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.

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.

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 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.

In this project we study the stars at infinity, a boundary structure defined by Anders Karlsson, on the horofunction boundary of polygonal norms in the plane. Much of the work of this project was done during a semester-long undergraduate research project run in Spring 2024 through the Madison Experimental Mathematics Lab at UW-Madison.