Topics at the Intersection of Geometry, Topology and Group Theory

几何、拓扑和群论交叉的主题

• 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

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提案摘要这个项目由三个主要部分组成。首先，Farb将继续研究映射类群和黎曼曲面的模空间。这个主题涉及数学的许多领域，从代数几何到低维拓扑学，再到弦理论和几何群论。法布将继续应用李群的离散子群的方法来理解这些物体，特别是“Torelli群”，它是理论中最古老但最不被理解的部分的一部分。对称性是数学中的核心概念。Farb将继续他与S.Weinberger在对所有空间(即黎曼流形)进行对称分类的广泛计划上的工作。到目前为止，这项工作中使用的思想包括调和映射理论、大规模几何理论和变换群理论。在第三个项目中，Farb将继续他与C.Hruska的工作，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-复形的自同构群中的格理论。这是Bass-Lubotzky的树格理论的二维扩展，其中可能会出现疯狂的新现象。在刚刚描述的每个项目中，Farb将继续与许多年轻的学生和研究人员合作并提供指导。