Topics at the Intersection of Geometry, Topology and Group Theory

几何、拓扑和群论交叉的主题

基本信息

批准号：
0604633
负责人：
Benson Farb
金额：
$ 33.36万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2006
资助国家：
美国
起止时间：
2006-07-01 至 2012-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604633&HistoricalAwards=false
关键词：
Topics Intersection Geometry Topology Group

项目摘要

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.

NSF提案的摘要0604633，这是该项目的三个主要组成部分。首先，Farb将继续研究映射课程组和Riemann表面的模量空间。该主题在于数学领域的许多领域，从代数几何到低维拓扑再到字符串理论再到几何群体理论。 Farb将继续采用谎言组离散亚组的方法，以了解这些对象，尤其是“ Torelli群体”，这是该理论中最古老但最不了解的部分的一部分。对称是数学中的核心思想。 Farb将继续与S. Weinberger合作，以用对称性对所有空间（即Riemannian流形）进行分类的广泛计划。到目前为止，这项工作中使用的想法包括谐波图，大规模几何形状和转型组的理论。在第三个项目中，Farb将继续与C. Hruska合作，从几何组理论中将技术和思想与谎言群体离散子组的技术汇总在一起，以建立2个复合体的自动形态群体中的晶格理论。这是低音 - 卢博兹基树格理论的二维扩展，可以在其中发生狂野的新现象。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。