Dynamics on the modular surface

Project outline

The goal of this project is to understand properties of the geodesic and horocycle flows on the modular surface. The modular surface is a hyperbolic surface that is of fundamental importance in number theory. Expressing number-theoretic problems in terms of these flows allows them to be studied using tools from ergodic theory and dynamical systems (and vice versa). 

We will read chapters from the book "Ergodic Theory: with a view towards number theory" by Einsiedler and Ward to familiarize ourselves with the key concepts we are studying, in particular:

• Hyperbolic geometry and the Poincaré upper half-plane model

• The group SL(2,Z) and fundamental domains

• The geodesic and horocycle flows

• The space of unimodular lattices in the plane

Based on their own interests, students will then explore further properties of dynamics on the modular surface and its connections with number theory, for example:

• The geodesic flow and continued fractions

• Dani's theorem on Diophantine approximation

• Dani's theorem on measure rigidity for the horocycle flow

• Closed geodesics and quadratic forms

• The prime geodesic theorem

Prerequisites: Analysis III, and either Geometry III or Number Theory III 

Corequisites: either Ergodic Theory IV or Topics in Algebra and Geometry IV.