Claremont Center for the Mathematical Sciences

The theory of harmonic measure describes the way in which a random particle in a domain first hits the boundary. It also, and equivalently, gives a description of the boundary behavior of solutions to Laplace's equation in a given, perhaps very complicated, domain.

A brief survey of harmonic measures in the plane and in space will be followed by a discussion of some analogs of classical theorems about exit behavior of Brownian motion in simply connected planar domains to the case of multiply connected planar domains and to some classes of "nice'' domains in three dimensions. There are many interesting open questions in the area and we will try to talk about a few of them. The talk will be accessible to undergraduates.