We review what is known, unknown, and expected about the mathematical properties of
Coulomb and Riesz gases. Those describe infinite configurations of points in R d interacting …

In this article we describe the crystallization conjecture. It states that, in appropriate physical
conditions, interacting particles always place themselves into periodic configurations …

Our goal is to provide an introduction to the study of minimal energy problems, particularly
from the perspective of generating point configurations that provide useful discretizations of …

There are a variety of needs for the dis-cretization of a manifold—statistical sampling,
quadrature rules, starting points for Newton's method, computeraided design, interpolation …

The extraordinary capabilities of large language models (LLMs) such as ChatGPT and GPT-
4 are in part unleashed by aligning them with reward models that are trained on human …

This survey discusses recent developments in the context of spherical designs and minimal
energy point configurations on spheres. The recent solution of the long standing problem of …

We investigate the energy of arrangements of N points on a rectifiable d-dimensional
manifold [Formula: see text] that interact through the power law (Riesz) potential V= 1/rs …

We survey known results and present estimates and conjectures for the next-order term in
the asymptotics of the optimal logarithmic energy and Riesz s-energy of N points on the unit …

As a means of improving analysis of biological shapes, we propose an algorithm for
sampling a Riemannian manifold by sequentially selecting points with maximum uncertainty …

In this paper we report on massive computer experiments aimed at finding spherical point
configurations that minimize potential energy. We present experimental evidence for two …