Schedule April-July 2021

I'll do my best to connect the topics in the title.

The density criteria of Nyman-Beurling and Báez-Duarte for location of zeros of Dirichlet L-functions will be discussed. In particular, these criteria provide necessary and sufficient conditions for the truth of the Riemann hypothesis. We analyze a finite dimensional extremal problem inspired by the Báez-Duarte criterion. The solution of the problem uses a sequence of Dirichlet orthogonal polynomials constructed recently by Doron Lubinsky. This is a joint work with Willian D. Oliveira.

Then we pose further open problems about certain determinants composed by Lubinsky's Dirichlet orthogonal polynomials and the corresponding kernel polynomials.

Large ensembles of points with Coulomb interactions arise in various settings of condensed matter physics, classical and quantum mechanics, statistical mechanics, random matrices and even approximation theory, and they give rise to a variety of questions pertaining to analysis, Partial Differential Equations and probability.

We will first review these motivations, then present the ”mean-field” derivation of effective models and equations describing the system at the macroscopic scale. We then explain how to analyze the next order behavior, giving information on the configurations at the microscopic level and connecting with crystallization questions, and finish with the description of the effect of temperature.

Smale’s 7th problem asks for an efficient description of N points in the unit sphere with quasioptimal logarithmic energy (to be though of as the repulsion energy one would expect if the points were equal-charged particles). In this talk I will explain the origins of this problem, the reason why it was included in Smale’s list, the state of the art in the research and my personal viewpoint on its destiny.

Fourier multipliers and pseudo-differential operators are defined by means of the Fourier transform and play an important role in the study of partial differential equations. In the same spirit,  Hermite pseudo-multipliers are associated to Hermite expansions and they represent the counterparts to pseudo-differential operators in the Hermite setting. 

After some preliminaries, we will present results on boundedness properties of pseudo-multipliers in function spaces associated to the Hermite operator. The main tools in the proofs involve new molecular decompositions and molecular synthesis estimates for Hermite Besov and Hermite Triebel-Lizorkin spaces, which allow to obtain boundedness results on spaces for which the smoothness allowed includes non-positive values. In particular, we obtain boundedness results for pseudo-multipliers on Lebesgue and Hermite local Hardy spaces.

The talk is based on joint work with Fu Ken Ly (The University of Sydney).

In harmonic analysis we study boundedness properties on various function spaces of for example maximal operators and singular integrals among many other operators. There is a parallel dyadic setting where often proofs are simpler. I would like to give you a panorama of this dyadic world and how it connects heuristically and nowadays very precisely to the non-dyadic world.