An Overview: Discretizing Manifolds via Particle Interactions

Abstract

The problem of distributing points on a manifold (or discretizing a manifold) arises in many contexts that are of interest to the scientific community as well as in applied fields—statistical sampling, quadrature rules, information theory, coding techniques, computer-aided design, interpolation schemes, finite element tessellations, ground states of matter—to name but a few. Our goal is to address this problem from the perspective of particle interactions; namely, starting from a given formula for the pairwise interactions of N particles (points) that are confined to a given manifold A in the Euclidean space \(\mathbb R^{p},\) we wish to describe the structure of those configurations that arise when the N particles reach an equilibrium state (a state of minimal energy).