Research

The boundaries between pure and applied mathematics are becoming increasingly blurred thanks to novel interdisciplinary connections and to explosions in computational capabilities. My primary research area is applied topology, an exciting and young field that is at the interface of algebraic and differential topology and computer science.

Despite the name ``applied'' topology, much of the research in the field is  theoretical work that focuses on constructing the conceptual foundations and techniques of the still very much evolving discipline, in addition to research that focuses on actually applying these techniques and implementing computational algorithms to analyze real data. I started working in the field as a postdoc at Duke University, when I became interested in how techniques from applied and computational topology and geometry can be leveraged and combined with machine learning methods to address questions about data. During my time at Union, my work in the field has shifted from a more application-oriented focus on topological data analysis to a more theoretical emphasis. In recent work, I have focused extensively on topological characterizations and summaries for metric graphs and simplicial complexes associated to them.  

My secondary research area is mathematical shape analysis via a geometric structure called the medial axis. My work with the medial axis originated in my (pure mathematics) dissertation, which specifically focused on the differential topology and singularity theory that characterizes the properties of the medial axis. Though the results in my dissertation were purely theoretical, the work was inspired by questions from medical image analysis, and that lit the spark for subsequent research of mine in the area of shape recognition and image analysis. Most recently, my work has focused on how the medial axis can be utilized to quantify compactness of legislative and congressional districts in the context of detecting gerrymandering.