Research



Research | Teaching | Talks | Outreach


Historically my research has primarily focused on outer automorphisms of free groups (i.e. the groups Out(F_r)), specifically aysmptotic dynamical conjugacy invariants and geodesics in Culler-Vogtmann Outer Space (the parameter space of weighted graphs). I am particularly interested in understanding the geodesic flow.

My focus now additionally includes mapping class groups, Teichmueller theory, foliations, geometric soccer modeling, and topological data analysis.

I am interested in understanding Culler-Vogtmann Outer Space from a variety of interconnected perspectives. This is a focus of my current year at the Institute for Advanced Study.


Publications:

20. A profinitely rigid hyperbolic free-by-cyclic group (Joint with Naomi Andrew, Paige Hillen, Robbie Lyman)
(arXiv id 2410.17817 )

19. Cannon-Thurston maps and singularity of measures on the two-sphere (Joint with Vaibhav Gadre, Thomas Haettel, Joseph Maher, Caglar Uyanik )
(preprint available upon request)

18. Collar Parameters for Teichmuller Space and Measured Foliations on a Surface (Joint with Daryl Cooper)
(preprint available upon request)

17. Out(F_r) train track automata I: Proper full fold decompositions
(arXiv id 2409.05599 )

16. Low complexity among principal fully irreducible elements of Out(F_3) (Joint with Naomi Andrew, Paige Hillen, Robbie Lyman)
(arXiv id 2405.03681 )

15. Collar Parameters for Teichmuller Space and Measured Foliations on a Surface Research Announcement (Joint with Daryl Cooper)
(arXiv id 2404.17081 )

14. Counting conjugacy classes of fully irreducibles: double exponential growth (Joint with Ilya Kapovich)
(Geometriae Dedicata, Volume 218, no. 39, (2024))

13. Taking the High-Edge Route Through Outer Space in Rank 3 (Joint with Damara Gagnier❄)
(International Journal of Algebra and Computation, Vol. 33, Issue 08, pp. 1659 - 1685, 2023)
Corresponding automaton

12. Controlling ball progression in soccer (Joint with Emily Hunter❄, Haozhi Hong❄, Daniel Forestell❄, Ari Fialkov❄, Zoey Drassinower❄, and Timothy Chan)
(Submitted, arXiv id 2210.16474 )

11. Random trees in the boundary of Outer space (Joint with Ilya Kapovich, Joseph Maher, Samuel J. Taylor)
(Geometry and Topology, 26 (2022) 127-162 )

10. Random outer automorphisms of free groups: Attracting trees and their singularity structures (Joint 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 (Joint 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 (Joint 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 (Joint 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) (Joint with Ilya Kapovich)
(International Journal of Algebra and Computation, Volume No.25, Issue No. 5. (2015), arXiv:1409.8044)

5. Lone Axes in Outer Space (Joint 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), arXiv:1311.4490)

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), arXiv:1301.7080)

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; arXiv:1301.6645)

1. Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs
(New York Journal of Mathematics Volume 21 (2015) 417-463, arXiv:1210.5762)

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

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 Bob Moses for supporting my 2024-2025 membership.


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