本项目研究低维流形几何拓扑学中有关曲面的几何结构形变空间的一些问题，具体地，包括曲面的SL(2,C)特征簇的几何刻画、拓扑学及动力学性质，有限秩自由群的SL(2,C)特征簇的使得映射类群作用为不连续的最大不变开集，Minsky所定义的本原稳定特征标集合与Bowditch特征标集合的关系，非Bowditch特征标集合的内点，非Bowditch特征标的端不变量，其他低维Spin群特征簇的参数化与几何拓扑性质，曲面Teichmuller空间的新参数化，曲面模空间上的极值问题，双曲曲面的简单长度谱的重数问题，曲面的简单长度谱的单调形变等问题。

In this project we shall investigate certain problems in low-dimensional topology that are related to the deformation spaces of geometric structures on surfaces. More precisely, these will include the characterization of SL(2,C) character varieties of surfaces and their topological as well as dynamical properties; finding out the largest open subset of the varieties on which the mapping class group acts properly discontinuously; the relationship between the subset of primitive stable characters defined by Minsky and the Bowditch subset; locating the interior points of the non-Bowditch subset; properties of the end invariants set of a non-Bowditch character; parametrization and geometric-topological properties of some other low-dimentional spin characters; new parametrization of Teichmuller spaces of surfaces; a minimization problem in the moduli space of a surface; multiplicities of the simple length spectrum, and the monotone deformation with repect to the simple length spectrum, and some other problems.