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Geometric Group Theory & Geometric Topology

I work with surfaces and their self-homeomorphisms. I am particularly interested in different simplicial complexes and graphs that can be defined using the information from surfaces, such as the curve complex. These graphs can then be used to gain insight about both surfaces and their self-homeomorphisms. Find more pictures here!

ladder surface.png
Complex Dynamics

One of the goals of my research is to understand the dynamics of polynomials -- that is, what happens if we start with a polynomial, and apply it many times? One way to understand polynomials is to organize them. My goal is to classify polynomials up to conjugation and isotopy.

complex dynamics.png

Figure: the complex plane under iteration of a polynomial

Publications & Preprints
  • Automorphisms of the fine 1-curve graph. With K. W. Booth and D. Minahan. Preprint.

  • A challenger to elliptic billiards fails: String construction over convex polygons and the Birkhoff-Poritsky conjecture. With L. Bunimovich. In progress.

  • An Alexander method for infinite-type surfaces. Appeared in New York Journal of Math.

  • Automorphisms of the k-curve graph. With S. Agrawal, T. Aougab, Y. Chandran, M. Loving, J. R. Oakley, X. Yang. Appeared in Michigan Math Journal. PDF.


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