Low Dimensional Topology and Hyperbolic Geometry

低维拓扑和双曲几何

• 批准号: 0504110
• 负责人: David Gabai
• 金额: --
• 依托单位: Princeton University
• 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: 2005-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0504110&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; Hyperbolic; Geometry

The principal investigator plans to investigate, in collaboration with others, several central issues in hyperbolic geometry and low dimensional topology. He will study the structure of low volume complete orientable hyperbolic 3-manifolds. He also plans to develop topological obstructions to geodesizing, via isotopy, simple curves in complete hyperbolic 3-manifolds and to investigate whether or not geometrically simply connected Schoenflies 4-balls are diffeomorphic to the standard 4-ball.Three-manifold topology is the study of objects which appear locally like the standard three dimensional space in which we live and two-manifold topology is the study of surfaces. The topology of two-dimensional manifolds, like the sphere and torus, was completely understood over 100 years ago. The geometry of two-dimensional manifolds is also very well understood. Despite the tremendous progress of the last 30 years some of the most basic issues of three-dimensional geometry and topology still must be understood. While we have experimental evidence predicting the low volume hyperbolic three-manifolds, we still do not even know the lowest volume closed ones. Given a hyperbolic three-manifold, say as a three-ball with face identifications, there is strikingly little understanding of how to pick out the shortest length paths. The last project is about understanding one of the most basic types of four-dimensional spaces.

首席研究人员计划与其他人合作调查双曲线几何学和低维拓扑的几个中心问题。 他将研究低体积完全取向双曲线3个体的结构。 He also plans to develop topological obstructions to geodesizing, via isotopy, simple curves in complete hyperbolic 3-manifolds and to investigate whether or not geometrically simply connected Schoenflies 4-balls are diffeomorphic to the standard 4-ball.Three-manifold topology is the study of objects which appear locally like the standard three dimensional space in which we live and two-manifold topology is the study of surfaces. 100年前，完全理解了二维流形的拓扑，例如球体和圆环。 二维歧管的几何形状也非常了解。 尽管过去30年取得了巨大进展，但仍必须了解三维几何学和拓扑的一些最基本问题。 虽然我们有实验证据预测低体积的双曲线三元素，但我们甚至仍然不知道最低的封闭式封闭情况。 鉴于双曲线三个manifold（例如三球带有面部识别），对如何挑选最短的长度路径的理解很少。 最后一个项目是了解四维空间的最基本类型之一。