Appears in collection : Hadamard Lectures 2019 - Maryna Viazovska - Automorphic Forms and Optimization in Euclidean Space

The goal of this lecture course, “Automorphic Forms and Optimization in Euclidean Space”, is to prove the universal optimality of the $E_8$ and Leech lattices. This theorem is the main result of a recent preprint “Universal Optimality of the $E_8$ and Leech Lattices and Interpolation Formulas”, written in collaboration with H. Cohn, A. Kumar, S.D. Miller and D. Radchenko. We prove that the $E_8$ and Leech lattices minimize energy of every potential function that is a completely monotonic function of squared distance (for example, inverse power laws or Gaussians).

This theorem implies recently proven optimality of $E_8$ and Leech lattices as sphere packings and broadly generalizes it to long-range interactions. The key ingredient of the proof is sharp linear programming bounds. To construct the optimal auxiliary functions attaining these bounds, we prove a new interpolation theorem.

At the last lecture, we will discuss open questions and conjectures which arose from our work.