Universal optimality of the E8 and Leech lattices.

I will talk about a joint work with Henry Cohn, Abhinav Kumar, Stephen D. Miller, and Maryna Viazovska.

We look at the problem of arranging points in Euclidean space in order to minimize the potential energy. In some
exceptional cases it might happen that a single configuration is a minimizer for all potentials that are completely monotone.
If this is the case, then such a configuration is calleduniversally optimal.

Until recently the only known example of a universally optimal configuration in Euclidean space was
the 1-dimensional lattice in R^1. We show that the E8 lattice and the Leech lattice are universally optimal in
dimensions 8 and 24 respectively.