Sign Uncertainty

Abstract

We will talk about the recent developments of the sign uncertainty principle and its relation with sphere packing and quadrature formulas. The talk will mainly be a report of the paper New Sign Uncertainty Principles, joint work with J. P. Ramos and D. Oliveira e Silva.

Sharp variance-entropy comparison for Gaussian quadratic forms

Abstract

We show that among nonnegative quadratic forms in n independent standard normal random variables, a diagonal form with equal coefficients maximizes differential entropy when variance is fixed. We also discuss some related open problems.

Reverse Euclidean and Gaussian isoperimetric inequalities for parallel sets with applications

Abstract

The r-parallel set of a measurable set A is the set of all points whose distance from A is at most r. In this talk, we discuss some recent results that establish upper bounds on the Euclidean and Gaussian surface areas of r-parallel sets. We also discuss a reverse form of the Brunn-Minkowski inequality for r-parallel sets, and as an aside a reverse entropy power inequality for Gaussian-smoothed random variables. We will conclude by presenting applications of our results to some problems of interest in theoretical machine learning.

Expected f-vector of the Poisson Zero Cell

Abstract

The Poisson hyperplane process describes, roughly speaking, infinitely many hyperplanes thrown uniformly at random into the d-dimensional Euclidean space. The hyperplanes dissect the space into countably many cells. The a.s. unique cell containing the origin is called the Poisson zero polytope. We prove an explicit combinatorial formula for the expected number of k-dimensional faces of the Poisson zero polytope. This number is expressed through the coefficients of the polynomial

$$

(1+ (d-1)^2 x^2) (1+(d-3)^2 x^2) (1+(d-5)^2 x^2) \ldots.

$$

We shall also discuss the analogue of the Sylvester four-point problem on the half-sphere as well as the following closely related problems. Sample n points $U_1,\ldots,U_n$ uniformly at random on the $d$-dimensional upper half-sphere. Let $C_n$ be the convex cone spanned by the vectors $U_1,\ldots,U_n$. What is the expected number of $k$-dimensonal faces of $C$? What is the expected solid angle of $C_n$?

A fuller picture of Brascamp--Lieb and Barthe-type inequalities

Abstract

This talk will mainly focus on the paper Euclidean forward-reverse Brascamp--Lieb inequalities, with Jingbo Liu. In particular, we show that a broad class of functional inequalities have Gaussian extremizers, and give necessary and sufficient conditions for finiteness of best constants. These results subsume many others in the literature; in particular, they unify and clarify the landscape of (Euclidean) Brascamp--Lieb and Barthe-type inequalities and the duality exhibited by sharp constants.

The contact process on random graphs

Abstract

The contact process is a model of the spread of diseases on networks. In this talk, we will discuss the phase transitions of the contact process on random graphs. In particular, we derive the necessary and sufficient conditions for the existence of the extinction phase and the first-order asymptotics for the extinction-survival threshold.

Matrix weights and finite rank perturbations

Abstract

The matrix-valued (and operator-valued, especially trace-class-valued) measures provide a natural language in the perturbation theory. They appeared in the earlier days of the spectral theory (de Branges, Kuroda), and were used, in particular, for one of the proofs of the Kato--Rosenbmlum theorem about preservation of the absolutely continuous spectrum.

Turns out that they are also quite useful for the investigation of the singular parts of the spectrum. Namely, the classical Aronszajn--Donoghue theorem states that the singular parts of the spectral measures of a self-adjoint operator and its rank one perturbation (by a cyclic vector) are mutually singular. While simple direct sum type examples would indicate that such result is impossible for the scalar spectral measures, it holds if one introduces the notion of vector mutual singularity of matrix-valued measures.

Two weight estimates with matrix weights and the matrix A₂​​ condition appear natively in this context, and will be used to prove the Aronszajn--Donoghue type theorem for finite rank perturbations.

The results can be generalized to the case of trace class perturbations.

The talk is based on a joint work with C. Liaw.

Some results in Banach space-valued time frequency analysis

Abstract

SIO (Singular Integral Operator) theory and, Calderón-Zygmund theory specifically, developed starting from the '60s, provides a vast array of tools for dealing with operators that resemble the Hilbert transform, an ubiquitous operator in Complex Analysis, semi-linear PDEs, and many other branches of mathematics. Results valid for complex-valued functions were extended to Banach spaces-valued functions thanks to Bourgain's groundbreaking work on the deep relation between Banach space geometry and boundedness properties of vector-valued SIOs. Scalar-valued bounds for multilinear SIOs, like the bilinear Hilbert transform, are classic in time-frequency-scale analysis. Banach-space valued results have appeared only in the last couple of years. The well understood connections with Banach space geometry from linear theory are just starting to be investigated. Open questions and generalizations to non-commutative analysis abound and would come hand-in-hand with progress in understanding SIOs with worse singularities than of Calderón-Zygmund type that can often be realized as SIO-valued CZ operators.

Zoom ID: 928 3597 1183 Link (opens 15 min prior to the meeting)
Passcode: PAW2020
Time: 3:00 pm (New York)

On some aspects of uncountable ergodic theory

Abstract

The talk aims at providing an introduction into some basic problems occurring in the ergodic theory of uncountable group actions and a setup and a few tools on how to resolve these issues. This part of the talk shall be accessible to anyone with a graduate-level background in probability and analysis. Towards the end of the talk some actual results in uncountable ergodic theory will be presented aimed at a more specialized audience. Most results will be drawn from recent preprints joint with Terence Tao, and if time permits I will also include some results from forthcoming papers by myself and with other co-authors in the area of multiple recurrence.

On the HRT Conjecture

Given a non-zero square-integrable function $g$ and $\Lambda=\{(a_k, b_k)\}_{k=1}^N \subset \mathbb{R}^2$ let $\mathcal{G}(g, \Lambda)=\{e^{2\pi i b_k \cdot}g(\cdot - a_k)\}_{k=1}^N.$ The Heil-Ramanathan-Topiwala (HRT) Conjecture is the question of whether $\mathcal{G}(g, \Lambda)$ is linearly independent. For the last two decades, very little progress has been made in settling the conjecture. In the first part of the talk, I will give an overview of the state of the conjecture focusing especially on the case $N\leq 4$. I will then describe some recent attempts in settling the conjecture for some special classes of functions and special sets $\Lambda$.

Random Cones

Let $U_1,\ldots,U_n$ be independent random vectors which are uniformly distributed on the unit sphere. The random hyperplanes $U_1^\perp,\ldots,U_n^\perp$ dissect the space into a collection of random cones. A uniform random cone $S_n$ from this collection is called the Schläfli random cone. In a classical paper of Cover and Efron (1967) it was proved that the expected number of $k$-dimensional faces of a cross section of $S_n$ converges to the number of $k$-dimensional faces of a cube. We investigate the question whether a similar convergence is true also for the random shape of these cross sections. This talk is based on joint work with Zachary Kabluchko and Daniel Temesvari.

Dimension free estimates for the discrete Hardy--Littlewood maximal functions

I will discuss recent progress on dimension-free estimates for the Hardy--Littlewood maximal functions in the continuous and discrete settings.

Marstrand's Theorem in general Banach spaces

We will discuss Marstrand's classical theorem concerning the interplay between density of a measure and the Hausdorff dimension of the measure's support in the context of finite-dimensional Banach spaces. This is joint work with David Bate and Tatiana Toro.

Weak-type Gagliardo spaces and applications

The celebrated Bourgain--Brezis--Mironescu formula enables us to recover Sobolev spaces in terms of limits of Gagliardo seminorms. Very recently, Brezis, Van Schaftingen and Yung have proposed an alternative methodology to approach Sobolev spaces via limits of weak-type Gagliardo functionals. In this talk we will show that the BvSY result is a special case of a more general phenomenon based on maximal inequalities and sequences of operators. In particular, we shall derive not only analogs of the BvSY theorem for different kinds of function spaces (Lebesgue, Calderon, higher-order Sobolev, …), but also applications to the PDE's, ergodic theory, Fourier series, etc This is joint work with Mario Milman.

Stochastic processes for Boolean profit

Not even influence inequalities for Boolean functions can escape the long arm of stochastic processes. I will present a (relatively) natural stochastic process which turns Boolean functions and their derivatives into jump-process martingales. There is much to profit from analyzing the individual paths of these processes: Using stopping times and level inequalities, we will reprove an inequality of Talagrand relating edge boundaries and the influences, and say something about functions which almost saturate the inequality. The technique (mostly) bypasses hypercontractivity.

Work with Ronen Eldan. For a short, animated video about the technique (proving a different result, don't worry).