Optimal and near optimal energy minimizing point configurations

Discrete and continuous energy problems that arise in a variety of scientific contexts are introduced, along with their fundamental existence and uniqueness results. Particular emphasis will be on Riesz and Gaussian pair potentials and their connections with best-packing and the discretization of manifolds. The latter application leads to the asymptotic theory (as N grows to infinity) for N-point configurations that minimize energy when the potential is hypersingular (short-range).

For fixed N, the determination of such minimizing configurations on the d-dimensional unit sphere is especially significant in a range of contexts that include coding theory, discrete geometry, and physics. We will review linear programming methods for proving the optimality of configurations on the sphere, including Cohn and Kumar’s theory of universal optimality.

The following reference will be made available during the workshop:

Discrete Energy on Rectifiable Sets, by S. Borodachov, D.P. Hardin and E.B. Saff, Springer Monographs in Mathematics, 2019