In order to join a seminar on a Saturday, click the link: Saturday link (the link opens half an hour before the talk)

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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.

On Saturdays, the lectures start one hour later, that is at:

8:30am in Los-Angeles, CA

9:30am in Edmonton, AB

10:30am in Columbia MO; College Station, TX; Chicago, IL

11:30am in Kent, OH; Atlanta, GA; Montreal; New York, NY

12:30pm in Rio de Janeiro, Buenos Aires

4:30pm (16:30) in London

5:30pm (17:30) in Paris, Milan, Budapest, Vienna

6:30pm (18:30) in Tel Aviv.

(We apologize in case we forgot to add someone to the e-mail list and very much appreciate you entering your e-mail!)

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

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

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

#### Abstract (click this link to view pdf)

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### 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.

#### Video of the talk

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### 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).

#### Video of the talk

### 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.

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### 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.

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### 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.

#### Video of the talk

### 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.#### Slides of the talk

#### Video of the talk

### 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.

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

### Andrea Colesanti, University of Florence, Italy

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

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

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

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

#### Topic: Valuations on Convex Functions

#### Abstract (click this link to view pdf)

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### 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.

#### Video of the talk

#### Notebook of the talk

#### Chat file from the talk

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

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

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

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

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### 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.

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### 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.

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### 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.

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

### Timothy Gowers, Cambridge University, UK

#### Topic: High-dimensional tennis balls

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

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### 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.

#### Video of the talk

### 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.

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

### Matthieu Fradelizi, Marne-la-Vallée, Paris, France

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

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

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### 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.

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### 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.

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

### Orli Herscovici, University of Haifa, Israel

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

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

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### 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.

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#### A cool video of the spectral measure evolution

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

### Elizaveta Rebrova, UCLA, Los-Angeles, CA, USA

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

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

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

### Peter Van Hintum, Cambridge University, UK

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

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

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#### Galyna Livshyts, Georgia Tech glivshyts6@math.gatech.edu

#### Liran Rotem, Technion lrotem@technion.ac.il

#### Dmitry Ryabogin, Kent State University ryabogin@math.kent.edu

#### Konstantin Tikhomirov, Georgia Tech ktikhomirov6@gatech.edu

#### Artem Zvavitch, Kent State University zvavitch@math.kent.edu