Sphere packing, energy minimization, and modular forms

Abstract:
In 2016 Viazovska settled the sphere packing problem in 8 dimensions,
showing that the E8 lattice gives the densest packing. The key new
ingredient in her stunningly elegant proof was the use of modular forms.
Shortly afterwards, the Leech lattice was shown to be the densest packing
in 24 dimensions, with a similar proof. Recently, we
(Cohn-Kumar-Miller-Radchenko-Viazovska) have put these methods into a
broader framework, using modular forms to derive new interpolation
formulae for radial Schwartz functions and show that E8 and the Leech
lattice are universally optimal - they minimize energy in their respective
dimensions for a large class of potential functions. I will describe
highlights and key ideas of these results.