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

基本信息

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

项目摘要

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.

低维拓扑研究四维或更少维的流形，是数学的一个中心、活跃领域。许多成功用于研究高维流形的数学工具并不适用于低维，尽管在过去四十年中取得了进步，但基本的、重要的问题仍未得到解决。低维拓扑是许多数学分支的纽带。几何、最小曲面理论和分析、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，低维拓扑的研究刺激了这些领域的进步。该研究项目涉及双曲几何、层状和叶理理论、Heegaard 理论和平滑 4 维流形理论等主题。正在研究的许多问题对于研究生和新研究人员来说都是可以理解的，其中一些人将参与该项目。研究人员将继续研究最终层压空间的拓扑结构、小尖点和低体积双曲 3- 的分类流形，以及 Heegaard 分裂的分类。通过一种新颖的方法，该项目将研究平滑的 4 维 Schoenflies 猜想：任何平滑嵌入标准 4 球体中的 3 球体都会限制一个平滑的标准 4 球体。