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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.

(In preparation, preliminary preprint available upon request)

11. Counting conjugacy classes of fully irreducibles: double exponential growth (Joint with Ilya Kapovich)

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10. Stable Strata of Geodesics in Outer Space (Joint with Yael Algom-Kfir and Ilya Kapovich)

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Talk on the paper

9. Normalizers and centralizers of cyclic subgroups generated by lone axis fully irreducible outer automorphisms (Joint with Yael Algom-Kfir)

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8. A Dense Geodesic Ray in the Out(F_r)-quotient of Reduced Outer Space (Joint with Yael Algom-Kfir)

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Talk on the paper

7. A Train Track Directed Random Walk on Out(F_r) (Joint with Ilya Kapovich)

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6. Lone Axes in Outer Space (Joint with Lee Mosher)

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5. Out(F_3) Index Realization

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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

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3. Ideal Whitehead Graphs in Out(F_r) III: Achieved Graphs in Rank 3

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2. Ideal Whitehead Graphs in Out(F_r) II: Complete Graphs in Every Rank

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1. Ideal Whitehead Graphs in Out(F_r) I: Some Unachieved Graphs

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0. Thesis.

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.

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