Title: Fourier Quasicrystals via Lee-Yang Polynomials

Abstract:

The concept of "quasi-periodic" sets, functions, and measures is
prevalent in diverse mathematical fields such as Mathematical Physics,
Fourier Analysis, and Number Theory. The Poisson summation formula provides a “Fourier characterization” for periodicity of discrete sets, and a Fourier Quasicrystals (FQ) generalizes this notion of periodicity: a counting measure of a discrete set is called a Fourier quasicrystal (FQ) if its Fourier transform is also a discrete atomic measure, together with some growth condition.  

Recently Kurasov and Sarnak provided a method for construction of one-dimensional FQs using the torus zero sets of multivariate Lee-Yang polynomials. In this talk, I will show that the Kurasov-Sarnak construction generates all sets which are FQs 1D.

I will also discuss the distribution of gaps between atoms in such 1D FQs, showing that there is a continuum of gaps, equidistributed on an interval, with a distribution given explicitly in terms of an ergodic dynamical system on the zero set of the above polynomial. 

In the last part I will introduce a recent result, generalizing the Kurasov-Sarnak construction to any dimension, by introducing Lee-Yang varieties.

The talk is aimed at a broad audience, no
prior knowledge in the field is assumed.

Based on joint works with Alex Cohen and Cynthia Vinzant.

Panopto Link: https://huji.cloud.panopto.eu/Panopto/Pages/Embed.aspx?id=82667e14-4498-4723-9052-b0e2008f0b9e