Colloquium – Simon Bortz, University of Washington
January 29 @ 11:00 am - 12:00 pm
Topic: Harmonic Measure and Geometry
Abstract: The harmonic measure for a domain provides an important link between the analytic properties of the domain and the geometry of the domain. This measure is canonically associated to a fundamental differential operator, the Laplacian. It can also be viewed as the `hitting probability’ for Brownian (random) motion. Combining these viewpoints reveals the aforementioned link between analytic properties and geometry. Studying the relationship between the behavior of the harmonic measure and the geometry of a domain has been a pursuit of considerable mathematical interest. While the topic has been an active area of study for over a century, characterizations along these lines have been furnished only recently. These breakthroughs can primarily be attributed to the amalgamation of ideas from harmonic analysis, partial differential equations and geometric measure theory.
One way to describe the harmonic measure is to compare it to another (canonical) measure with the goal of sensing the geometry of the domain through a particular relationship. This talk will concern two of my collaborations in this area. We will compare the harmonic measure to the “natural surface measure” in quite extreme generality (joint work with Akman, Hofmann and Martell). Then we will compare the harmonic measure of a domain to the harmonic measure of its “exterior domain” (joint work with Engelstein, Goering, Toro and Zhao). This second work provides a characterization of boundary flatness from a specific relationship between these two harmonic measures.