本项目将研究曲率与拓扑的关系，特别是曲率非负流形的几何拓扑。在低维拓扑方面，本项目将运用Seiberg-Witten理论, 特别是Bauer-Furuta不变量以及 Seiberg-Witten-Taubes 理论研究四维流形上的群作用, 讨论一些四维流形上的局部线性群作用的拓扑分类, 实现问题以及是否可以光滑化的问题. 我们还将研究Kähler曲面上的Ricci流，包括起点模型的分类，局部手术以及长期解的Cheeger-Gromov收敛. 在三维流形的拓扑的研究中, 我们将致力于Heegaard 分解的粘合映射的特征, 与相应的 Heegaard 分解以及对应的三维流形的拓扑性质和几何结构之间的联系. 我们以曲面和几何三维流形为主, 研究自同胚以及一般映射的不动点与周期点问题, 寻找描述周期点集不变量, 并分析这些不变量与空间拓扑之间的关系.

In this proposal we will study the interaction between curvature and topology, in particular the geometric topology of manifolds with nonnegative curvature. In low dimensional topology, we shall study the group actions on 4-dimensional manifolds, by using Seiberg-Witten theory, especially the Bauer-Furuta invariant and Seiberg-Witten-Taubes theory. We shall concern with the topological classification of locally linear group actions on certain $4$-dimensional manifolds. Corresponding realization and smoothlization are also addressed. We also study the geometry of Kähler surfaces by the method of Ricci flow, including the classification of singularity models, local surgeries and the long time limits in the Cheeger-Gromov topology. For 3-dimensional manifolds, we shall focus on the character of the gluing maps of a Heegaard splitting, and the relation with the topological and geometric structure of Heegaard splitting itself and the corresponding manifold. We shall study the fixed points and periodic points of homeomorphisms or general maps, mainly on surfaces and 3-dimensional manifolds with geometric structures. We shall ask for new invariants describing periodic point sets, and also the relations between these invariants and the topologies of underline spaces.