Energy minimization and formal duality of periodic point sets

11/14/2017 - 12:15pm
11/14/2017 - 1:10pm
Achill Schürmann (University of Rostock and CMC)

Point configurations that minimize energy for a given pair potential function occur in diverse contexts. In this talk we discuss recent observations and results about periodic point configurations which minimize such energies. We are in particular interested in universally optimal periodic sets, which minimize energy for all completely monotonic potential functions. Using a new parameter space for m-periodic point sets, numerical simulations have revealed yet unexplained phenomena: at least in low dimensions energy minimizing point configurations appear to satisfy a ''formal duality relation'' which generalizes the familiar duality notion for lattices. Universally optimal periodic sets appear to exist in dimensions 2,4,8,24 and somewhat surprisingly in dimension~9. For the first four cases we can prove a local version of universal optimality for corresponding lattices. In dimension~9, so far, we can only prove a weaker version of local optimality for the non-lattice set $\mathsf{D}_9^+$. A crucial role in these results is played by the fact that sets of vectors of a given length (shells) form spherical 3- or 4-designs.

Millikan 2099, Pomona College