Research

Research profile
My research interests are in geometric analysis, geometric group theory, and geometric topology. Current research topics include:

Geometric evolution equations: Ricci flow and mean curvature flow.
Quasi-isometric rigidity.
Quasisymmetric structure of self-similar spaces, such as boundaries of hyperbolic groups.
Bilipschitz embedding problems.
Analysis on singular spaces, e.g. Carnot groups, and spaces satisfying Poincare inequalities.

Selected papers:

Rigidity for quasi-isometries of symmetric spaces and Euclidean buildings,
with Bernhard Leeb ps pdf
Rigidity of quasi-isometries for symmetric space and Euclidean buildings
(announcement), with Bernhard Leeb postscript
Groups quasi-isometric to symmetric spaces,
with Bernhard Leeb ps pdf
Quasi-isometries and the de Rham decomposition,
with Misha Kapovich and Bernhard Leeb ps pdf
Separated nets in Euclidean space,
with Dmitri Burago ps pdf
Boundaries of nonpositively curved spaces,
with Chris Croke ps pdf
Hyperbolic groups with low dimensional boundary,
with Misha Kapovich ps pdf
The local structure of length spaces with curvature bounded above ps pdf
The structure of the stable norm for metrics on tori,
with Dmitri Burago and Sergei Ivanov ps pdf
Coarse Alexander duality and duality groups,
with Misha Kapovich ps pdf
The geodesic flow of a nonpositively curved graph manifold,
with Chris Croke ps pdf
Rigidity for Quasi-Mobius group actions (formerly Rigidity for convergence group actions),
with Mario Bonk ps pdf
Rectifying separated nets,
with Dmitri Burago ps pdf
Van Kampen's embedding obstruction for discrete groups,
with Mladen Bestvina and Misha Kapovich ps pdf
Quasisymmetric parametrizations of two-dimensional metric spheres,
with Mario Bonk ps pdf
Review of 3 books on nonpositive curvature ps pdf
Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary,
with Mario Bonk, second version (July 16, 2003) ps pdf
Quasi-hyperbolic planes in hyperbolic groups,
with Mario Bonk ps pdf
Notes on Perelman's Ricci flow papers,
with John Lott Click here
Singularity structure in mean curvature flow of mean convex sets,
with Toby Colding ps pdf
The weak hyperbolization conjecture for 3-dimensional CAT(0) groups,
with Michael Kapovich ps pdf
Hadamard spaces with isolated flats,
with Chris Hruska ps pdf
Rigidity of invariant convex sets in symmetric spaces
with Bernhard Leeb, pdf
The asymptotic geometry of negatively curved spaces: uniformization, geometrization, and rigidity,
2006 ICM Proceedings ps pdf
On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces,
with Jeff Cheeger ps pdf
Differentiating maps into L^1, and the geometry of BV functions,
with Jeff Cheeger ps pdf
Generalized differentiation and bi-Lipschitz nonembedding in L^1 (announcement),
with Jeff Cheeger, ps pdf
The asymptotic geometry of right-angled Artin groups I,
with Mladen Bestvina and Michah Sageev, ps pdf
A new proof of Gromov's theorem on groups of polynomial growth, ps pdf

Characterization of the Radon-Nikodym Property in terms of inverse limits,
with Jeff Cheeger, ps pdf
Geometry and rigidity of mapping class groups,
with Jason Behrstock, Yair Minsky, and Lee Mosher, ps pdf
Induced quasi-actions: a remark
with Bernhard Leeb, ps pdf
Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane
with Luigi Ambrosio and Enrico Le Donne, ps pdf
Quasiflats in CAT(0) complexes
with Mladen Bestvina and Michah Sageev, ps pdf
Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property,
with Jeff Cheeger, ps pdf
Metric differentiation, monotonicity and maps to L^1
with Jeff Cheeger, ps pdf
Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups
with Marc Bourdon, ps pdf
Compression bounds for Lipschitz maps from the Heisenberg group to L_1
with Jeff Cheeger and Assaf Naor, arXiv
Rigidity of Schottky sets
with Mario Bonk and Sergei Merenkov, pdf
Differentiable structures om metric measure spaces: A primer
with John Mackay, pdf
Realization of metric spaces as inverse limits, and bilipschitz embedding in L_1
with Jeff Cheeger, pdf
Some applications of l_p-cohomology to boundaries of Gromov hyperbolic spaces
with Marc Bourdon, pdf

This material is based upon work supported by the National Science Foundation under Grant No. DMS-0701515. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.