Time: 3:00pm (New York time zone)

Zoom Meeting ID:  970 0901 9678

Password: last 4 digits of the zoom meeting ID in reverse order

Threshold for the expected measure of random polytopes

Abstract: 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 $K_N$.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  930 0738 6410

Password: last 4 digits of the zoom meeting ID in reverse order

Laplacians and Polynomials on Spheres: the Large-N Limit

Abstract: We discuss the behavior of the Laplacian and its eigenfunctions for the sphere in N-dimensions of radius square-root of N.  We also discuss purely algebraic aspects of classical polynomials over spheres.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  974 9137 6197

Password: last 4 digits of the zoom meeting ID in reverse order

On the Lp dual Minkowski problem for −1 < p < 0

The Lp dual curvature measure was introduced by Lutwak, Yang, and Zhang in 2018. The associated Minkowski problem, known as the Lp dual Minkowski problem, asks about existence of a convex body with prescribed Lp dual curvature measure. This question unifies the previously disjoint Lp Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this talk, we will discuss the existence of a solution to the Lp dual Minkowski problem for the case of q < p + 1, −1 < p < 0, and p ̸= q for even measures.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  449 762 7644

Password: the last 4 digits of the zoom  ID in reverse order

Orthogonal projections and sumset estimates in convex geometry

In this talk we will discuss old and not so old inequalities on the volume of the orthogonal projections (sometimes called local Loomis-Whitney type estimates).  We will explore connections of those inequalities to inequalities for Mixed Volumes.  We will also show  the links between those inequalities and  a number of   interesting inequalities in Convex Geometry which are  inspired by sumsets estimates in additive combinatorics and classical facts from the  information theory. This is a joint work with Matthieu Fradelizi, Mokshay Madiman and Mathieu Meyer.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  449 762 7644

Password: the last 4 digits of the zoom  ID in reverse order

Moments, concentration, and entropy of log-concave distributions

In this talk I will present a simple mechanism, combining log-concavity and majorization in the convex order to derive moments, concentration, and entropy inequalities for random variables that are log-concave with respect to a reference measure.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  449 762 7644

Password: the last 4 digits of the zoom  ID in reverse order

On a converse of Fatou's theorem

Fatou's theorem states that a bounded analytic function in the unit disc has radial limits a.e. on the unit circle $T$. We prove the following theorem in the converse direction.

Theorem 1. Let $E$ be a subset on the unit circle $T$. There exists a bounded analytic function in the unit disc which has no radial limits on $E$ but has unrestricted limits at the remaining points of $T$ if and only if $E$ is an $F_\sigma$ set of measure zero.

The sufficiency part of this theorem immediately implies a known theorem of Lohwater and Piranian. The proof of Theorem 1 is based on Fatou's interpolation theorem, for which the author has recently suggested a new simple proof. 

It turns out that for the Blaschke products, a well-known subclass of bounded analytic functions, Theorem 1 takes the following form.   

Theorem 2. Let $E$ be a subset on the unit circle $T$. There exists a Blaschke product which has no radial limits on $E$ but has unrestricted limits at all the remaining points of $T$ if and only if $E$ is a closed set of measure zero.

(Theorem 2 is a joint result with Spyros Pasias.) Even the proof of the necessity part of Theorem 2 uses a new argument in the classical topic of boundary behavior of Blashchke products.

References.

1. A. A. Danielyan, On Fatou's theorem, Anal. Math. Phys. V. 10, Paper no. 28, 2020.

2. A. A. Danielyan, A proof of Fatou's interpolation theorem, J. Fourier Anal. Appl., V. 28, Paper no. 45, 2022.

3. A. A. Danielyan and S. Pasias, On a boundary property of Blaschke products, to appear in Anal. Mathematica

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  939 0054 5443

Password: the last 4 digits of the zoom  ID in reverse order

Conditioning of random matrices and polytopes, 

and the smoothed complexity of Frank-Wolfe methods

This talk is about condition measures of random matrices and polytopes. These are generalizations of the condition number of a matrix, that is, numbers that quantify the ill-posedness of a problem based on a matrix or polytope from the viewpoint of numerical stability or algorithmic complexity. Our main motivation is the analysis of Frank-Wolfe methods, popular for optimization over a polytope. The main condition measure we consider for a polytope is its vertex-facet distance (Beck and Shtern), namely the minimum distance between any vertex and the affine hull of any facet not containing the vertex. We show that for the convex hull of random Gaussian points it is exponentially small as a function of the dimension d, even for as few as cd points for c>1.

Our argument for polytopes is a refinement of an argument that we develop to study the conditioning of random matrices. The basic argument shows that for c>1 a d-by-n random Gaussian matrix with n ≥ cd has a d-by-d submatrix with minimum singular value that is exponentially small with high probability. This also has consequences on known results about the robust uniqueness of tensor decompositions, the complexity of the simplex method and the diameter of polytopes.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  930 0738 6410

Password: the last 4 digits of the zoom  ID in reverse order

The dyadic and the continuous Hilbert transforms with values in Banach spaces

We show that the Hilbert transform with values in a Banach space and the dyadic Hilbert transform have comparable $L^p$ norms, with two-sided  linear relation. The dyadic Hilbert transform is a specific dyadic shift of complexity one, sharing a lot of similarities with the Hilbert transform. The two sides of the linear estimate use very different techniques. One side uses elaborations on Bourgain’s result that Banach spaces with $L^p$ bounded Hilbert transform are UMD. The other side uses stochastic numerical analysis for a well crafted random process.

Time: 3:00pm (New York time zone)
Zoom Meeting ID:  619 2497 0285

Password: the last 4 digits of the zoom  ID in reverse order

An interface between Fourier Analysis and Analytic Number Theory

In this talk, we will discuss some optimization problems in Fourier analysis, and their applications in number theory. In particular, we will talk about certain proportions related to the zeros of the Riemann zeta-function and L-functions. This is based on joint work with E. Carneiro and M. B. Milinovich.

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  993 4521 4670

Password: the last 4 digits of the zoom  ID in reverse order

How far apart can centroids be?

The orthogonal projection of the centroid (barycenter, center of mass) of a convex body K onto a hyperplane H, and the centroid of the projection of K onto H coincide if K is centrally-symmetric. In general, this is not the case for non-symmetric convex bodies. In this talk, we investigate how far apart these points can be with respect to the width in the direction of the segment connecting them. The optimizers are described as well. The talk is based on the joint work with K. Tatarko and V. Yaskin (https://arxiv.org/abs/2212.14456).

Time: 3:00pm (New York time zone)

Zoom Meeting ID:  930 0738 6410

Password: the last 4 digits of the zoom  ID in reverse order

Szemeredi regularity, matrix decompositions, and covariance loss

We will discuss a new kind of weak Szemeredi regularity lemma. It allows one to decompose a positive semidefinite matrix into a small number of "flat" matrices, up to a small error in the Frobenius norm. The proof utilizes randomized rounding based on Grothendieck’s identity. The regularity lemma can be interpreted as a probabilistic statement about "covariance loss" – the amount of covariance that is lost by taking conditional expectation of a random vector. This talk is based on a joint work with March Boedihardjo and Thomas Strohmer.

An integer parallelotope with small surface area

We will prove that there exists an n-dimensional convex body whose surface area is at most √n times a lower order factor, yet its translates by the integer lattice tile space.

Joint work with Oded Regev.

Endpoint sparse domination for oscillatory Fourier multipliers 

Sparse domination was first introduced in the context of Calderón--Zygmund theory. Shortly after, the concept was successfully extended to many other operators in Harmonic Analysis, although many endpoint situations have remained unknown. In this talk, we will present new endpoint sparse bounds for oscillatory and Miyachi-type Fourier multipliers using Littlewood—Paley theory. Furthermore, the results can be extended to more general dilation-invariant classes of multiplier transformations via Hardy space techniques, yielding results, for instance, for multi-scale sums of radial bump multipliers.

This is joint work with Joris Roos and Andreas Seeger

 Thickness and a Gap Lemma in R^d 

A general problem that comes up repeatedly in geometric measure theory, dynamics and analysis is understanding when two or more "small" compact sets intersect. In the real line, the classical Gap Lemma of S. Newhouse, based on the notion of thickness, gives an easily checkable robust condition for two Cantor sets to intersect, but it is strongly based on the order structure of the reals. I will discuss some recent extensions of the notion of thickness, and the Gap Lemma, to higher dimensions. Applications to patterns in fractals will be discussed.