Discrete Math Seminar: Shuxing Li

In Euclidean space, a periodic configuration is a union of finitely many translations of a lattice. In particular, energy-minimizing periodic configurations are those which possess minimum energy. Finding energy-minimizing periodic configurations is an interesting problem, not only because of its theoretical significance in physics, but also its connection with the famous sphere packing problem. On the other hand, the search of energy-minimizing periodic configurations is notoriously difficult and very few theoretical results are known. Nevertheless, an insightful idea due to Cohn, Kumar, Reiher and Schurmann, enables us to study the energy-minimizing periodic configurations from a combinatorial viewpoint. Roughly speaking, among pairs of energy-minimizing periodic configurations, they revealed a remarkable symmetry named formal duality. Furthermore, they translated the formal duality into a purely combinatorial context, where the corresponding configuration was called a formally dual pair, which is a pair of subsets in a finite abelian group satisfying a subtle difference-set-like property. In this talk, we will give an overview and present some recent results of formally dual pairs, which involve constructions, nonexistence results and characterizations.

This is joint work with Alexander Pott and Robert Schuller.