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Time: 3:00 pm (New York)

On Lp-Brunn-Minkowski type and Lp-isoperimetric type inequalities for measures.

Abstract [PDF]

On the distribution of Randomly Signed Sums and Tomaszewski’s Conjecture

A Rademacher sum X is a random variable characterized by real numbers a_1, ..., a_n, and is equal to

X = a_1 x_1 + ... + a_n x_n, where x_1, ..., x_n are independent signs (uniformly selected from {-1, 1}).

A conjecture by Bogusław Tomaszewski, 1986: all Rademacher sums X satisfy Pr[ |X| <= sqrt Var(X) ] >= 1/2.

We prove the conjecture, and discuss other ways in which Rademacher sums behave like normally distributed variables.

Joint work with Nathan Keller.

On an inequality somewhat related to the Log-Brunn-Minkowski conjecture

I shall prove a neat inequality concerning convex bodies and semi-norms, and show that cylinders (i.e. direct products of intervals with convex bodies of lower dimension) give equality cases for it. I shall explain its connections to the Log-Brunn-Minkowski conjecture. If time permits, I will discuss related results and surrounding questions. Based on a joint work with A. V. Kolesnikov https://arxiv.org/pdf/2004.06103.pdf. The talk will be whiteboard.

Lieb-Thirring bounds and other inequalities for orthonormal functions

We discuss extensions of several inequalities in harmonic analysis to the setting of families of orthonormal functions. While the case of Sobolev-type inequalities is classical, newer results concern the Strichartz inequality, the Stein-Tomas inequality and Sogge’s spectral cluster estimates, among others. Of particular interest is the dependence of the constants in the resulting bounds on the number of functions and we will present some optimal results.

The talk is based on joint work with Julien Sabin.

Universality for the minimum modulus of random trigonometric polynomials

We consider the restriction to the unit circle of random degree-n polynomials with iid coefficients (Kac polynomials). Recent work of Yakir and Zeitouni shows that for Gaussian coefficients, the minimum modulus (suitably rescaled) follows a limiting exponential distribution. We show this is a universal phenomenon, extending their result to arbitrary sub-Gaussian coefficients, such as Rademacher signs. Our approach relates the joint distribution of small values at several angles to that of a random walk in high-dimensional phase space, for which we obtain strong central limit theorems. The case of discrete coefficients is particularly challenging as the distribution is then sensitive to arithmetic structure among the angles. Based on joint work with Hoi Nguyen.

Extension technique for non-symmetric operators, or boundary traces of non-symmetric diffusions

The classical Dirichlet-to-Neumann operator maps the boundary values of a harmonic function in a domain to the normal derivative along the boundary. It is a classical result that the Dirichlet-to-Neumann operator for the half-plane is the square root of the 1-D Laplace operator. Caffarelli and Silvestre found a similar description of a general fractional power of the Laplace operator, by using solutions of an appropriate elliptic problem in the half-plane. Probabilistic counterpart of the former result is the following fact, due to Spitzer: the trace left by the reflected planar Brownian motion in a half-plane on the boundary coincides with the Cauchy process (a symmetric stable Lévy process with index 1). Similarly, Caffarelli–Silvestre extension technique corresponds to identification of symmetric stable Lévy processes with traces of appropriate diffusions in a half-plane by Molchanov and Ostrovski. Various extensions of these results to more general operators and diffusions have been studied. However, no results have been available for essentially non-symmetric operators or Lévy processes; for example, a representation of non-symmetric stable Lévy processes have been missing. In my talk I will discuss the results of two recent preprints: arXiv:1907.11444 and arXiv:1912.00072, which provide a complete description of 1-D non-local operators, or 1-D Lévy processes, that arise as Dirichlet-to-Neumann maps for appropriate elliptic operators, or boundary traces of appropriate diffusions in a half-plane. A key ingredient in the existence part is played by a beautiful result by Eckhardt and Kostenko.

Three Candidate Plurality is Stablest for Small Correlations

Suppose we model n votes in an election between two candidates as n i.i.d. uniform random variables in {-1,1}, so that 1 represents a vote for the first candidate, and -1 represents a vote for the other candidate. Then, for each vote, we flip a biased coin (with fixed probability larger than 1/2 of landing heads). If the coin lands tails, the vote is changed to the other candidate. In an election where each voter has a small influence on the election's outcome, and each candidate has an equal chance of winning, the majority function best preserves the election's outcome, when comparing the original election vote to the corrupted vote. This Majority is Stablest Theorem was proven in 2005 by Mossel-O'Donnell-Oleszkiewicz. The corresponding statement for elections between three or more voters has remained open since its formulation in 2004 by Khot-Kindler-Mossel-O'Donnell. We show that Plurality is Stablest for elections with three candidates, when the original and corrupted votes have a sufficiently small correlation. In fact, this result is a corollary of a more general structure theorem that applies for elections with any number of candidates and any correlation parameter.

(joint with Alex Tarter)

Dyadic analysis (virtually) meets number theory

Abstract: In this talk we discuss two ways in which dyadic analysis and number theory share a rich interaction. The first involves a complete classification of "distinct dyadic systems". These are sets of grids which allow one to compare any Euclidean ball nicely with any dyadic cube, and allow for showing that a large number of continuous objects and operators can be "replaced" with their easier dyadic counterparts. Secondly, we define and make progress on showing the (failure) of a "Hasse principle" in harmonic analysis; specifically, we discuss the interplay between number theory and dyadic analysis that allows us to construct a measure that is "p-adic" doubling for any prime p (in a finite set of primes), yet not doubling overall.

Metric Influence inequalities

Talagrand's influence inequality (1994) is an asymptotic improvement of the classical Poincaré inequality on the Hamming cube with numerous applications to Boolean analysis, discrete probability theory and geometric functional analysis. In this talk, we shall introduce a metric space-valued version of Talagrand's inequality and show its validity for some natural classes of spaces. Emphasis will be given to the probabilistic aspects of the proofs. We will also explain a geometric application of this metric invariant to the bi-Lipschitz embeddability of a natural family of finite metrics and mention related open problems. The talk is based on joint work with D. Cordero-Erausquin.

Stochastic functional inequalities and shadow systems

I will discuss stochastic geometry of random concave functions. In particular, I will explain how a "local" stochastic dominance underlies several functional inequalities. Emphasis will be on a notion of shadow systems for s-concave functions and their interplay with functional inequalities. Based on joint works with J. Rebollo Bueno.

Generalized probabilistic theories and tensor products of normed spaces

Generalized Probabilistic Theories (GPTs) form an abstract framework to describe theories of nature that have probabilistic features. A GPT must specify the set of states purporting to represent the physical reality, the allowable measurements, the rules for outcome statistics of the latter, and the composition rules describing what happens when we merge subsystems and create a larger system. Examples include classical probability and quantum theory.

The composition rules alluded to above usually involve tensor products and, in some formulations, normed spaces. Among tensor products of normed spaces that have operational meaning in the GPT context, the projective and the injective product are the extreme ones, which leads to the natural question "How much do they differ?" considered already by Grothendieck and Pisier (in the 1950s and 1980s). Surprisingly, no systematic quantitative analysis of the finite-dimensional case seems to have ever been made. We show that the projective/injective discrepancy is always lower-bounded by the power of the (smaller) dimension, with the exponent depending on the generality of the setup. Some of the results are essentially optimal, but others can be likely improved. The methods involve a wide range of techniques from geometry of Banach spaces and random matrices.

Joint work with G. Aubrun, L. Lami, C. Palazuelos, A. Winter.

On the real Davies' conjecture

We show that every $n \times n$ real matrix $A$ is within distance $\delta \|A\|$ in the operator norm of an $n\times n$ real matrix $A'$ whose eigenvectors have condition number $\tilde{O}(\text{poly}(n)/\delta)$. In fact, we show that with high probability, an additive i.i.d. sub-Gaussian perturbation of $A$ has this property. Up to log factors, this confirms a speculation of E.B. Davies.

Based on joint work with Ashwin Sah (MIT) and Mehtaab Sawhney (MIT)

On the polynomial Szemer\'edi theorem and related results

In this talk, I'll survey recent progress on problems in additive combinatorics, harmonic analysis, and ergodic theory related to Bergelson and Leibman's polynomial generalization of Szemer\'edi's theorem.

Trilinear embedding theorem for elliptic partial differential operators in divergence form with complex coefficients

We introduce the notion of p-ellipticity of a complex matrix function and discuss basic examples where it plays a major role, as well as the techniques that led to the introduction of the notion. In the second part of the talk we focus on a so-called trilinear embedding theorem for complex elliptic operators and its corollaries. The talk is based on collaboration with Andrea Carbonaro (U. Genova) and Vjekoslav Kovač and Kristina Škreb (U. Zagreb).

Discrete analogues of maximally modulated singular integrals of Stein-Wainger type

Stein and Wainger introduced an interesting class of maximal oscillatory integral operators related to Carleson's theorem. The talk will be about joint work with Ben Krause on discrete analogues of some of these operators. These discrete analogues feature a number of substantial difficulties that are absent in the real-variable setting and involve themes from number theory and analysis.