Abstract: For more than a century, the interplay between harmonic functions/measure and the geometry of `rough’ domains has been a topic of considerable interest. The earliest developments were focused on the plane, in geometric settings amenable to complex analysis tools (e.g. simply connected domains). In higher dimensions, the situation becomes quite a bit more complicated, where the state of the art methods are adapted from the study of singular integrals on rough sets.

This talk will survey some of the remarkable progress that has been made in recent years on connecting, or even characterizing, the geometry of a domain by properties of harmonic functions/measure (in any dimension). Along these lines, I will describe some recent work with Max Engelstein and Steve Hofmann.