The proposal concerns the interrelated analytic study of billiards in rational polygons, moduli spaces of abelian and quadratic differentials, and the dynamics of the action by the group of two-by-two matrices on these moduli spaces. In recent work with M.Mirzakhani, 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 billiards are a set of measure zero in the moduli space, and ergodic theorems which hold at every point are needed to prove results about billiards. 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. The PI proposes additional research in this direction.
Some natural phenomena are ``chaotic'' (i.e. unpredictable). These are often studied by statistical methods. Others are ``integrable'' (i.e. predictable and regular). Other phenomena fit somewhere in between. The polygonal billiard system, which is one of the main subjects of study of the proposal, is a good model of intermediate behavior. As such it has been studied extensively in physics as well, in particular in connection to``quantum chaos''. The PI believes that the new techniques and tools introduced will have applications in these and other fields.
