This project deals with a long-standing problem related to randomness. Consider a ball on a frictionless table. If the ball is set in motion, it will travel forever, making perfectly elastic collisions with the walls. If the table is a square or an equilateral triangle, there are only two possible behaviors: either the ball repeats the same periodic path forever, or it travels completely randomly in the entire polygon, eventually visiting every part of the table. This project is directed toward the basic mathematical problem of understanding the behavior of a ball when the table is a more general polygon. This is a basic problem arising in physics and statistical mechanics.
The project concerns the interrelated analytic study of trajectories on rational polygonal tables, moduli spaces of abelian and quadratic differentials, and the dynamics of the action of the group of two-by-two matrices on these moduli spaces. In recent work with M. Mirzakhani and in part with A. Mohammadi the PI was able to prove some dynamical rigidity results for this action, which allow one to understand every (and not just almost every) orbit. This is important for several reasons. In particular, the surfaces which arise from table trajectories are a set of measure zero in the moduli space, and ergodic theorems which hold at every point are needed to prove results about the trajectories. Many of the results and techniques are based on a loose analogy with the theory of unipotent flows on locally symmetric spaces (e.g. Ratner's theorem). However, the moduli spaces of differentials are substantially different and new ideas were needed. We propose developing these ideas further, both in the context of dynamics on moduli space and also in the context of other group actions.
