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 and here is a current research statement.


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


I have advised the PhDs of Nick Vlamis, Tommaso Cremaschi, Cristina Mullican and Sangsan (Tee) Warakkagun. I'm currently advising Mujie Wang and Matthew Zevenbergen.

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.


Here are some scripts I wrote for an IBL Introduction to Proof course at Boston College, on logic, sets, induction, number theory, graph theory, functions and relations, and cardinality. If you want to use these notes for a class, email me and I'll send you the .tex files.


Here is a set of notes on ergodic theory that I wrote for a 2023 topics class. It also includes some applications to geometry, beginning with ergodicity of the geodesic flow on finite volume hyperbolic manifolds and ending with brief discussions of lattice point and curve counting, and the Kahn-Markovic surface subgroup theorem.

Papers, in reverse chronological order
Convergence of normalized Betti numbers in nonpositive curvature.
Joint with Miklos Abert, Nicolas Bergeron and Tsachik Gelander.
Duke Mathematical Journal, (2023). 54 pp.
Thick hyperbolic 3-manifolds with bounded rank.
Joint with Juan Souto.
To appear in Memoirs of the AMS. 225 pp.
Unimodular measures on the space of all Riemannian manifolds
Joint with Miklos Abert.
Geometry and Topology (2023). 81 pp.
Invariant random subgroups of semidirect products
Joint with Lewis Bowen and Omer Tamuz.
Ergodic Theory and Dynamical Systems (2019). 14 pp.
Determining the finite subgraphs of the curve graph
Joint with Tarik Aougab and Jonah Gaster.
Groups, Geometry and Dynamics (2018). 12 pp.
Metrizing the Chabauty topology.
Geometriae Dedicata (2017). 3 pp.
Nielsen equivalence in mapping tori over the torus.
Conformal Geometry and Dynamics (2017). 6 pp.
Intersection growth in groups
Joint with Khalid Bou-Rabee, Martin Kassabov and Francesco Matucci.
To appear in Transactions of the AMS (2017). 20 pp.
On the growth of L2-invariants for sequences of lattices in Lie groups
Joint with Miklos Abert, Nicolas Bergeron, Tsachik Gelander, Nikolay Nikolov, Jean Raimbault and Iddo Samet.
Annals of Mathematics (2017). 64 pp.
An announcement, published in Comptes Rendus Mathematique (2012), is here.
Ends of unimodular random manifolds
Joint with Jean Raimbault.
Proceedings of the AMS (2017). 8 pp.
Automorphisms of the compression body graph
Joint with Nicholas Vlamis.
Journal of the London Mathematical Society (2017). 25 pp.
Unimodularity of invariant random subgroups
Joint with Omer Tamuz.
Transactions of the AMS (2016). 23 pp.
Rank of mapping tori via the curve complex
Joint with Juan Souto.
Journal fur die reine und angewandte Mathematik (2015). 21 pp.
Extending pseudo-Anosov maps to compression bodies
Joint with Yair Minsky and Jesse Johnson.
Journal of Topology (2013). 29 pp.
A finiteness theorem for closed hyperbolic 3-manifolds
Joint with Juan Souto.
Journal of the London mathematical society (2011). 20 pp.
Geometry and Rank of Fibered Hyperbolic 3-Manifolds
Algebraic and Geometric Topology (2009). 13 pp.
The Three Gap Theorem and Riemannian Geometry
Joint with Benjamin Schmidt.
Geometriae Dedicata, vol. 136 (2008). 18 pp.
Algebra vs. geometry in hyperbolic 3-manifolds
PhD thesis, University of Chicago. 89 pp.