The SL(2,R) action on moduli space

本文研究了有理多边形中台球的相关解析问题、阿贝尔微分和二次微分的模空间，以及二乘二矩阵群在这些模空间上作用的动力学问题。在最近与M.Mirzakhani的合作中，PI能够证明这种作用的一些动力刚性结果，这使得人们能够理解每一个（而不仅仅是几乎每一个）轨道。这一点很重要，原因有几个。特别地，由台球产生的曲面是模空间中测度为零的集合，并且需要在每一点上成立的遍历定理来证明关于台球的结果。许多结果和技术是基于与局部对称空间上的单幂流理论（如拉特纳定理）的松散类比。然而，微分的模空间本质上是不同的，需要新的思想。PI建议在这个方向上进行更多的研究。有些自然现象是“混沌的”（即不可预测的）。这些通常用统计方法来研究。其他则是“可积的”（即可预测和规则的）。其他现象介于两者之间。作为本文主要研究对象之一的多边形台球系统是一种很好的中间行为模型。因此，它在物理学中也得到了广泛的研究，特别是在“量子混沌”方面。PI相信，引入的新技术和工具将在这些领域和其他领域得到应用。

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.