CAREER: Ergodic Geometry Beyond Compactness and Uniform Hyperbolicity

This project explores questions in ergodic geometry, an interdisciplinary field connecting ergodic theory and geometry. Ergodic theory, a branch of dynamical systems, investigates systems that evolve over time, often exhibiting unpredictable and chaotic behaviors. Examples of such systems include planetary motion, weather patterns, and stock markets.

Geometry, on the other hand, studies the shape and structure of objects. Ergodic geometry combines these perspectives, using tools from dynamical systems to address geometric problems. For instance, the shape of an object can influence the complexity of certain dynamical systems occurring on that object.

This project aims to deepen our understanding of the relationship between an object’s shape and key dynamical quantities associated with it, such as entropic quantities, which serve as indices of complexity. In addition to advancing mathematical knowledge, the PI will establish the GW Experimental Mathematics Lab, creating a collaborative and vertically integrated research environment for students at George Washington University.

This lab will foster hands-on learning and mentorship, preparing students to engage in cutting-edge mathematical research.

In ergodic geometry, significant milestones have been achieved when the underlying dynamical systems are compact and uniformly hyperbolic. This project aims to extend these findings to geometric systems that are non-compact, nonuniformly hyperbolic, or both, using thermodynamic formalism. The research focuses on three objectives: studying correlations in higher-rank cusped geometric structures through orbital distribution analysis; examining the metric geometry of deformation spaces of these structures via Thurston’s asymmetric metric; and analyzing orbital correlations in nonuniformly hyperbolic geometric structures.

Progress in these areas will involve developing new tools in thermodynamic formalism combined with geometric insights, potentially advancing our understanding of the interplay between geometry and dynamics in more general settings.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.