We shall talk about energy optimization problems of the following type: let Ω be a (compact) metric space, F(x,y) be a symmetric kernel on Ω×Ω, and μ a Borel probability measure on Ω. The energy of μ with respect to the interaction given by F is defined as
IF (μ) = ∫Ω∫Ω F(x,y) dμ(x) dμ(y),
and one is interested in the properties of measures μ which minimize (or maximize) these energies. Such objects (as well as their discrete analogs) arise naturally in discrete and metric geometry, mathematical physics, signal processing, and are directly related to numerous topics in analysis, such as positive definiteness, convexity, orthogonal polynomials, embeddings of metric spaces. A particularly interesting example (connected to the latter topic) arises when F(x,y) is a power of the distance.
We shall discuss several examples, the interplay between geometry and energy minimizers, with a special emphasis on the peculiar situations when minimizers become discrete (or are supported on small subsets of Ω), as well as connections to various topics and problems: the Fejes Toth conjecture on the sum of angles between lines, rendezvous numbers, tight frames, equiangular lines, and mutually unbiased bases.
https://univienna.zoom.us/j/67922750549?pwd=Ulh5L1QxNFhBOC9PUjlVdG9hc0tmUT09