The classical theory of compact translation surfaces is organized through the stratification of moduli spaces according to the orders of the zeros of the associated Abelian differentials. In the setting of tame translation surfaces of infinite topological type, no analogous notion of strata has been systematically developed. In this article, we introduce a stratification theory for tame translation surfaces based on their singularity data. To each non-compact tame translation surface, we associate a sequence recording the

cardinalities of finite-angle cone singularities of each possible cone angle together with the cardinality of the set of infinite-angle singularities. We call this sequence the singularity data of the surface.

We study the realization problem for singularity data on the Loch Ness monster. Our first main result shows that there is no tame translation structure on the Loch Ness monster having only a finite number of

finite cone angle singularities. We then prove that this is the only obstruction: every singularity data not excluded by the previous result is realized by a tame translation structure on the Loch Ness monster. As a consequence, we obtain a complete characterization of the strata of tame translation structures on this surface in terms of their singularity data. These results provide the first realization theorem for strata of non-compact tame translation surfaces.

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.

Contributing to the growing  body of work on properties of random surfaces, we estimate the expected covering radius in the SL(2,R)-orbits of regular 2n-gon translation surfaces. We show that the elements of the orbit with a specific combinatorial Delaunay triangulation form a fundamental domain under a group action which preserves covering radius. We then numerically approximate the covering radius for these orbits.

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. Some 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.