The 2017 Excelsior Lectures
Harmonic Measure and Rectifiability
Svitlana Mayboroda (University of Minnesota)
Abstract: Over the past century an effort to understand dimension and structure of the harmonic measure spanned many spectacular developments in Analysis and in Geometric Measure Theory. Uniform rectifiability emerged as a natural geometric condition, necessary and sufficient for classical estimates in harmonic analysis, boundedness of the harmonic Riesz transform in L2 , and, in the presence of some background topological assumptions, for suitable scale-invariant estimates on harmonic functions closely related to the solvability of the Dirichlet problem. While many of geometric and analytic problems remain relevant in sets of higher co-dimension (e.g. a curve in 3-dimensional Euclidean space), the concept of the harmonic measure is notoriously missing.
In the first lecture, we will discuss the state of the art in co-dimension one, concentrating on the recent breakthroughs which finally revealed a picture of equivalence of scale-invariant, geometric, analytic, and PDE properties of sets.
In the second lecture, we will introduce a new notion of a "harmonic" measure, also associated to a linear PDE, which serves the higher co-dimensional sets. We discover its basic properties and give large strokes of the argument to prove that our measure is absolutely continuous with respect to the Hausdorff measure on all uniformly rectifiable sets.