Organizational Work |
Historically my research has primarily focused on outer automorphisms of free groups (i.e. the groups Out(F_r)), specifically conjugacy invariants and their realization, determining generic elements of Out(F_r), the Out(F_r) conjugacy problem, and the geometry of Culler-Vogtmann Outer Space.
More recently my focus has shifted to include mapping class groups, Teichmueller theory, outer automorphisms of right-angled Artin groups, and the space of convex projective structures on a surface.
I also have ambitions of in the future taking inspiration from "Geometry of the Space of Phylogenetic Trees" and using my understanding of deformation spaces to study other systems arising in nature. One may interpret this as an invitation to talk with me on this subject.
13. Random outer automorphisms of free groups: Attracting trees and their singularity structures (Joint with Ilya Kapovich, Joseph Maher, Samuel J. Taylor)
12. From Measured Foliations to Teichmüller Space: Using the right trousers (Joint with Daryl Cooper)
Counting conjugacy classes of fully irreducibles: double exponential growth (Joint with Ilya Kapovich)
10. Stable Strata of Geodesics in Outer Space (Joint with Yael Algom-Kfir and Ilya Kapovich)
(International Mathematics Research Notices, To Appear)
Talk on the paper
9. 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)
8. A Dense Geodesic Ray in the Out(F_r)-quotient of Reduced Outer Space
(Joint with Yael Algom-Kfir)
(Groups, Geometry, & Dynamics, To Appear)
Talk on the paper
7. 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)
6. Lone Axes in Outer Space
(Joint with Lee Mosher)
(Algebraic & Geometric Topology 16 (2016), no. 6, 3385-3418. )
5. 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.
4. Ideal Whitehead Graphs in Out(F_r) IV: Building ideal Whitehead graphs in higher ranks and ideal Whitehead graphs with cut vertices
(Topology Proceedings, Subject to Revisions)
3. Ideal Whitehead Graphs in Out(F_r) III: Achieved Graphs in Rank 3
(Journal of Topology & 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)
I have only listed above papers for which I have a minimum of a complete draft. If you are interested in the other projects that I am currently working on, thinking about, planning on thinking about, etc, my research statement is available upon request.
This research has in large part been supported by the U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 ``RNMS: Geometric structures And Representation varieties'' (the GEAR Network). 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.