Title: Sphere packing, Fourier interpolation, and real-variable Kloosterman sums
Abstract: In 2017, Viazovska et al. proved that the E8 and Leech lattices are the densest sphere packings in dimensions 8 and 24, respectively. Later in 2022, the same authors proved the universal optimality of these lattices. For these breakthrough works, Viazovska was awarded the Fields Medal. One of the key ingredients in these works is an interpolation formula allowing a reconstruction of radial Schwartz Fourier eigen-functions from their values at the corresponding lattices. These interpolation formulas was constructed from integrals of vector-valued modular forms. Following these results, the connection between Fourier interpolation problems and modular forms has been extensively studied by Bondarenko, Radchenko, Ramos, Seip, Sousa, Stoller, and many others. In this talk, we present an explicit construction of Fourier interpolation in dimensions 3 and 4 using Maass-Poincaré type series and real-variable Kloosterman sums. This is joint work with Danylo Radchenko.