Geometry and dynamics of moduli spaces and mapping class groups of surfaces

本课题的主题是研究双曲曲面模空间上流动的几何和动力学，以及映射类群对表示变体的作用。本文讨论的许多问题和所采用的方法都是由负弯曲流形和齐次空间上的类似陈述所激发的。项目中需要解决的一些问题如下：1)理解模空间上束上的旋回和地震流的动力学。我们的目的是研究这些流与齐次空间上的单幂流之间的类比；2)曲面群在SL（n，R）中的Hitchin表示空间的几何性质；3)映射类群作用于紧复李群表面群表示的遍历性质。本课题的主要目标是进一步探索和理解曲面上的不同几何结构和相应模空间上的自然诱导动力系统。本课题提出的问题和方法可能会对几何结构在表面上的变形，以及生成随机黎曼曲面的不同模型之间的关系提供新的见解和结果。

The theme of this project is to study the geometry and dynamics of flows on moduli spaces of hyperbolic surfaces, and actions of the mapping class group on representation varieties. Many questions discussed in this proposal and the methods applied are motivated by analogous statements on negatively curved manifolds and homogeneous spaces. Some of the problems to be addressed within the project are the following: 1) understanding the dynamics of the horocycle and earthquake flow on bundles over the moduli space. We aim to study the analogies between theses flows and unipotent flows on homogeneous spaces; 2) geometric properties of the space of Hitchin representations of surface groups in SL(n,R); and 3) Ergodic properties of the actions of the mapping class group on the representations of surface groups in to a compact complex lie group.The main 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. The questions and methods in this project might provide new insights and results about deformations of geometric structures on surfaces, and the relationship between different models for generating random Riemann surfaces.