Hyperbolic Geometry and 3-Dimensional Topology

双曲几何和三维拓扑

• 批准号: 0504975
• 负责人: Peter Shalen
• 金额: --
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504975&HistoricalAwards=false
• 关键词: Hyperbolic; Geometry; Dimensional; Topology

The work supported by this grant is bringing to bear on the study ofhyperbolic 3-manifolds methods from the analytic theory of Kleiniangroups, geometric and algebraic topology, algebra and combinatorics.One group of projects aims at understanding, in a quantitative way,how the volume of a hyperbolic manifold reflects its underlyingtopology. Another project studies the relationship between thealgebraic rank of a hyperbolic 3- manifold and its Heegaardgenus. A third group of projects addresses the construction ofhyperbolic manifolds by the Dehn filling construction.A hyperbolic manifold is a space which is locally modelled on thenon-euclidean geometry of Lobachevsky, Bolyai and Gauss, in which thesum of the angles of a triangle is less than pi. Besides being offundamental importance for classical geometry and number theory,hyperbolic manifolds have long been known to play a central role inthree-dimensional topology. This has been newly confirmed byPerelman's announcement of a proof of Thurston's geometrizationconjecture.

该基金支持的工作是从Kleiniang群、几何和代数拓扑、代数和组合学的分析理论研究双曲3-流形方法。一组项目旨在以定量的方式理解双曲流形的体积如何反映其基础拓扑。 另一个项目研究了双曲三维流形的代数秩与其Heegaard亏格之间的关系。第三组项目解决了由Dehn填充构造双曲流形的构造问题。双曲流形是一个空间，它局部地以Lobachevsky、Bolyai和Gauss的非欧几何为模型，其中三角形的角之和小于π。 除了对经典几何学和数论的重要性之外，双曲流形在三维拓扑学中扮演着重要的角色。这一点最近被佩雷尔曼宣布的瑟斯顿几何化猜想的证明所证实。