Research profile

My research interests are in geometric analysis, geometric group theory, and geometric topology. These areas involve a mixture of ideas from geometry, analysis, topology, and algebra. Current research topics include:

 
 
 

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



This material is based upon work supported by the National Science Foundation under Grant Nos. DMS-1405899, DMS-1406394, DMS-1711556, and DMS-2005553.   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.