|Date: Tuesday, March 21, 2006
Title: Recent Progress in Sphere Packing
Abstract: The problem of densely packing non-overlapping congruent spheres in Euclidean space of various dimensions N has a long history and rich connections with various aspects of modern pure and applied mathematics. For example, the Leech lattice yields a remarkably dense and symmetrical packing in dimension N=24, conjectured to be the densest possible in this dimension. In this packing, each sphere touches 196,560 others. It has been known for some time that this ``kissing configuration' is optimal. We review the proof of this fact and explain why the method cannot be used directly to study the sphere-packing problem. We then outline recent work on the sphere-packing problem, culminating in the proof by H.Cohn and A.Kumar that the conjectured optimality of the Leech packing holds under the condition that the centers of the spheres lie on a lattice (previously the densest lattice packing was known only for N=1 through N=8). Without the lattice condition, they show that any packing cannot improve on the Leech density by more than 1 part in 10-30. They also give strong evidence for the existence of mysterious functions of a positive real variable that would prove the optimality of the Gosset (E_8) and Leech packings of spheres for N=8 and N=24.
Speaker: Noam Elkies