申请人代表性的工作是提出一整套方法研究Teichmüller测地流上的Lyaounov指数。申请人和合作者在Teichmüller测地流领域证明了几个Kontsevich-Zorich关于Lyapunov指数的猜想; 申请人单独提出比较代数几何稳定性和动力系统稳定性的猜想，此猜想被Eskin-Kontsevich-Möller-Zorich专文证明。在本项目中拟对以下问题展开研究: (1) 猜想等号达到情形与的Thin群的关系。（2)在一般表示中定义Lyapunov指数，以及寻找类似的猜想。（3）Teichmüller测地流这一套方法可否应用在Bridgeland稳定条件空间上。(4)建立Teichmüller测地流与映射类群之间的关系，给出一些可能在低维拓扑中的应用。

The representative work of the applicant is to propose a set of methods to study the Lyapunov exponents of Teichmüller geodesic flows. The applicant and the collaborator prove several conjectures about Lyapunov exponents of Teichmüller geodesic flows. The applicant separately proposes a conjecture about the relationship betwwen stability in algebraic geometry and stability in dynamical system, which is porved by Eskin-Kontsevich-Möller-Zorich. In this project, it is proposed to do research in the following problems: (1)The relationship between the equalities and the thin groups. (2) How to define the Lyapunov exponents in general representations, and find similar conjectures. (3)Does the methords of Teichmüller geodesic flows can be applied to the spaces of Bridgeland stablility condeitions. (4)Establish a relationship between Teichmüller geodesic flows and mapping class groups, and give some applications to low dimension topology.