My main fields of study are 3-dimensional topology, hyperbolic geometry, Riemann surface theory (Teichmüller theory) and mapping class groups. I've also dabbled in holomorphic dynamics and combinatorial group theory.
For a (somewhat out of date) collection of pictures of the kind of objects I am interested in, see these lecture slides.
My PhD students
Papers, roughly in reverse chronological order:
Stable cubulations, bicombings and barycenters. with M. Durham and A. Sisto. Arxiv.
A polynomial invariant for veering triangulations. with M. Landry and S. Taylor. Arxiv.
Bottlenecks for Weil-Petersson geodesics. with B. Modami. Arxiv
Weil-Petersson translation length and manifolds with many fibered fillings. with C. Leininger, J. Souto and S. Taylor. Arxiv
Surface groups are flexibly stable. with N. Lazarovich and A. Levit. Arxiv
Skinning bounds along thick rays. with K. Bromberg and A. Kent. To appear, J. Top. Anal. Arxiv
Local topology in deformation spaces of hyperbolic 3-manifolds II. with J. Brock, K. Bromberg, R. Canary and C. Lecuire. Groups, Geom. and Dyn. 13 (2019) no. 3, 767--793. Arxiv
Windows, cores and skinning maps. with J. Brock, K. Bromberg and R. Canary. Arxiv. To appear, Ann. Sci. Ec. Norm. Super.
Bounded combinatorics and uniform models for hyperbolic 3-manifolds. with J. Brock, H. Namazi and J. Souto. J. Topol. (2016). Arxiv
A brief introduction to mapping class groups. Moduli spaces of Riemann Surfaces, 5-44, IAS/Park City Math. Ser., 20. Amer. Math. Soc. (2013). Preprint
Thick-skinned 3-manifolds. with A. Kent. Geom. Funct. Anal. 24 (2014) 1981-2001. Arxiv
Convergence properties of end invariants. with J. Brock, K. Bromberg and R. Canary. ArXiv. Geom. Topol. 17 (2013) 2877–2922.
Picture-hanging puzzles. with E. Demaine, M. Demaine, J. Mitchell, R. Rivest and M. Patrascu. ArXiv. Theory Comput. Syst. 54 (2014), no. 4, 531-550.
Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges. with B. Weiss. ArXiv. Ann. Sci. Ec. Norm. Super. (4) 47 (2014), no. 2, 245-284.
The dynamics of Aut(F_n) on redundant representations. with T. Gelander. ArXiv. Groups, Geometry and Dynamics 7 (2013), 557-576.
Extending pseudo-Anosov maps to compression bodies. with I. Biringer and J. Johnson. ArXiv. J. Topol. 6 (2013), no. 4, 1019-1042.
Discrete primitive-stable representations with large rank surplus. with Y. Moriah. ArXiv. Geom. Topol. 17 (2013) 2223-2261.
Asymptotics of Weil-Petersson geodesics II: bounded geometry and unbounded entropy. with J. Brock and H. Masur. ArXiv. Geom. Funct. Anal. 21 (2011), 820-850.
Heegaard splittings with large subsurface distances. with J. Johnson and Y. Moriah. ArXiv. Algebr. Geom. Topol. 10 (2010), 2251-2275.
Local topology in deformation spaces of hyperbolic 3-manifolds. with J. Brock, K. Bromberg and R. Canary. ArXiv Geom. Topol. 15 (2011), 1169-1224.
On dynamics of Out(Fn) on PSL(2,C) characters. Arxiv. Israel J. Math. 193 (2013), 47-70.
Centroids and the rapid decay property in mapping class groups. with J. Behrstock. Arxiv. J. Lond. Math. Soc. (2) 84 (2011), 765--784.
Geometry and rigidity of mapping class groups. with J. Behrstock, B. Kleiner and L. Mosher. Arxiv. Geom. Topol. 16 (2012), 781-888.
High distance knots. with Y. Moriah and S. Schleimer. ArXiv. Algebraic & Geometric Topology 7 (2007), 101–999.
Curve complexes, surfaces and 3-manifolds. Proceedings of the ICM 2006. PDF
The classification of Kleinian surface groups II: The Ending Lamination Conjecture. with J. Brock and R. Canary. PDF. Ann. of Math. (2) 176 (2012), 1-149.
End invariants and the classification of hyperbolic 3-manifolds. Lecture notes from the Current Developments in Mathematics conference, Harvard, Nov. 2002. PDF.
Quasiconvexity in the curve complex. with H. Masur. In the Tradition of Ahlfors and Bers, III (W. Abikoff and A. Haas, eds.), Contemporary Mathematics 355, Amer. Math. Soc. (2004), 309-320. PDF.
The classification of Kleinian surface groups I: models and bounds. math.GT/0302208. Ann. of Math. (2) 171 (2010), 1-107.
Combinatorial and Geometrical Aspects of Hyperbolic 3-Manifolds. PDF. in Kleinian Groups and Hyperbolic 3-Manifolds (Y. Komori, V. Markovic and C. Series, Eds.) London Math. Soc. Lec. Notes, 299 (2003), 3-40.
Nondivergence of horocyclic flows on Moduli space, with B. Weiss. IMS Preprint 2000-08. J. Reine Angew. Math. 552 (2002), 131-177.
Short geodesics and end invariants. Expository article, Comprehensive Research on Complex Dynamical Systems and Related Fields, RIMS Kokyuroku #1153 (2000) 1-20, M. Kisaka and S. Morosawa, eds. Preprint.
Rank one phenomena in mapping class groups, with B. Farb and A. Lubotzky. Duke Math. J. 106 (2001), 581-597. PDF.
Unstable quasi-geodesics in Teichmuller space, with H. Masur. In the tradition of Ahlfors and Bers: Proceedings of the first Ahlfors-Bers colloquium, I. Kra and B. Maskit, eds. AMS Contemp. Math. 256 (2000), 239-241. Preprint.
Kleinian groups and the complex of curves, Geometry and Topology, 4 (2000), 117-148.
The classification of punctured-torus groups, Annals of Math. 149 (1999), 559-626. May 1998 Version.
Laminations in holomorphic dynamics, with M. Lyubich. J. Diff. Geometry. 47 (1997), 17-94. PDF.
A geometric approach to the complex of curves on a surface,Proceedings of the Taniguchi Symposium, Finland 1995. Preprint.
Spectral theory, Hausdorff dimension and the topology of hyperbolic 3-manifolds, with R. Canary and E. Taylor. J. Geom. Analysis 9 (1999), 18-40. PDF
On Thurston's ending lamination conjecture, Proceedings of the Knoxville Conference on Low-Dimensional Topology, 1992. International Press 1994. Preprint.
Extremal length estimates and product regions in Teichmuller space, Duke Math. J. 83 (1996), 249-286. Preprint.
On limits of tame hyperbolic 3-manifolds, with R. Canary, J. Differential Geom. 43 (1996), 1-41. PDF
Quasi-projections in Teichmuller space, J. Reine Angew. Math. 473 (1996), 121-136. Preprint.
On rigidity, limit sets and end invariants of hyperbolic 3-manifolds, J. Amer. Math. Soc. 7 (1994), 539-588. PDF
Teichmuller geodesics and ends of hyperbolic 3-manifolds, Topology 32 (1993), 625-647. PDF
Harmonic maps into hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 332 (1992), 607-632. PDF
Harmonic maps, length and energy in Teichmuller space, J. Differential Geom. 35 (1992) 151-217. PDF
Some of the work shown here was supported by grants from the National Science Foundation and a fellowship from the Alfred P. Sloan foundation. 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.