Topics at the Intersection of Geometry, Topology and Group Theory

NSF提案0604633摘要本项目有三个主要组成部分。 首先，法布将继续调查映射类组和模空间的黎曼曲面。 这个主题是在许多数学领域的交叉点，从代数几何到低维拓扑学到弦论到几何群论。 法布将继续适用的方法，从离散子群的李群，以了解这些对象，特别是“Torelli组”，这是一个部分的最古老的，但最不了解的一部分理论。 对称性是数学的核心思想。 法布将继续他的工作与S。Weinberger对所有空间（即黎曼流形）进行对称性分类的广泛计划。 到目前为止，在这项工作中使用的思想包括调和映射，大规模几何和变换群的理论。 在第三个项目中，Farb将继续他与C的工作。Hruska汇集的技术和思想，从几何群论与那些从离散子群的李群，以建立理论的格自同构群的2-复合物。 这是巴斯-卢博茨基的树格理论的二维扩展，在树格中可以出现狂野的新现象。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.