The investigator and his colleagues study the theory and applications of discrete minimum energy problems that are of significance in the discretization of compact metric spaces. By utilizing minimal weighted Riesz energy configurations, one can generate sequences of points on surfaces that approximate prescribed density functions on that surface. Such constructions have a myriad of possible applications to the physical and biological sciences. Specifically, the team is studying (i) separation and fill-radius estimates for optimal N-point weighted energy configurations; (ii) geometrical properties of optimal and near optimal weighted energy configurations on 2-dimensional surfaces; (iii) algorithms for the fast generation of points distributed on a manifold in accordance with a prescribed density; (iv) properties of greedy energy points; and (v) near optimal discretizations for problems of importance in geophysical applications.
The investigator and his colleagues study stable (minimum energy) states of charged particle configurations on curved surfaces and solid materials, especially when such particles are restricted in density or under the influence of external forces. One goal of the project is to determine efficient numerical methods for the generation of optimal or near optimal distributions of point charges that can be used to help understand the convection of the Earth's mantle, for the modeling of the geodynamo, and to gain a better determination of the Earth's gravitational field. This project addresses in several different contexts the fundamental problem of how best to convert from analog to digital.