Minimal Energy Problems and Applications: Ping Pong Balayage and Convexity of Equilibrium Measures

This talk has two parts. Initially, I will present a survey of various minimal energy problems in potential theory having applications to different areas of research not only in mathematics, but in physics, biology, chemistry, etc. In addition to being a preview of a sequence of more detailed lectures to be given at subsequent colloquia and analysis seminars, the survey will introduce the necessary preliminaries to present the main result. Let E be the union of finitely many intervals or arcs on the unit circle. In a joint work with David Benko we prove that the equilibrium measure of E has a convex density. This is true for both the classical logarithmic kernel, and the Riesz kernel. This seems to be a fundamental result in potential theory, maybe one for the books. The electrostatic interpretation is the following: if we have a finite union of subintervals on the real line, or arcs on the unit circle, the electrostatic distribution of \u22many electrons\u22 will have convex density on every subinterval