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.

接触和辛拓扑是由物理学，特别是经典力学和热力学推动的数学分支。三维流形是对我们居住的空间进行建模的，接触结构在三维流体流动的研究中自然出现。物理流体流动的数学结构产生了接触拓扑。这个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三维拓扑的研究。三维流形研究中最重要的工具之一是对其支持的余维结构的分析。 这些包括表面、叶状结构和接触结构。 当这些结构分别不可压缩、拉紧和紧密时，它们最能揭示周围的结构。 当人们深入了解这些结构如何相互作用时，该领域取得了一些重大进展。 Gabai 和 Thurston 在 20 世纪 80 年代在表面和叶状结构方面取得了重大进展。 Giroux 在 20 世纪 90 年代发现的凸面和接触拓扑之间的相互作用非常有用。 1998 年，Eliashberg 和 Thurston 发现了叶状结构和接触拓扑之间令人惊讶的联系，这种联系非常有影响力，也是拟议研究的起点。 PI 建议更好地理解叶理和接触拓扑之间的关系。 这包括扩展近似定理的适用性，并锐化此类定理的结论。这还包括对拉紧的叶状结构和紧密接触结构的存在性和唯一性问题的调查。