My broad research interests are in geometric measure theory, but my focus is the study of rectifiable sets, which are a measure theoretic analogue of smooth manifolds: while a manifold is locally and smoothly parametrized by Euclidean space, a rectifiable set but that it be covered, up to measure zero, by just Lipschitz images (which are only almost everywhere differentiable). I use techniques from geometric measure theory and harmonic analysis to study their properties and find geometric characterizations, usually with a quantitative point of view in mind.
In applications, this class of sets is sometimes more convenient as many theorems stated for smooth surfaces hold more generally for rectifiable sets, and in some cases this is exactly the broadest class of sets where the result can hold. For example, the only sets of finite and positive length in the plane for which there are nonconstant bounded analytic functions on their complement must contain a rectifiable subset.
Recently I have also been studying harmonic measure for Euclidean domains, which is the measure that computes the probability that a Brownian motion beginning at a fixed pole in the domain will first hit the boundary in a given subset of the boundary. The analytic properties of this measure dictate interesting geometric information about the boundary and vice versa. For example, if it is absolutely continuous with respect to surface measure, then the boundary must contain a rectifiable set (that is, this analytic property of absolute continuity guarantees that a portion of the boundary is "almost smooth".)
Current and Recent PhD Students
Michele Villa Matthew Hyde Jean Burnazyan
I got my degree in mathematics from the University of Nebraska-Lincoln in 2006 and my PhD from UCLA in 2011. Afterwards, I held postdoctoral positions at the University of Washington and the Universitat AutÃ²noma de Barcelona, as well as short fellowships at the Mathematical Sciences Research Institute and the Institute for Pure and Applied Mathematics.