Discrete Minimal Energy Configurations and Related Problems in Potential Theory

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The research will primarily focus on three areas: (i) the analysis of minimal energy point configurations; (ii) the noninvasive detection of faults (cracks) in homogeneous media; and (iii) limiting behaviors of certain basis functions over planar regions. Regarding (i), we shall analyze certain equilibrium (or "ground state") configurations for N particles interacting via a pairwise repulsive interaction V and confined to a fixed set (such a sphere or torus). Such configurations are useful for purposes of sampling data, computer graphics, best-packing, and understanding the physics of self-assembling materials. Concerning (ii), we shall continue with the development of rational approximation methods that can determine the existence and location of cracks inside a conducting material. Regarding (iii), our focus will be on polynomials orthogonal over planar regions and how their behavior can be used to solve problems related to tomography (the recovery of the underlying planar region).