The sphere packing problem in dimensions 8 and 24

par Maryna Viazovska (EPFL (Lausanne))

lundi 27 nov. 2017, 16:15 → 17:15 Europe/Paris

Fokko du Cloux (UCBL-Braconnier)

The sphere packing problem is to find an arrangement of non-overlapping unit spheres in the d-dimensional Euclidean space in which the spheres fill as large a proportion of the space as possible. In this talk we will present a solution of the sphere packing problem in dimensions 8 and 24. In 2003 N. Elkies and H. Cohn proved that the existence of a real function satisfying certain constrains leads to an upper bound for the sphere packing constant. Using this method they obtained almost sharp estimates in dimensions 8 and 24. We will show that functions providing exact bounds can be constructed explicitly as certain integral transforms of modular forms. Therefore, the sphere packing problem in dimensions 8 and 24 is solved by a linear programming method.