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# Threshold for the expected measure of random polytopes

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Let $\mu$ be a log-concave probability measure on ${\mathbb R}^n$ and for any $N>n$ consider the random polytope $K_N={\rm conv}\{X_1,\ldots ,X_N\}$, where $X_1,X_2,\ldots $ are independent random points in ${\mathbb R}^n$ distributed according to $\mu $. We study the question if there exists a threshold for the expected measure of $K_N$.

For zoom ID see

https://sites.google.com/view/paw-seminar

# Universality in nonasymptotic random matrix theory

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Nonasymptotic random matrix theory aims to estimate spectral statistics (such as the extreme eigenvalues) of rather general random matrix models in a quantitative fashion. Such results are often first established under Gaussian or sub-Gaussian assumptions, and much work is then devoted to extending such bounds to more general situations. In this talk I will discuss a very different perspective on such problems: under remarkably weak structural assumptions, one can show in a precise nonasymptotic manner that the behavior of random matrices is accurately captured by that of an associated Gaussian model, regardless of the behavior of the Gaussian model itself. When combined with recent developments in the understanding of Gaussian random matrices, this nonasymptotic universality principle yields a powerful "black box" tool for understanding the behavior of extremely general nonhomogeneous and non-Gaussian random matrix models. If time permits, I will discuss applications to random graphs, spiked models, sample covariance matrices, and/or free probability theory. (Based on joint work with Tatiana Brailovskaya.)

# Multi-Bubble Isoperimetric Problems - Old and New

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The classical isoperimetric inequality in Euclidean space R^n states that among all sets ("bubbles") of prescribed volume, the Euclidean ball minimizes surface area. One may similarly consider isoperimetric problems for more general metric-measure spaces, such as on the n-sphere S^n and on n-dimensional Gaussian space G^n (i.e. R^n endowed with the standard Gaussian measure). Furthermore, one may consider the "multi-bubble" isoperimetric problem, in which one prescribes the volume of p ≥ 2 bubbles (possibly disconnected) and minimizes their total surface area -- as any mutual interface will only be counted once, the bubbles are now incentivized to clump together. The classical case, referred to as the single-bubble isoperimetric problem, corresponds to p=1; the case p=2 is called the double-bubble problem, and so on.

In 2000, Hutchings, Morgan, Ritoré and Ros resolved the double-bubble conjecture in Euclidean space R^3 (and this was subsequently resolved in R^n as well) -- the boundary of a minimizing double-bubble is given by three spherical caps meeting at 120-degree angles. A more general conjecture of J. Sullivan from the 1990's asserts that when p ≤ n+1, the optimal multi-bubble in R^n (as well as in S^n) is obtained by taking the Voronoi cells of p+1 equidistant points in S^n and applying appropriate stereographic projections to R^n (and backwards).

In 2018, together with Joe Neeman, we resolved the analogous multi-bubble conjecture for p ≤ n bubbles in Gaussian space G^n -- the unique partition which minimizes the total Gaussian surface area is given by the Voronoi cells of (appropriately translated) p+1 equidistant points. In the talk, I will describe our approach in that work, as well as recent progress we have made on the multi-bubble problem on R^n and S^n. In particular, we show that minimizing bubbles in R^n and S^n are always spherical when p ≤ n, and we resolve the latter conjectures when in addition p ≤ 5 (e.g. the triple-bubble conjectures when n ≥ 3 and the quadruple-bubble conjectures when n ≥ 4).

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# Time-frequency localization operators, their eigenvalues and relationship to elliptic PDE

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In the classical realm of time-frequency analysis, a classical object of interest is the short-time Fourier transform of a function. This object is a modified Fourier transform of a signal f(x), modified by a certain 'window function', in order to make joint time-frequency analysis of functions more feasible.

Since the pioneering work of Daubechies, time-frequency localisation operators have been of extreme importance in that analysis. These are defined through V^∗1_Ω V f=P_Ω f, where V denotes the short-time Fourier transform with some fixed window. These operators seek to measure how much a function concentrates in the time-frequency plane, and thus the study of their eigenvalues and eigenfunctions is intimately connected to the previous questions.

In this talk, we will explore the case of a Gaussian window function φ(x)=e−πx^2, and the operators thus obtained. We will discuss some classical and recent results on domains of maximal time-frequency concentration, their eigenvalues, and inverse problems associated with such properties. During this investigation, we shall see that many of these problems possess some rather unexpected connections with overdetermined elliptic boundary value problems and free boundary problems in general. This is based on recent joint work with Paolo Tilli.