This research project focuses on the study of discretizations of compact manifolds in Euclidean space via minimal energy points, with special emphasis on Riesz energy kernels. Previous work on the discrete equilibrium configurations for the Riesz s-energy (generalized Thomson problem) showed the utility of investigating the dependence of behavior on the parameter s. Of particular interest is the critical value that occurs when s equals the Hausdorff dimension of the manifold and a transition occurs from long range to short range interactions. This project will investigate: (i) finer asymptotics for the minimal energy and its connection with the curvature and smoothness properties of the manifold; (ii) for dimensions 2, 8, and 24, the determination (or estimation) of constants arising in the minimal energy expansion in terms of the zeta functions in s for special lattices existing in these dimensions; (iii) asymptotic results for energy on self-similar sets; (iv) the behavior of "greedy energy points," especially in the presence of an external field; (v) the determination of the limiting support of discrete long range minimal energy configurations on surfaces of revolution; and (vi) development and analysis of algorithms for the fast generation of uniformly distributed points on manifolds.
This research project focuses on the mathematics of how charged particles on a curved surface arrange themselves in a stable configuration when interacting through two-particle repulsive interactions. This study of the ordering of matter will broaden the understanding of the physics of membranes and films and has applications to the design of new materials with novel optical and electronic properties. A related aspect of the project is the rapid generation of data sampling points on curved surfaces (such as the earth) which can be used to measure a variety of physical properties. Such methods for point generation are also useful for testing detection devices such as radar systems. This research addresses in several different contexts the fundamental problem of how best to convert from analog to digital.
