Mathilde Gerbelli-Gauthier : Sphere-Packing, Fourier Interpolation, and the Segal--Shale--Weil Representation

Posted by Philip Engel , part of the Departmental Colloquium.

At

Feb. 28, 2025, 3 p.m.

In

636 SEO

Abstract

In 2016, Viazovska proved that the $E_8$​ lattice provides the optimal sphere-packing in dimension 8, and soon after, Cohn--Kumar--Miller--Radchenko--Viazovska proved the analogous result for the Leech lattice in dimension 24. Viazovska's breakthrough came through the solution of a Fourier interpolation problem: she constructed a function $f$ such that $f$, its Fourier transform, and their first derivatives take on specific values at square roots of natural numbers. Prior to this, Radchenko and Viazovska had solved a "toy case"—a Fourier interpolation result for even Schwartz functions on the real line. In this talk, I will explain how this one-dimensional version can be understood through the lens of the Segal--Shale--Weil representation, an infinite-dimensional representation that originally arose in the context of quantum mechanics.