Education

Research Interests

My recent research has been focused mainly on two areas.

The first involves the quantitative study of singular sets for various geometric nonlinear partial differential equations including, Einstein manifolds and their Gromov-Hausdorff limits, minimizing harmonic maps, minimizing hypersurfaces and others. In this work, we obtain sharp estimates on the volume of the set of points away from which the solution has any definite amount of regularity, as measured by the so-called "regularity scale". The estimates make effective the classical bounds on the Hausdorff dimension of the corresponding singular sets. The basic techniques were developed in joint work with Aaron Naber; later we were joined by Robert Haslhofer and Daniele Valtorta.

The second area of my recent research involves the study of metric measure spaces which have enough regularity to do first order differential calculus, even though they need not be infinitesimally Euclidean, and indeed may have fractional Hausdorff dimension. These spaces are known as PI spaces, or more generally, as Lipschitz differentiability spaces. Basic issues involve understanding the infinitesimal structure of such spaces, the construction of large families of examples, as well as quantitative bi-Lipschitz embedding and nonembedding theorems for various Banach space targets. During the past 10 years, this has been joint work with Bruce Kleiner and Assaf Naor.

Selected Publications