Collaborative Research: Taut foliations and contact topology

接触拓扑学和辛拓扑学是受物理学，特别是经典力学和热力学启发的数学分支。三维流形以我们生活的空间为模型，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学。这个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三维拓扑的研究。研究三维流形最重要的工具之一是分析它们所支持的余维结构。这些包括表面、叶理和接触结构。当这些结构分别处于不可压缩状态、紧绷状态和紧绷状态时，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年代在表面和叶理方面取得了重大进展。吉鲁在20世纪90年代发现的凸面和接触拓扑之间的相互作用是非常有用的。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，这一联系非常有影响力，并且是拟议研究的起点。pi的提出是为了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，并强化这些定理的结论。这也包括对紧叶理和紧接触结构的存在性和唯一性问题的调查。

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.