I am currently a Hale Visiting Assistant Professor at Georgia Tech.
Broadly speaking, my research interests lie in low-dimensional topology and geometric group theory. More specifically, I like to think about mapping class groups of surfaces, pseudo-Anosov mapping classes, translation surfaces, curve and arc complexes, hyperbolic 3-manifolds, as well as related algorithms and computation.
Fast Nielsen-Thurston classification (with Dan Margalit and Öykü Yurttas)
The goal of this project is to give a framework for fast computation in mapping class groups. We show that there is a quadratic-time algorithm that computes the Nielsen-Thurston type of a mapping class (finite order, pseudo-Anosov or reducible). The algorithm also finds the reducing curves, the stretch factors and invariant foliations on pseudo-Anosov components, invariant train tracks and veering triangulations of the pseudo-Anosov components, all in quadratic time.
Here is a poster in about the project.
An implementation of the algorithm above. One of the main goals is to create a program that works for closed surfaces in addition to punctured surfaces, a feature currently existing programs are lacking. The implementation of our Nielsen-Thurston classification is not yet done, but Macaw already has some functionality for closed surfaces. Here is the project website.
In the fall of 2017, Vignesh Raman and Jonathan Chen, undergraduate students at Georgia Tech, are joining in to help make progress on Macaw.
Fibering of 3-manifolds and translation length in the arc complex
The goal of this project is to study the asymptotic translation length in the arc complex of monodromies corresponding to different fibrations of a hyperbolic 3-manifold. Fried showed in 1982 that the stretch factors of the pseudo-Anosov monodromies vary nicely: the appropriately normalized stretch factor function is convex and continuous on a fibered face of the 3-manifold. We show that for the arc complex, the situation is in a sense worse: the analogous function is generally discontinuous (probably at every point). However, in another sense, the function is even nicer than the normalized stretch factor function: when taking appropriate limits, we obtain a function which is not only convex and continuous but also rational and depends (up to a constant factor) only on the shape of the fibered face.