Seminar: Energy-Optimal Measure Distributions on Fractal Sets

Abstract: The Riesz energy of a measure gives a great deal of information about the size and arrangement of the underlying support; it is closely related to other geometric quantities including Favard length and analytic capacity. Physically, it can be thought of as the amount of energy that is required to concentrate a unit of electric charge into a particular region; one may then ask for an energy-optimal charge distribution given a geometric constraint. In this talk we will discuss a solution to this problem for neighborhoods of fractal 1-sets, including the generations of the four-corner Cantor set. The key tool used will be a mass-redistribution algorithm applied to a discretized variant of the Riesz energy. We will also discuss ongoing work towards solving related extremal problems in complex analysis.

Faculty Host: Alan Chang