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

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 $K_N$.

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

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). 

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

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

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.

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