• 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:

1. Rigidity for quasi-isometries of symmetric spaces and Euclidean buildings
2. Rigidity of quasi-isometries for symmetric space and Euclidean buildings (announcement)
3. Groups quasi-isometric to symmetric spaces
4. Quasi-isometries and the de Rham decomposition
5. Separated nets in Euclidean space
6. Boundaries of nonpositively curved spaces
7. Hyperbolic groups with low dimensional boundary
8. The local structure of length spaces with curvature bounded above
9. The structure of the stable norm for metrics on tori
10. Coarse Alexander duality and duality groups
11. The geodesic flow of a nonpositively curved graph manifold
12. Rigidity for Quasi-Mobius group actions (formerly Rigidityfor convergence group actions)
13. Rectifying separated nets
14. Van Kampen's embedding obstruction for discrete groups
15. Quasisymmetric parametrizations of two-dimensional metric spheres
16. Review of 3 books on nonpositive curvature
17. Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary
18. Quasi-hyperbolic planes in hyperbolic groups
19. Notes on Perelman's Ricci flow papers (arxiv)
20. Singularity structure in mean curvature flow of mean convex sets
21. The weak hyperbolization conjecture for 3-dimensional CAT(0) groups
22. Hadamard spaces with isolated flats
23. Rigidity of invariant convex sets in symmetric spaces
24. The asymptotic geometry of negatively curved spaces: uniformization, geometrization, and rigidity
    2006 ICM Proceedings
25. On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces
26. Differentiating maps into L^1, and the geometry of BV functions
27. Generalized differentiation and bi-Lipschitz nonembedding in L^1 (announcement)
28. The asymptotic geometry of right-angled Artin groups I
29. A new proof of Gromov's theorem on groups of polynomial growth
30. Characterization of the Radon-Nikodym Property in terms of inverse limits
31. Geometry and rigidity of mapping class groups
32. Induced quasi-actions: a remark
33. Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane
34. Quasiflats in CAT(0) complexes
35. Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon Nikodym property
36. Metric differentiation, monotonicity and maps to L^1
37. Combinatorial modulus, the Combinatorial Loewner Property, and Coxeter groups
38. Compression bounds for Lipschitz maps from the Heisenberg group to L_1 (arXiv)
39. Locally Collapsed 3-Manifolds
40. Geometrization of Three-Dimensional Orbifolds via Ricci Flow
41. Rigidity of Schottky sets
42. Differentiable structures on metric measure spaces: A primer
43. Realization of metric spaces as inverse limits, and bilipschitz embedding in L_1
44. Some applications of l_p-cohomology to boundaries of Gromov hyperbolic spaces
45. Mean curvature flow of mean convex hypersurfaces
46. Inverse limit spaces satisfying a Poincare inequality
47. Mean curvature flow with surgery
    
48. Singular Ricci flows I
    
49. Infinitesimal structure of differentiability spaces, and metric differentiation
    
50. PI spaces with analytic dimension 1 and arbitrary topological dimension
    
51. Groups quasi-isometric to RAAG's
    
52. Rectifiability of planes and Alberti representations
    
53. Uniqueness and stability of Ricci flow through singularities
    
54. Ricci flow and diffeomorphism groups of 3-manifolds
    
55. Higher rank hyperbolicity
    
56. On the rotational symmetry of 3-dimensional kappa-solutions
    
57. Ricci flow and contractibility of spaces of metrics