Problems with a lot of Potential: Energy Optimization on Compact Spaces

Abstract

In this dissertation, we provide a survey of the author’s work in energy optimization oncompact spaces with continuous potentials. We present several new results relating the positive definiteness of a potential, convexity of its energy functional, and properties of the minimizing measures of the energy, first in general spaces, then specifically on two-point homogeneous spaces, and especially on spheres. We also obtain sufficient conditions for the existence, and in some cases uniqueness, of discrete minimizers for a large class of energies. We discuss the Stolarsky Invariance Principle, which connects discrepancy and energy, as well as some analogues and generalizations of this phenomenon. In addition, we investigate some particularly interesting optimization problems, such as determining the maximum sum of pairwise angles between N points on the sphere S^(d-1) and the maximum sum of angles between N lines passing through the origin, both of which are related to conjectures of Fejes Tóth. We also study the p-frame energies, which are related to signal processing and quantum mechanics. We show that on the sphere, the support of any minimizer of the p-frame energy has empty interior whenever p is not an even integer, and, moreover, that tight designs are the unique minimizers for certain values of p, among other results. We complete this paper by developing the theory of minimization for energies with multivariate kernels, i.e. energies for which pairwise interactions are replaced by interactions between triples, or more generally, n-tuples of particles. Such objects arise naturally in various fields and present subtle difference and complications when compared to the classical two-input case. We introduce appropriate analogues of conditionally positive definite kernels, establish a series of relevant results in potential theory, and present a variety of interesting examples, including some problems in probabilistic geometry which are related to multivariate versions of the Riesz s-energies.