[Geometry, Topology and Dynamics Seminar] Viazovska's work on the sphere packing problem and some recent developments in number theory by Dr. Dan Romik (UC Davis)

Abstract: The sphere packing problem in d dimensions asks for the densest packing of spheres in d-dimensional Euclidean space R^d. Prior to 2016, the problem had only been solved in dimensions 2 and 3 (with the solution in dimension 3 being Hales's famously complicated solution to Kepler's conjecture), and solutions in most other dimensions were thought to be out of reach, except for dimensions 8 and 24 where overwhelming numerical and theoretical evidence supported the conjectures pointing at specific lattices as the optimal packings. In a breakthrough in 2016, Maryna Viazovska published a stunning proof for the 8-dimensional case, followed up shortly with a proof for the 24-dimensional case with several coauthors. The solutions work by using a previous reduction of the problem (due to Henry Cohn and Noam Elkies) to a problem in harmonic analysis, and ingeniously solving that analysis problem using number-theoretic tools - specifically, the mathematics of modular forms.

 

In this talk I will give a quick survey of these developments and the beautiful new challenges and opportunities that they open up for attacking this classical geometry problem. I will also discuss my own recent results on infinite series representations for the Riemann xi function and explain how Viazovska's work, which appears not strongly related to the problem I was studying, nonetheless provided me with useful inspiration.