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.

首席研究员计划与其他人合作，研究双曲几何和低维拓扑中的几个核心问题。他将研究小体积完全可定向双曲3-流形的结构。他还计划通过同位素，在完全双曲3流形中的简单曲线中开发测地线化的拓扑障碍，并研究几何上单连通的schoen蝇4球是否与标准4球微分同构。三流形拓扑研究的是局部出现的物体，就像我们生活的标准三维空间一样，二流形拓扑研究的是表面。二维流形的拓扑结构，如球面和环面，在100多年前就被完全理解了。二维流形的几何也很容易理解。尽管在过去的30年里取得了巨大的进步，三维几何和拓扑的一些最基本的问题仍然必须理解。虽然我们有实验证据预测低体积双曲三流形，但我们仍然不知道最小体积的闭合流形。给定一个双曲三流形，比如一个带有人脸识别的三球，人们对如何挑选出最短路径的理解少得惊人。最后一个项目是关于理解四维空间的最基本类型之一。