Hyperbolic Geometry, Heegaard Surfaces, Foliation/Lamination Theory, and Smooth Four-Dimensional Topology

双曲几何、Heegaard 曲面、叶状/层状理论和平滑四维拓扑

• 批准号: 1607374
• 负责人: David Gabai
• 金额: $ 66.67万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2022-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607374&HistoricalAwards=false
• 关键词: Hyperbolic; Geometry; Heegaard; Surfaces; Foliation

Low-dimensional topology, which studies manifolds of four or fewer dimensions, is a central, active area of mathematics. Many of the mathematical tools successfully used to study high-dimensional manifolds do not apply in low dimensions, and despite advances in the last forty years, fundamental, important problems remain unresolved. Low-dimensional topology is at the nexus of many branches of mathematics. Methods from geometry, minimal surface theory and analysis, group theory, number theory, dynamical systems, and theoretical computer science have contributed to its development, and conversely, research in low-dimensional topology stimulates advances in these areas. This research project addresses topics in hyperbolic geometry, lamination and foliation theory, Heegaard theory, and smooth 4-dimensional manifold theory. Many of the questions under study are accessible to graduate students and new researchers, several of whom will be involved in the project.The investigator will continue research on the topology of ending lamination space, the classification of both small cusped and low volume hyperbolic 3-manifolds, and the classification of Heegaard splittings. Through a novel approach, the project will investigate the smooth 4-dimensional Schoenflies conjecture: any smoothly embedded 3-sphere in the standard 4-sphere bounds a smoothly standard 4-ball.

低维拓扑学，研究四维或更少维的流形，是数学的一个中心，活跃的领域。许多成功用于研究高维流形的数学工具并不适用于低维，尽管在过去的40年里取得了进展，但基本的重要问题仍然没有解决。低维拓扑是数学许多分支的核心。几何学、极小曲面理论和分析、群论、数论、动力系统和理论计算机科学的方法促进了它的发展，相反，低维拓扑学的研究促进了这些领域的发展。这个研究计画的主题包括双曲几何、层理与叶理理论、Heegaard理论与光滑四维流形理论。许多正在研究的问题是研究生和新的研究人员，其中一些人将参与该项目的访问。调查员将继续研究结束层空间的拓扑结构，小尖和低容量双曲3-流形的分类，以及Heegaard分裂的分类。 通过一种新的方法，该项目将研究光滑的4维Schoenflies猜想：任何光滑嵌入标准4-球的3-球都有一个光滑的标准4-球。