Collaborative Research: Taut Foliations and Contact Topology

接触和辛拓扑是数学的分支，其动机是物理学，特别是经典力学和热力学。三维流形是在我们生活的空间中建模的，接触结构在三维流体流动的研究中自然出现。正是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学的研究中。三维流形研究中最重要的工具之一是分析它们所支持的余维1结构。 这些包括表面、叶理和接触构造。 当这些结构分别是不可压缩的、绷紧的和紧的时，它们最能揭示环境结构。 当人们深入了解这些结构如何相互作用时，该领域的一些重大进展已经取得。 Gabai和Thurston在20世纪80年代在表面和叶理方面取得了重大进展。 Giroux在20世纪90年代发现凸曲面和接触拓扑之间的相互作用非常有用。 1998年，Eliashberg和Thurston发现了叶理和接触拓扑之间令人惊讶的联系，这是非常有影响力的，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。 这包括扩展逼近定理的适用性，并锐化这些定理的结论。这也包括调查的存在性和唯一性问题，为两个紧张叶理和紧密接触的结构。

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.