Mathematical Sciences: Mapping Class Groups and Teichmueller Spaces

小行星9704817 这个项目涉及曲面的映射类群的代数结构和Teichmueller空间的几何结构。 众所周知，它们是密切相关的，也与曲面曲线的复合体的组合结构有关。 研究人员尼古拉·伊万诺夫过去曾成功地利用了这些相互关系，并打算继续这一研究方向。 首先，他计划探索映射类群是拟等距刚性的可能性，这是由映射类群和算术群之间的类比以及R关于算术群的最新结果所指导的。施瓦茨，B。Farb，A.埃斯金，B。Kleiner和B。Leeb. 该项目的另一部分是关于通常由伪Anosov元素生成的映射类组的子组的结构。 希望是Teichmueller空间是“充分双曲的”（如最近的H. Masur和Y. Minsky关于双曲复曲线），以便这样的子群存在的丰度。 最后，研究人员正计划探索两个最近的几何结构Teichmueller空间，即瑟斯顿半度量和Kerckhoff几何。 特别是，他计划找到这些结构的所有自同构。 全球对曲面及其上各种几何结构的研究可以追溯到上个世纪，其基础是高斯、莫比乌斯和黎曼。 映射类组编码了所有可能的曲面对称性，而Teichmueller空间将曲面上特定类型的所有可能几何结构收集在一起。 事实证明，曲面理论的这两个方面是紧密相关的，该项目侧重于这种关系的几个方面。 在过去的十年里，表面理论由于一个有希望的物理学“万有理论”，即所谓的弦理论而获得了额外的意义。 在这个理论中，物质的基本成分不是点状粒子，而是在运动过程中扫过表面的环（“弦”）的集合。 事实证明，曲面上的各种几何结构以及它们的对称性在这里都具有根本的重要性。 ***

9704817 Ivanov This project is concerned with the algebraic structure of the mapping class groups of surfaces and the geometric structure of Teichmueller spaces. As is well known, they are deeply related, and also related to the combinatorial structure of complexes of curves of surfaces. The investigator, Nikolai Ivanov, has successfully exploited these interrelations in the past, and intends to continue this line of research. First, he is planning to explore the possibility that the mapping class groups are quasi-isometrically rigid, guided by the analogy between mapping class groups and arithmetic groups and recent results about arithmetic groups due to R. Schwartz, B. Farb, A. Eskin, B. Kleiner and B. Leeb. Another part of the project is concerned with the structure of subgroups of the mapping class groups normally generated by pseudo-Anosov elements. The hope is that the Teichmueller spaces are "sufficiently hyperbolic" (as evidenced by the recent theorem of H. Masur and Y. Minsky about hyperbolicity of complexes of curves) in order for such subgroups to exist in abundance. Finally, the investigator is planning to explore two more recent geometric structures on Teichmueller spaces, namely, the Thurston semi-metric and the Kerckhoff geometry. In particular, he is planning to find all automorphisms of these structures. The global study of surfaces and of various geometric structures on them goes back to the last century, with the foundations laid down by Gauss, Moebius, and Riemann. The mapping class groups encode all possible symmetries of surfaces, while Teichmueller spaces collect together all possible geometric structures of a particular sort on a surface. It turns out that these two aspects of the theory of surfaces are deeply interrelated, and the project focuses on several aspects of this relationship. In the last decade, the theory of surfaces acquired an additional significance as a result of a promising physical "theory of everything ," namely the so-called string theory. In this theory, the elementary constituents of matter are not point-like particles, but rather collections of loops ("strings") sweeping a surface during their motion. It turns out that both various geometric structures on surfaces, as well as their symmetries, are of fundamental importance here. ***