Sharp sphere packings

In this talk we will speak about recent progress on the sphere packing problem. The packing problem can be formulated for a wide class of metric spaces equipped with a measure. An interesting feature of this optimization problem is that a slight change of parameters (such as the dimension of the space or radius of the spheres) can dramatically change the properties of optimal configurations. We will focus on those cases when the solution of the packing problem is particularly simple. Namely, we say that a packing problem is sharp if its density attains the so-called linear programming bound. Several such configurations have been known for a long time and we have recently proved that the E8 lattice sphere packing in ℝ8 and the Leech lattice packing in ℝ24 are sharp. Moreover, we will discuss common unusual properties of shared by such configurations and outline possible applications to Fourier analysis.