Speaker:  Andrew Salmon  (MIT)

Title:  Semidefinite progrramming bounds for sphere packing

Abstract:

We discuss work, joint with Henry Cohn and David de Laat, that gives bounds on sphere packing stronger than the linear programming bound for sphere packing. We show that the original Cohn-Elkies linear programming bound is the first level of two hierarchies: a Lasserre hierarchy that theoretically converges to the optimal sphere center density, as well as a hierarchy of k-point bounds. The 3-point bound for sphere packing gives well-defined finite-dimensional semidefinite programms that bound sphere packing density. By computing feasible solutions to these semidefinite programs, we give new upper bounds on sphere packing density in dimensions 4-7 and 9-16, and the only obstruction to giving a better bound in a typical dimension seems to be computational power. If time permits, we will also discuss some work in preparation (joint with Henry Cohn) on asymptotic bounds for the dual linear program for sphere packing stronger than Torquato-Stillinger's bound, as well as some open problems.