For you Simpsons fans out there, I like to think of myself as a hyperbolic topologist. The terminology comes from the following Professor Frink quote: "Well, it should be obvious to even the most dim-witted individual who holds an advanced degree in hyperbolic topology, that Homer Simpson has stumbled into... the third dimension!"

Research Interests

  • low-dimensional topology
  • mapping class groups; geometric and topological group theory
  • curve graphs; combinatorics
  • hyperbolic geometry; identities on hyperbolic manifolds
  • Riemann surfaces; Teichmüller theory (both finite- and infinite-dimensional)
  • convex real projective structures; Hitchin representations
  • Notes on the topology of mapping class groups (pdf)
    • These notes stem from discussions at the AIM workshop "Surfaces of infinite type".  In particular, they give proofs that big mapping class groups are not locally compact, not compactly generated, homeomorphic to the irrational numbers, and give an infinite family of mapping class groups that are generated by coarsely bounded sets (and hence have a well-defined quasi-isometry class).

  1. Big mapping class groups acting on homology (arXiv)
  2. Joint with Federica Fanoni and Sebastian Hensel.
  3. The first integral cohomology of pure mapping class groups (arXiv | pdf*)
  4. Joint with Javier Aramayona and Priyam Patel.
*A pdf link for a preprint indicates that there is a new version available that is not currently posted on the arXiv. 

  1. Thurston norms of tunnel number-one manifolds (arXiv)
  2. Joint with Natalia Pacheco-Tallaj and Kevin Schreve.
    J. Knot Theory Ramifications, to appear.
  3. Algebraic and topological properties of big mapping class groups (journalarXiv)
  4. Joint with Priyam Patel.
    Algebr. Geom. Topol. 18 (2018), no. 7, 4109-4142.
  5. Coloring Curves on Surfaces (journalarXiv)
    Joint with Jonah Gaster and Josh Greene.
    Forum Math. Sigma 6 (2018), e17.
  6. Graphs of curves on infinite-type surfaces with mapping class group actions (journalarXiv)
  7. Joint with Matthew Durham and Federica Fanoni
    Ann. Inst. Fourier (Grenoble) 68 (2018), no. 6, 2581-2612.
  8. The Bridgeman-Kahn identity for hyperbolic manifolds with cusped boundary (journalarXiv)
    Joint with Andrew Yarmola.
    Geom. Dedicata 194 (2018), 81-97.
  9. Automorphisms of the compression body graph (journalarXiv)
    Joint with Ian Biringer.
    J. London Math. Soc. (2) 95 (2017), no. 1, 94-114.
  10. Basmajian identities in higher Teichmüller-Thurston theory (journalarXiv)
    Joint with Andrew Yarmola.
    J. Topology 10 (2017), no. 3, 744-764.
  11. Moments of a length function on the boundary of a hyperbolic manifold (journalarXiv
    Algebr. Geom. Topol. 15 (2015), no. 4, 1909-1929.
  12. Quasiconformal homogeneity and subgroups of the mapping class group (journal | arXiv)
    Michigan Math. J. 64 (2015), no. 1, 53-75.
(You can also see a list of all my publications/preprints on the arXiv here.)

Denotes an undergraduate author.

Recorded Talks

My talk "Algebraic and topological properties of big mapping class groups" at the Groups, Geometry, and Dynamics seminar at UIUC was recorded and can be found here (the link will take you to YouTube).