The sphere packing problem in dimensions 8 and 24

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.