Description of Research

My recent work has concerned four topics: geometric embeddings, existence of Poincare inequalities, self-improvement properties of Poincare inequalities and the local geometry of Lipschitz differentiability spaces. The first topic is covered in this paper. There we were able to construct bi-Lipschitz embeddings for subsets of manifold, orbifolds and group quotients and bound their distortion and target dimension using diameter and a both-sided bound on sectional curvature. The latter three topics are coverered here. A simple re-proof with better bounds for the classical result of Keith and Zhong is contained in here. The most important aspect of these papers is a novel classification of Poincare inequalities using a new type of quantitative connectivity. This is then applied to show that different spaces admit Poincare inequalities, and that, remarkably, RNP-differentiability spaces introduced by Cheeger and Kleiner (and named by Bate and Li) are rectifiable in terms of spaced admitting Poincare inequalities.

I am very much interested in questions related to this previous work, as well as analysis on metric spaces as a whole. The goal of analysis on metric spaces is to extend analytic tools from Euclidean space to possibly very fractal like spaces. Many of the fundamental tools in this theory remain somewhat poorly understood and my goal is to contribute to understanding them better.

I am also engaging in research problems with computer scientists in computational geometry. The goal here is to develop algorithms for discrete and geometric problems. See my CV for a list of papers and contributions in this topic.