Collaborative Research: Taut Foliations and Contact Topology

合作研究：张紧的叶状结构和接触拓扑

• 批准号: 1612475
• 负责人: Rachel Roberts
• 金额: $ 22.12万
• 依托单位: Washington 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: 2016-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1612475&HistoricalAwards=false
• 关键词: Collaborative; Research; Taut; Foliations; Contact

Contact and symplectic topology are branches of mathematics that are motivated by Physics, specifically by classical mechanics and thermodynamics. Three-dimensional manifolds are modeled on the space we live in, and contact structures arise naturally in the study of three-dimensional fluid flows. It is the mathematical structure of physical fluid flows that gives rise to contact topology. This National Science Foundation funded project seeks to extend the application of physical phenomena to the study of three-dimensional topology.One of the most important tools in the study of three-dimensional manifolds is an analysis of the codimension one structures they support. These include surfaces, foliations, and contact structures. These structures are most revealing of the ambient structure when they are, respectively, incompressible, taut, and tight. Some of the major advances in the field have been made when people gained insight into how these structures interact. Gabai and Thurston made major advances relating surfaces and foliations in the 1980's. Giroux's discovery of the interplay between convex surfaces and contact topology in the 1990's has been extremely useful. In 1998, Eliashberg and Thurston discovered a surprising connection between foliations and contact topology that has been very influential and is the starting point for the proposed research. The PIs propose to better understand the relationship between foliations and contact topology. This includes extending the applicability of approximation theorems, and sharpening the conclusions of such theorems. This includes also an investigation of existence and uniqueness questions, for both taut foliations and tight contact structures.

接触和符号拓扑是数学的分支，这些分支是由物理学动机的，特别是由经典的力学和热力学。三维歧管是在我们居住的空间上建模的，并且在研究三维流体流中自然出现了接触结构。物理流体流的数学结构引起了接触拓扑。这项由国家科学基金会资助的项目旨在将物理现象的应用扩展到研究三维拓扑的研究。研究三维流形的最重要工具之一是分析其支持的编成拟态度的结构。 这些包括表面，叶子和接触结构。 这些结构分别是不可压缩，绷紧和紧绷的周围结构最揭示的。 当人们深入了解这些结构如何相互作用时，该领域的一些主要进步已经取得了成就。 Gabai和Thurston在1980年代就表面和叶子方面取得了重大进展。 Giroux在1990年代发现了凸表面与接触拓扑之间的相互作用的发现非常有用。 1998年，埃里亚斯贝格（Eliashberg）和瑟斯顿（Thurston）发现了叶子和接触拓扑之间的令人惊讶的联系，这是非常有影响力的，并且是拟议研究的起点。 PI建议更好地了解叶子和接触拓扑之间的关系。 这包括扩展近似定理的适用性，并提高此类定理的结论。这还包括对绷紧的叶子和紧密的接触结构的存在和独特性问题的研究。