Matthew de Courcy-Ireland: Six-dimensional sphere packing and linear programming

Abstract

A sphere packing is a collection of equal-sized balls in Euclidean space, disjoint except for tangencies. One would like the largest possible fraction of volume to be occupied by the packing. The method of linear programming developed by Cohn and Elkies gives an upper bound on the packing fraction that can be achieved, based on linear inequalities involving an auxiliary function and its Fourier transform. The method leads to a sharp bound in dimension 8 by Viazovska and in dimension 24 by Cohn-Kumar-Miller-Radchenko-Viazovska. The goal of the talk is to survey some of these developments, and to sketch joint work in progress with Maria Dostert and Maryna Viazovska where we try to understand why the bound is not sharp in other dimensions. In particular, in dimension 6, we construct a "fake packing" that tricks the linear programming bound. This generalizes an approach via modular forms that Cohn and Triantafillou had proposed for dimensions divisible by 4.