The goal of this proposal is to further explore and understand different geometric structures on surfaces and natural induced dynamical systems on the corresponding moduli spaces. We intend to study the dynamics of natural geometric flows on the cotangent bundle of the moduli space of Riemann surfaces of a given genus. Most of the results and questions in this area are based on analogies between these flows and unipotent flows on homogeneous spaces. We also plan to investigate the asymptotic behavior of intersection pairings of tautological line bundles on the moduli spaces of Riemann surfaces of genus g as g goes to infinity. This could shed some light on geometric properties of random hyperbolic surfaces of large genus in different models.
The main problems motivating our proposed project are centered around the geometry of Riemann surfaces and their deformations. Moduli spaces of Riemann surfaces appear naturally in various contexts, for example in algebraic geometry and string theory. This research proposal is interdisciplinary and has connections to different branches of mathematics, especially dynamical systems, low dimensional topology, algebraic geometry, hyperbolic geometry and geometric group theory.