Relating Structure and Function in Dynamical Systems with Spectral and Topological Methods.

Speaker: Tobias Timofeyev (University of Vermont)

Date: 1/5/26

Abstract: In this talk, I will explore how graph-theoretic and topological methods can be used to relate structure and function in applied dynamical systems. When modeling real-world systems, both observations and interactions are inherently discrete, giving rise to natural combinatorial structure. A central question is how dynamical behavior can be understood or characterized through such structure. I will address this question in two complementary settings. First, we will consider collective dynamics in networked systems, showing how spectral properties of interaction graphs link mesoscopic structure, such as community organization, to cluster synchronization in the Kuramoto model. We will motivate this topic with applications in neuroscience and power systems. Second, we will consider the combinatorial structure of time series data, and how we may relate periodic behavior to persistent features in a topological Dowker persistence framework. This addresses the challenge of distinguishing qualitative shifts in high dimensional time series data, such as those gathered in weather system observations.