10:30AM (New York, NY time) (two talks 20 min each)

#### 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).

#### 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-Gutirrez and Mara A. Hernndez Cifre.

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

### Dima Faifman, Tel Aviv University, Israel

#### Topic: Crofton formulas in isotropic pseudo-Riemannian spaces.

#### Abstract: The length of a curve in the plane can be computed by counting the intersection points with a line, and integrating over all lines.
More generally, the intrinsic volumes (quermassintegrals) of a subset of Euclidean space can be computed by Crofton integrals, bringing forth their fundamental role in integral geometry.
In spherical and hyperbolic geometry, such formulas are also known and classical.
In pseudo-Riemannian isotropic spaces, such as de Sitter or anti-de Sitter space, one can similarly ask for an integral-geometric formula for the volume of a submanifold, or more generally for the intrinsic volumes of a subset, which were introduced only recently. I will explain how to obtain and apply such formulas, and how in fact there is a universal Crofton formula depending on a complex parameter extending the Riemannian Crofton formulas, for which all indefinite signatures are distributional boundary values. This is a joint work in progress with Andreas Bernig and Gil Solanes.

### 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.

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

#### Topic: Existence of potentials for non-traditional cost functions

#### Abstract: In this talk, we will present a new approach to the problem of existence of a potential for the optimal transport problem and apply it to non-traditional cost functions (i.e. costs that may attain infinite values). As a by-product, we give a new transparent proof of Rockafellar-Ruschendorf theorem. As an example of a non-traditional cost, we discuss the polar cost, which is particularly interesting as it induces the polarity transform and the class of geometric convex functions. This is joint work with S. Artstein-Avidan and S. Sadovsky.

### Tuesday, December 1, 10:30AM (New York, NY time)

### Shay Sadovsky, Tel Aviv University, Israel

#### Topic: Existence of potentials for non-traditional cost functions (part 2)

#### Abstract: In this talk, we present a constructive method for finding solutions to Monge's problem of mass-transport between two measures with respect to the polar cost. Our costruction, generalizing an idea of Keith Ball, utilizes a new notion of 'Hall polytopes', which we introduce. Our method applies to non-traditional transport problems, i.e. those with costs which can attain the value infinity, as well as the classical case. Based on joint work with Shiri Artstein-Avidan and Kasia Wyczesany.

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

#### Topic: Sharp Isoperimetric Inequalities for Affine Quermassintegrals

#### Abstract: The affine quermassintegrals associated to a convex body in $\R^n$ are affine-invariant analogues of the classical intrinsic volumes from the Brunn--Minkowski theory, and thus constitute a central pillar of affine convex geometry. They were introduced in the 1980's by E. Lutwak, who conjectured that among all convex bodies of a given volume, the $k$-th affine quermassintegral is minimized precisely on the family of ellipsoids. The known cases $k=1$ and $k=n-1$ correspond to the classical Blaschke--Santal\'o and Petty projection inequalities, respectively. In this work we confirm Lutwak's conjecture, including characterization of the equality cases, for all values of $k=1,\ldots,n-1$, in a single unified framework. In fact, it turns out that ellipsoids are the only \emph{local} minimizers with respect to the Hausdorff topology. In addition, we address a related conjecture of Lutwak on the validity of certain Alexandrov--Fenchel-type inequalities for affine (and more generally $L^p$-moment) quermassintegrals. The case $p=0$ corresponds to a sharp averaged Loomis--Whitney isoperimetric inequality. Based on joint work with Amir Yehudayoff.

### Tuesday, December 15, 10:30AM (New York, NY time)

### Pierre Youssef, NYU Abu Dhabi, United Arab Emirates

#### Topic: Mixing time of the switch chain on regular bipartite graphs.

#### Abstract: Given a fixed integer d, we consider the switch chain on the set of d-regular bipartite graphs on n vertices equipped with the uniform measure. We prove a sharp Poincar and log-Sobolev inequality implying that the mixing time of the switch chain is at most O(n log^2n) which is optimal up to a logarithmic term. This improves on earlier results of Kannan, Tetali, Vempala and Dyer et al. who obtained the bounds O(n^13 log n) and O(n^7 log n) respectively. This is a joint work with Konstantin Tikhomirov.

## Schedule Spring 2021:

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

### Yuansi Chen, Duke University, Durham, NC, USA

#### Topic: Recent progress on the KLS conjecture and the stochastic localization scheme of Eldan

#### Abstract: Kannan, Lovasz and Simonovits (KLS) conjectured in 1993 that the Cheeger isoperimetric coefficient of any log-concave density is achieved by half-spaces up to a universal constant factor. This conjecture also implies other important conjectures such as Bourgain's slicing conjecture (1986) and the thin-shell conjecture (2003). In this talk, first we briefly survey the origin and the main consequences of these conjectures. Then we present the development and the refinement of the main proof technique, namely Eldan's stochastic localization scheme, which results in the current best bounds of the Cheeger isoperimetric coefficient in the KLS conjecture.

### Tuesday, January 12, 2021:

### No seminar. This winter school is happening at the time of the seminar.

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

### Mark Agranovsky, Bar Ilan University, Ramat Gan, Israel

#### Topic: On integrable domains and surfaces

#### Abstract: Integrability of domains or surfaces in R^n is defined in terms of sectional or solid volume functions, evaluating the volumes of the intersections with affine planes or half-spaces. Study of relations between the geometry of domains and types of their volume functions is motivated by a problem of V.I. Arnold about algebraically integrable bodies, which in turn goes back to celebrated Newton's Lemma about ovals. The talk will be devoted to a survey of some recent works in this area.

### 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.

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

#### 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.

### 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 Voronois geometric theory of positive definite quadratic forms and is related to many questions in mathematical crystallography including Hilberts 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
### 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
### 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)

#### 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
### 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
### 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
### 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
### 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
### 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.

### 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.

### 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)

#### 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-Nicols and M. Trraga.

### 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.

### 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.

### 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 Boaz Klartag

## Schedule Fall 2021:

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

#### 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 Lwner 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.

### 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.

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

#### 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.

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

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

#### 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.

### 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 Fabrcio Caluza Machado.

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

#### 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

### 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.

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

### Eli Putterman, Weizmann Institute of Science, Rehovot, Israel

#### Topic: TBA

#### Abstract:

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

#### 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.

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

#### Topic: TBA

#### Abstract:

### 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..

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

#### 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.

### Organizers: