Research

Research interests

I am broadly interested in the Teichm├╝ller space, and its isometrically embedded submanifolds, as well as subgroups of the mapping class group and their action on these spaces. A particular example of such a isometrically embedded submanifold is the Teichm├╝ller space of non-orientable surfaces, which embeds as a totally real submanifold, and whose stabilizer is the mapping class group of non-orientable surfaces. The action of the stabilizer has surprising dynamical properties, and I'm interested in understanding how the dynamics differ from the orientable version.

I have also of late been thinking about hierarchically hyperbolic spaces, and about the dynamics of hierarchicically hyperbolic groups acting on them.

Papers

  1. The limit set of non-orientable mapping class groups. (To appear in Journal of Modern Dynamics; arXiv:2110.00037) (abstract)
  2. Pseudo-Anosov homeomorphisms of punctured non-orientable surfaces with small stretch factor. With Caleb Partin and Becca Winarski. (arXiv, pdf, abstract) (To appear in Algebraic & Geometric Topology; arXiv:2107.04068)

Computer experiments

I also write computer programs to assist in visualization of exotic geometric objects, as well as testing small conjectures. I usually upload my cleaned up code on Github, but if it's a small one-off program, I might not make it public. In case you would like access to the code anyways, let me know, and I will be happy to share it.

Limit set of a non-orientable mapping class group

limit-set-picture

A part of the limit set of the mapping class group of a non-orientable surface of genus 6 when restricted to a 3-dimensional stratum of projective measured foliations. Unlike in the orientable case, the limit set is not the entire stratum, but rather a nowhere dense measure 0 subset. The Hausdorff dimension of the limit set restricted to a dimension n-stratum is bounded between n-1 and n-2, where the upper bound is sharp. Locally, the limit set is contained in a hyperplane, and the image is a rendering of the limit set when restricted to one such hyperplane.

Resolving intersections of parallel copies of intersecting curves

Given two two-sided curves on a non-orientable surface with positive intersection, we can take m and n parallel copies of the curves and resolve all the intersections in a consistent manner to get a multicurve. This code resolves all the intersection, and counts the number of one-sided and two-sided components of the resulting multicurve. If there are no one-sided components, the program plots the point $(m+m, \frac{m}{m+n})$ on a scatter plot. The fact that the projection to the y-axis is not dense in $[0,1]$ seems to suggest that a convex combination of two minimal foliations that are approximable by two-sided multicurves may not itself be approximable by two-sided multicurves.

scatter-plot-picture