June 23, 2022. Michael A. Tsfasman (IITP, IUM, and CNRS). Dense sphere packings.

How to place equal balls in the N-dimensional Euclidean space in the densest possible way? The problem is non-trivial even in dimension N = 2. In dimension 3 the answer was obtained at the very end of 20th century, and for N = 4 it is yet unknown.

In the introduction I shall present the history, the statement of the problem, and some well-known results. The main part is devoted to astounding results of Maryna Viazovska, who solved the problem in dimensions 8 and 24 using quasimodular forms. At the end I plan to recall some results of mine for very large dimensions (N → ∞), based on algebraic geometry and number theory.

List of talks at previous sessions of the seminar.