Research

My general interests include:

A 360° look at a sub-Finsler sphere in the Heisenberg group

Research papers

On the horofunction boundaries of homogeneous groups, preprint. (preprint).

We investigate general properties of horofunction boundaries of homogeneous metrics in graded groups. In particular, we can describe characteristics of horofunctions of a general family of metrics on Carnot groups. We also explore families which generalize the 3-dimensional Heisenberg group, namely, filiform groups and the higher Heisenberg groups, and answer questions about the dimension and topology of their horofunction boundaries.

Sub-Finsler horofunction boundaries of the Heisenberg group, with Sebastiano Nicolussi Golo. Analysis and Geometry in Metric spaces, 2021. arXiv link

Link to video

We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics---that is, those that arise as asymptotic cones of word metrics---on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.

Stars at infinity in Teichmuller space, with Moon Duchin. Geometriae Dedicata, 2021.    arXiv link   
Links to videos: Part 1 and Part 2

We investigate a metric structure on the Thurston boundary of Teichmüller space. To do this, we develop tools in sup metrics and apply Minsky's theorem.

In preparation

Expected covering radius of doubled slit tori, with Anthony Sanchez and Sunrose Shrestha.

In this project we consider the covering radius function on the moduli space of translation surfaces, which gives the radius of the largest immersed disk in a surface. The asymptotic averages of this function were studied by Masur-Rafi-Randecker, which was partly inspired by a similar question in the context of random hyperbolic surfaces studied by Mirzakhani. We obtain exact values for the expected covering radius for a specific class of translation surfaces called doubled slit tori, making use of Delaunay triangulations and a natural coordinate system in the moduli space. We are also currently working to generalize our results to other families of translation surfaces.

Other writing and projects

In Spring 2022, with graduate student Caitlyn Booms I led an undergraduate research project with the Madison Experimental Mathematics Lab. The group of 5 students learned about random walks in groups and discovered new trends in random walks on the Heisenberg group. You can see the poster they made here.

When writing my Ph.D. dissertation, I decided to include a chapter talking about geometry and my research using language that (I hope!) my friends and family could understand. See it here.

Here is a set of notes on nilpotent groups and their geometry which I co-wrote with Moon Duchin while in grad school.

 For my candidacy exam in grad school in December 2017, I studied curvature in complex hyperbolic space as well as the horofunction boundary of complex hyperbolic space. After the project, I wrote up a brief set of notes, including an original exploration into its curvature. You can find those notes here.

Current and past collaborators/mentors