  Building on Viazovska's recent solution of the sphere packing problem in
eight dimensions, we prove that the Leech lattice is the densest packing of
congruent spheres in twenty-four dimensions and that it is the unique optimal
periodic packing. In particular, we find an optimal auxiliary function for the
linear programming bounds, which is an analogue of Viazovska's function for the
eight-dimensional case.
