Curves, Complexity, and Configurations

The objectives of this project are to answer a series of fundamental questions about the geometry and dynamics of surfaces, motivated by questions originating in physics and materials science, in particular the theories of kinetic motion and of electron transport.The intellectual merit and broader impacts in this proposal inform each other via engagement with students at all levels and a commitment to diversity in teaching, mentoring, public engagement, and professional service activities associated to this project. This project has three main themes, centering on the geometry and dynamics of surfaces. The first is to develop a theory of Eisenstein series and cusp forms for the moduli space of translation surfaces, drawing inspiration from classical number theory, to unify and strengthen important geometric and dynamical results for renormalization dynamics on the moduli spaces of translation surfaces. The second thread, inspired by the study of special trajectories on polygonal billiards, focuses on the relationship between combinatorial complexity (informally, the number of bounces for a billiard trajectory) and geometric length of saddle connections on translation surfaces, and in particular, proving precise asymptotic results for new families of translation surfaces beyond flat tori. The third thread considers the general geometry and counting problems associated to configurations of curves and graphs on translation and hyperbolic surfaces. 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.