1/6 Automorphic Forms and Optimization in Euclidean Space
Apparaît également dans la 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.
![[1116] Zéros de fonctions aléatoires gaussiennes](/media/cache/video_light/uploads/video/video-e4285a60a12befed4eeedd15d9b54c79.jpg)
![[1196] Recent progress on bounds for sets with no three terms in arithmetic progression](/media/cache/video_light/uploads/video/Capture%20d%E2%80%99%C3%A9cran%202024-04-15%20%C3%A0%2012.07.44.png)
![[1143] Des points vortex aux équations de Navier-Stokes](/media/cache/video_light/uploads/video/video-92f04444d5e2221d8426388c1b4a4b28.jpg)
