Topology and Geometry of Complex Systems

Abstract: Complex systems with interesting topology and geometry abound in the physical and mathematical sciences. In my research, I develop methods in topological and geometric data analysis to study these systems, and apply them to structures in materials science, physics, and biology. Here, I describe two major topics of my current and future research. First, I discuss joint work with J. K. Mason and D. Rodney to develop a language to describe the local structure of silica glasses. The relationship between the local structure and global properties of these materials is surprisingly poorly understood, in part due to the lack of an appropriate structural descriptor. Second, I present results on the relationship between fractal dimension and persistent homology. This work includes limit theorems showing that the fractal dimension of a measure can be recovered from the persistent homology of random samples as well as computational studies (joint with J. Jaquette) demonstrating that a persistent homology-based dimension estimation algorithm performs as well or better than classical techniques.