Research

Most of my work is in hyperbolic geometry and low-dimensional topology, although I am also interested in combinatorial, measurable and geometric group theory, foliations, and manifolds of nonpositive curvature. Here is my CV.


I co-organize the Geometry/Topology seminar at Boston College.
Teaching

I am working on an online book Geometry in Two Dimensions that is suitable for students with only a multivariable calculus background. Currently, it discusses Euclidean, spherical and hyperbolic geometry, and touches on additional topics like scissors congruence, isoperimetric inequalities, and tilings.


Papers
Algebra vs. geometry in hyperbolic 3-manifolds
PhD thesis, University of Chicago. 89 pp.
The Three Gap Theorem and Riemannian Geometry
Joint with Benjamin Schmidt.
Geometriae Dedicata, vol. 136 (2008). 18 pp.
Geometry and Rank of Fibered Hyperbolic 3-Manifolds
Algebraic and Geometric Topology (2009). 13 pp.
Algebraic and Geometric Convergence of Discrete Representations into PSL(2,C)
Joint with Juan Souto.
Geometry and Topology (2010). 45 pp.
A finiteness theorem for closed hyperbolic 3-manifolds
Joint with Juan Souto.
Journal of the London mathematical society (2011). 20 pp.
Extending pseudo-Anosov maps to compression bodies
Joint with Yair Minsky and Jesse Johnson.
Journal of Topology (2013). 29 pp.
Rank of mapping tori via the curve complex
Joint with Juan Souto.
Journal fur die reine und angewandte Mathematik (2015). 21 pp.
Unimodularity of invariant random subgroups
Joint with Omer Tamuz.
Transactions of the AMS (2016). 23 pp.
Automorphisms of the compression body graph
Joint with Nicholas Vlamis.
Journal of the London Mathematical Society (2017). 25 pp.
Ends of unimodular random manifolds
Joint with Jean Raimbault.
Proceedings of the AMS (2017). 8 pp.
On the growth of L2-invariants for sequences of lattices in Lie groups
Joint with M. Abert, N. Bergeron, T. Gelander, N. Nikolov, J. Raimbault and I. Samet.
Annals of Mathematics (2017). 64 pp.
An announcement, published in Comptes Rendus Mathematique (2012), is here.
Intersection growth in groups
Joint with Khalid Bou-Rabee, Martin Kassabov and Francesco Mattucci.
To appear in Transactions of the AMS (2017). 20 pp.
Nielsen equivalence in mapping tori over the torus.
Conformal Geometry and Dynamics (2017). 6 pp.
Metrizing the Chabauty topology.
Geometriae Dedicata (2017). 3 pp.
Packing curves on surfaces with few intersections.
Joint with Tarik Aougab and Jonah Gaster.
To appear in IMRN. 8 pp.
Determining the finite subgraphs of the curve graph
Joint with Tarik Aougab and Jonah Gaster.
To appear in Groups, Geometry and Dynamics. 12 pp.
On the growth of L2-invariants of locally symmetric spaces, II: exotic invariant random subgroups in rank one
Joint with M. Abert, N. Bergeron, T. Gelander, N. Nikolov, J. Raimbault and I. Samet.
Submitted. 26 pp.
Unimodular measures on the space of all Riemannian manifolds
Joint with Miklos Abert.
Submitted. 80 pp.
Invariant random subgroups of semidirect products
Joint with Lewis Bowen and Omer Tamuz.
Submitted. 14 pp.
Thick hyperbolic 3-manifolds with bounded rank.
Joint with Juan Souto.
Submitted. 170 pp.
Iteration in Schottky space
Joint with Cyril Lecuire. (in preparation)