Research Interests

In pure mathematics my research has concentrated on function spaces and geometric measure theory. I did much of my early work on bounded analytic functions in planar domains and the space of functions of bounded mean oscillation (BMO) on Euclidean space. I found a constructive method of producing bounded solutions to the d-bar problem when the data was a Carleson measure, and this later led to my proof with John Garnett of the corona problem for Denjoy domains. I also worked on various problems concerning Sobolev spaces, including a proof that locally uniform domains are extension domains for all classical Sobolev spaces. My work in potential theory includes a theorem proven with Tom Wolf: On every planar domain, the harmonic measure is supported on a set of Hausdorff dimension at most one. This extended a result due to N.G. Makarov for the simply connected case. My work in geometric measure theory has centered on results in quantitative rectifiability. I proved that in the plane one can give a geometric multi-scale condition for solving the analyst's traveling salesman problem: When is a set contained in a curve (connected set) of finite length? One half of that proof works in any dimension; the other half was later proven in all dimensions by Kate Okikiolu. With Chris Bishop I proved several results on Hausdorff dimension of limit sets of Kleinian groups. Recently I have been working on problems related to diffusion geometry, a method used for discovery in data sets. With Mauro Maggioni and Raanan Schul I have shown that on Riemannian manifolds with minimal smoothness, one can use eigenfunctions to efficiently put coordinate systems on embedded balls.

Membership Type


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Primary Section

Section 11: Mathematics

Secondary Section

Section 32: Applied Mathematical Sciences