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.

低维拓扑学研究了四个或更少的维度，是数学的中心，活跃的领域。许多成功用于研究高维歧管的数学工具并不适用于低维度，尽管过去四十年的进步，基本的，重要的问题仍未解决。低维拓扑是在许多数学分支的联系。来自几何，最小表面理论和分析，群体理论，数量理论，动力学系统和理论计算机科学的方法有助于其发展，相反，低维拓扑的研究刺激了这些领域的进步。该研究项目介绍了双曲线几何学，层压和叶面理论，Heegaard理论和光滑的4维流形理论的主题。研究生和新研究人员都可以解决许多研究的问题，其中一些人将参与该项目。研究人员将继续研究结束层压空间的拓扑，小凹口和低体积的三曲3个manifolds的分类以及Heegaard划分的分类。通过一种新的方法，该项目将研究平滑的4维型schoenflies猜想：在标准的4个球体边界中，任何平滑的3速率都可以平滑标准的4球。