Georgia Institute of Technology College of Sciences

The classic sphere packing problem is to determine the densest possible packing of non-overlapping congruent spheres in Euclidean space.  The problem is trivial in dimension 1, straightforward in dimension 2, but a major challenge or mystery in higher dimensions, with the only other solved cases being dimensions 3, 8, and 24.  The hard sphere model is a classic model of a gas from statistical physics, with particles interacting via a hard-core pair potential.  It is believed that this model exhibits a crystallization phase transition in dimension 3, giving a purely geometric explanation for freezing phenomena in nature, but this remains an open mathematical problem. The sphere packing problem and the hard sphere model are closely linked through the following rough rephrasing of the phase transition question: do typical sphere packings at densities just below the maximum density align with a maximum packing or are they disordered?  

 

I will present results on high-dimensional sphere packings and spherical codes and new bounds for the absence of phase transition at low densities in the hard sphere model.  The techniques used take the perspective of algorithms and optimization and can be applied to problems in extremal and enumerative combinatorics as well.