# Trimester Seminar Series

# Talks

## Time: every Thursday, 4 p.m. (CET)

Zoom data will be provided to all participants of the trimester program and all persons who registered for the winter school or one of the two workshops in due time.

**April 15, 2021**

**Kavita Ramanan (Brown University)**

## Applications of sharp large deviation estimates in asymptotic convex geometry

Recent work has shown the utility of large deviations techniques in the study of certain questions in asymptotic convex geometry. In this talk, we will describe sharp large deviation estimates in the spirit of Bahadur-Ranga Rao, and show how they can be used to obtain refined estimates for random projections, norms and volumes of Orlicz balls.

This talk is based on joint works with Yin-Ting Liao.

**April 8, 2021**

**Shiri Artstein-Avidan (University of Tel Aviv)**

## On optimal transport with respect to non traditional costs

After a short review of the topic of optimal transport, introducing the c-transform and c-subgradients, we will dive into the intricacies of transportation with respect to a cost which can attain infinite values ("non-traditional") and discuss necessary and sufficient conditions on pairs of measures for a transport plan supported on a c-subgradient (a Brenier-type map) to exist between them. A main example will be the polar cost underlying the polarity transform for functions. Joint work with Shay Sadovsky and Kasia Wyczesany.

**April 1, 2021**

**Keith Ball (University of Warwick)**

## Restricted Invertibility

I will briefly discuss the Kadison-Singer problem and then explain a beautiful argument of Bourgain and Tzafriri that I included in an article commissioned in memory of Jean Bourgain.

**March 11, 2021**

**Sergey Bobkov (University of Minnesota)**

## Concentration functions and entropy bounds for discrete log-concave distributions

We will be discussing two-sided bounds for concentration functions and Renyi entropies in the class of discrete log-concave probability distributions. They are used to derive certain variants of the entropy power inequalities. The talk is based on a joint work with Arnaud Marsiglietti and James Melbourne.

**March 4, 2021**

**Masha Gordina (University of Connecticut)**

## Stochastic analysis and geometric functional inequalities

We will survey different methods of proving functional inequalities for hypoelliptic diffusions and the corresponding heat kernels. Some of these methods rely on geometric methods such as curvature-dimension inequalities (due to Baudoin-Garofalo), and some are probabilistic such as couplings. If time permits we will also mention recent applications to ergodicity for Langevin dynamics. This is based on joint work with F. Baudoin, B. Driver, T. Melcher, Ph. Mariano et al.

**February 18, 2021**

**Alexander Litvak (University of Alberta)**

## A remark on the minimal dispersion

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.

**February 11, 2021**

**Daniel Hug (Karlsruher Institut für Technologie)**

## Random tessellations in hyperbolic space - first steps

Random tessellations in Euclidean space are a classical topic and highly relevant for many applications. Poisson hyperplane tessellations present a particular model for which mean values and variances for functionals of interest have been studied successfully and a central limit theory has been developed. In recent years, similar results have been obtained in spherical space. The purpose of this presentation is to discuss a new dimension dependent phenomenon which arises for Poisson hyperplane tessellations in hyperbolic space. In particular, we consider the $k$-volume of the $k$-skeleton induced by such a tessellation within a geodesic ball of radius $r$ and ask whether it satisfies a CLT. If $r$ is fixed and the intensity $t$ of the underlying Poisson process is increasing, the answer is yes. However, if $t$ is fixed and $r\to\infty$, then the situation is very different and is in contrast to corresponding results in Euclidean space.

Joint work with Felix Herold and Christoph Thäle.

**February 4, 2021**

**Alexander Kolesnikov (National Research Institute, Higher School of Economics, Moscow)**

## Blaschke-Santalo inequality for many functions and geodesic barycenters of measures

Motivated by the geodesic barycenter problem from optimal transportation theory, we prove a natural generalization of the Blaschke-Santalo inequality for many sets and many functions. We derive from it an entropy bound for the total Kantorovich cost appearing in the barycenter problem.

The talk is based on joint work with Elisabeth Werner.

**January 28, 2021**

**Andrea Colesanti (University of Florence) **

## An overview on a young research topic: valuations on spaces of functions

I will start from the theory of valuations on convex bodies, which for me was the main motivation to study corresponding functionals in an analytic setting. Then I will devote some time to the notion of valuations on a space of functions. After a general review on this topic, I will describe in some detail recents results containing classifications of valuations on the space of Lipschitz functions and on the space of convex functions.

**January 21, 2021**

**Ramon van Handel (Princeton University)**

## The mysterious extremals of the Alexandrov-Fenchel inequality

The Alexandrov-Fenchel inequality is a far-reaching generalization of the classical isoperimetric inequality to arbitrary mixed volumes. It is one of the central results in convex geometry, and has deep connections with other areas of mathematics. The characterization of its extremal bodies (i.e., its equality cases) is a long-standing open problem that dates back to the original works of Minokwski (1903) and Alexandrov (1937). The known extremals are already numerous and strikingly bizarre, and a fundamental but incomplete conjecture on their general structure, due to Loritz and Schneider, has remained wide open except in some very special cases.

Significant new progress on these problems was made in joint works with Yair Shenfeld. In particular, we recently succeeded to characterize all extremals of the Alexandrov-Fenchel inequality for convex polytopes, which completely settles the combinatorial aspect of the problem. In this talk, I aim to describe this result and some key insights that appear in the proof. The talk will be nontechnical and will assume minimal background.

If time permits, I may discuss a nontrivial application to extremal combinatorics, and indicate some tantalizing questions that arise from that. I will also outline what seems to be the key analytic difficulty in passing from the combinatorial setting to general convex bodies.