Topics at the Intersection of Geometry, Topology and Group Theory

NSF提案0604633本项目主要由三个部分组成。首先，Farb将继续研究映射类群和黎曼曲面的模空间。这个主题是许多数学领域的交叉点，从代数几何到低维拓扑，从弦理论到几何群论。Farb将继续运用李群的离散子群的方法来理解这些对象，特别是“Torelli群”，这是理论中最古老但最不为人所知的部分。对称是数学中的一个核心思想。Farb将继续与S. Weinberger一起研究对所有空间（即黎曼流形）进行对称分类的广义程序。到目前为止，在这项工作中使用的思想包括谐波映射理论、大尺度几何和变换群。在第三个项目中，Farb将继续与C. Hruska合作，将几何群论的技术和思想与李群的离散子群的技术和思想结合起来，以建立2-复合体自同构群的格理论。这是Bass-Lubotzky的树格理论的二维扩展，在树格理论中可以出现各种各样的新现象。在刚刚描述的每个项目中，Farb将继续与许多年轻学生和研究人员合作并指导他们。

ABSTRACT FOR NSF PROPOSAL 0604633There are three main components to this project. First, Farb will continue to investigate mapping class groups and the moduli space of Riemann surfaces. This topic lies at the intersection of many areas of mathematics, from algebraic geometry to low-dimensional topology to string theory to geometric group theory. Farb will continue to apply methods from discrete subgroups of Lie groups in order to understand these objects, especially the "Torelli group", which is a part of the oldest but least understood part of the theory. Symmetry is a core idea in mathematics. Farb will continue his work with S. Weinberger on the broad program of classifying all spaces (that is Riemannian manifolds) with symmetry. The ideas used so far in this work have included the theories of harmonic maps, large-scale geometry, and transformation groups. In a third project, Farb will continue his work with C. Hruska on bringing together techniques and ideas from geometric group theory with those from discrete subgroups of Lie groups in order to build the theory of lattices in automorphism groups of 2-complexes. This is a 2-dimensional extension of Bass-Lubotzky's theory of tree lattices, where wild new phenomena can occur. Throughout each of the projects just described, Farb will continue to work with and mentor many young students and researchers.