Homeomorphism Groups of One-manifolds: Rigidity and Regularity

一维流形是数学中的基本对象，其结构和性质长期以来一直受到数学家的关注。该项目的重点是一维流形的连续对称群的结构，包括对称群的局部结构和光滑性对代数性质的影响。该项目将解决关于这些群的结构的几个基本开放问题，并研究新的代数现象。该项目的思想和方法将广泛应用于其他领域，并将位于拓扑学、几何群论和动力系统的交界处。一流形的同胚群的结构，特别是紧致一流形的结构，多年来一直受到人们的关注。近年来，这一领域得到了长足的发展。该项目将通过研究闭曲面群的表示空间到圆的同胚射的局部拓扑来促进这些进展，而圆的局部连通性是一个长期未解决的问题。此外，该项目还讨论了正则性对紧流形上有限生成的同胚群的子群的代数结构的影响，并将构造能够以指定的正则性水平作用但不能光滑到更高正则性水平的群。还将讨论Thompson群F的作用的正则性，目的是建立紧致流形上F的光滑作用的Brin-Sapir猜想。

One dimensional manifolds are fundamental objects in mathematics, and their structure and properties have attracted the attention of mathematicians for a long time. The focus of the project is the structure of the group of continuous symmetries of one-manifolds, including the local structure of the group of symmetries and the effect of smoothness on algebraic properties. The project will resolve several basic open questions about the structure of these groups and investigate new algebraic phenomena. The ideas and methods in the project will be broadly applicable to other fields, and will lie at the interface of topology, geometric group theory, and dynamical systems.The structure of the homeomorphism groups of one-manifolds, especially compact one-manifolds, has received much attention over the years. Recently, this field has experienced tremendous development. The project will contribute to these advances by studying the local topology of the space of representations of a closed surface group into homeomorphisms of the circle, the local connectivity of which is a long standing open problem. Moreover, the project addresses the effect of regularity on the algebraic structure of finitely generated subgroups of homeomorphism groups on compact manifolds, and will construct groups which can act with prescribed levels of regularity but which are not smoothable to higher levels of regularity. The regularity of actions of Thompson's group F will also be addressed, with the goal of establishing the Brin-Sapir conjecture for smooth actions of F on compact manifolds.