本项目研究与低维流形的几何结构形变空间相关的一些几何拓扑问题，具体地，包括穿孔环面的SL(2,C)特征标簇的Bowditch集合的连通性，映射类群在曲面的PSL(2,R)特征标簇的非极端分支上作用的遍历性及相关的Bowditch问题，穿孔环面在相对Teichmuller空间中的一个极小化问题及与透镜空间的联系，由对称多项式所定义的代数簇的自同构群在该代数簇上的作用，相对离散群概念的建立与基本性质，Atiyah的线性无关性猜测等。本项目的研究有希望带来有意义的新想法与研究结果，并引发进一步的研究工作。

This project will focus on some problems in geometric topology related to deformation spaces of geometric structures on low-dimensional manifolds, which include the connectivity of Bowditch set of SL(2,C) characters of a once-punctured torus, the ergodicity of the actions of the mapping class group on the non-extremal components of PSL(2, R) characters of a surface and a related problem of Bowditch, a minimizing problem of a once-punctured torus over its fixed relative Teichmuller space and its relation with lens spaces, the action of automorphisms of an algebraic variety defined by a symmetric polynomial, the notion of a relatively discrete group and its fundamental properties, and Atiyah's linear independence conjecture. We hope that the research of this project will bring in useful new ideas and results, and inspire further research work.