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,…

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…

I will discuss recent progress on dimension-free estimates for the Hardy--Littlewood maximal functions in the continuous and discrete settings. Website: https://sites.google.com/view/paw-seminar Host: Paata Ivanisvili  pivanis@ncsu.edu

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. Website: https://sites.google.com/view/paw-seminar Host: Paata Ivanisvili  pivanis@ncsu.edu

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…

Carleson embedding theorem is a building block for many singular integral operators and the main instrument in proving ``Leibniz rule" for fractional derivatives (Kato--Ponce, Kenig). It is also an essential step in all known ``corona theorems’’. Multi-parameter embedding is a tool to prove more complicated Leibniz rules that are also widely used in well-posedness questions…

In 2011, Lutwak, Yang and Zhang extended the definition of the Lp-Minkowski convex combination (p ≥ 1) from convex bodies containing the origin in their interiors to all measurable subsets in R n , and as a consequence, extended the Lp-Brunn-Minkowski inequality to the setting of all measurable sets. In this talk, I will present…

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 >= 1/2. We…

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…

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…

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…

Zoom ID: 939 3643 7199 (opens 15 min prior to the meeting) Passcode: 9917

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…

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…

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.

Zoom ID: 936 5420 1948 Password: the last four digits of the Zoom ID in reverse order

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…

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.   Zoom ID: 980 3116 7550 Passcode: last 4 digits of the Zoom ID in reverse order

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…

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.   Zoom…