Research

Thank you to the Institute for Advanced Study for my 2024-2025 membership!
Thank you to the Institut des Hautes Etudes Scientifiques for my April-May 2026 Invited Researcher position!

Research Overview

Historically my research has primarily focused on outer automorphisms of free groups (i.e. the groups Out(F_r)), specifically asymptotic dynamical conjugacy invariants and geodesics in Culler-Vogtmann Outer Space (the parameter space of weighted graphs). I am currently particularly interested in understanding the geodesic flow and connections between metric graphs, metric surfaces, symbolic dynamics, theoretical physics, and computer science. I have also worked with mapping class groups, Teichmueller theory, foliations, geometric soccer modeling, and topological data analysis.

Publications in Progress

(drafts available upon request)

22. An enumerative and combinatorial description of Teichmuller space and the space of measured foliations
(with Victoria Schleis)
21. Collar Parameters for Teichmuller Space and Measured Foliations on a Surface
(with Daryl Cooper)

Publications

20. Singularity of Cannon-Thurston maps (with Vaibhav Gadre, Joseph Maher, Caglar Uyanik)
arXiv id 2510.04350
19. A "cubist" decomposition of the Handel-Mosher axis bundle (with Chi Cheuk Tsang)
arXiv id 2503.16360
18. Collar Parameters for Teichmuller Space and Measured Foliations on a Surface Research Announcement (with Daryl Cooper)
Comptes Rendus - Serie Mathematique, to appear
17. Out(F_r) train track automata I: Proper full fold decompositions
Journal of Topology and Analysis, to appear
16. A profinitely rigid hyperbolic free-by-cyclic group (with Naomi Andrew, Paige Hillen, Robbie Lyman)
Glasgow Mathematical Journal, to appear
15. Low complexity among principal fully irreducible elements of Out(F_3) (with Naomi Andrew, Paige Hillen, Robbie Lyman)
Algebraic and Geometric Topology, to appear
14. Counting conjugacy classes of fully irreducibles: double exponential growth (Joint with Ilya Kapovich)
Geometriae Dedicata, Volume 218, no. 39, 2024
13. Controlling ball progression in soccer (with Emily Hunter†, Haozhi Hong†, Daniel Forestell†, Ari Fialkov†, Zoey Drassinower†, and Timothy Chan)
SIAM Undergraduate Research Online, Vol 18, pp. 45-65, 2025
12. Taking the High-Edge Route Through Outer Space in Rank 3 (with Damara Gagnier†)
International Journal of Algebra and Computation, Vol. 33, Issue 08, pp. 1659 - 1685, 2023
Corresponding automaton
11. Random trees in the boundary of Outer space (with Ilya Kapovich, Joseph Maher, Samuel J. Taylor)
Geometry & Topology, 26 (2022) 127-162
10. Random outer automorphisms of free groups: Attracting trees and their singularity structures (with Ilya Kapovich, Joseph Maher, Samuel J. Taylor)
Transactions of the American Mathematical Society, 375 (2022), 525-557
9. Stable Strata of Geodesics in Outer Space (with Yael Algom-Kfir and Ilya Kapovich)
International Mathematics Research Notices, Vol. 2019, Issue 14, July 2019, pp. 4549-4578.
Talk on the paper
8. Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms (with Yael Algom-Kfir)
New York Journal of Mathematics 23 (2017), 365-381
7. A Dense Geodesic Ray in the Out(F_r)-quotient of Reduced Outer Space (with Yael Algom-Kfir)
Groups, Geometry, & Dynamics, 12 (2018), 571 - 613
Talk on the paper
6. A Train Track Directed Random Walk on Out(F_r) (with Ilya Kapovich)
International Journal of Algebra and Computation, Volume No.25, Issue No. 5. 2015
5. Lone Axes in Outer Space (with Lee Mosher)
Algebraic and Geometric Topology 16 (2016), no. 6, 3385-3418.
4. Out(F_3) Index Realization
Mathematical Proceedings of the Cambridge Philosophical Society, volume 159, issue 03, pp. 445-458. 2015
The following corresponding sage worksheet shows how I produced the index list -1/2 via a random automorphism generation process. The other index lists are not produced in this manner. However, it may be interesting that, while this particular example was difficult to produce by hand, our hypothesis that this list might be generic had some supporting evidence in the fact that (once we used enough generators), the example showed up immediately.
3. Ideal Whitehead Graphs in Out(F_r) III: Achieved Graphs in Rank 3
Journal of Topology and Analysis DOI: 10.1142/S1793525316500084 (2015), Volume No.08, Issue No. 02 2016
2. Ideal Whitehead Graphs in Out(F_r) II: Complete Graphs in Every Rank
Journal of Homotopy and Related Structures: Volume 10, Issue 2 (2015), Page 275-301
1. Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs
New York Journal of Mathematics Volume 21 (2015) 417-463
0. Thesis

Other Preprints

Ideal Whitehead Graphs in Out(F_r) IV: Building ideal Whitehead graphs in higher ranks and ideal Whitehead graphs with cut vertices

† Undergraduate research student

Funding and Support

This research has in large part been supported by an NSERC Discover Grant, an NSERC Discovery Launch Supplement, the U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric structures And Representation varieties" (the GEAR Network), and CANSSI Collaborative Research Team Project: Sports Analytics. It has also been supported by research funds granted via a Ky Fan Visiting Assistant Professorship, by the ARCHIMEDE Labex (ANR-11-LABX- 0033) and the A*MIDEX project (ANR-11-IDEX-0001-02) funded by the "Investissements d'Avenir," managed by the ANR, and by the CRC701 grant of the DFG, supporting the projects B1 and C13 in Bielefeld. Thank you to the Institute for Advanced study for their hospitality and the Bob Moses Fund for supporting my 2024-2025 membership.