03/20/2024
By Joris Roos

The Department of Mathematics & Statistics invites you to attend a colloquium lecture by Lior Alon from Massachusetts Institute of Technology.

The talk will be delivered in person, but it is also possible to attend via Zoom. Everyone is welcome.

Title: Fourier Quasicrystals via Lee-Yang Polynomials
Date: Wednesday, March 27
Time: 11 a.m. to noon
Room: Southwick 350W

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 constructing one-dimensional FQs as the intersections of an irrational line in the torus with the zero set of a multivariate Lee-Yang polynomial. In this talk, I will show that the Kurasov-Sarnak construction generates all one-dimensional FQs. I will also discuss the distribution of gaps between atoms in such FQs, showing that the countably many gaps equidistribute on an interval, with a distribution given explicitly in terms of ergodic dynamical systems on tori. In the last part, I will present a generalization of 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, Cynthia Vinzant, Mario Kummer, and Pavel Kurasov.

For future events in this colloquium series, visit the Math website