Collaborative Research: Taut foliations and contact topology

接触和辛拓扑是数学的一个分支，是由物理学，特别是经典力学和热力学所推动的。三维流形是以我们所处的空间为模型的，接触结构在三维流体流动的研究中自然产生。正是物理流体流动的数学结构导致了接触拓扑。这项国家科学基金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协维一结构。这些包括表面、叶理和接触结构。当它们分别是不可压缩的、紧的和紧的时，这些结构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。加拜和瑟斯顿在20世纪80年代的S中取得了与曲面和叶层相关的重大进展，吉鲁在90年代的S中发现了凸曲面和接触拓扑之间的相互作用，这一发现非常有用。1998年，埃利亚什伯格和瑟斯顿发现了叶理和接触拓扑学之间令人惊讶的联系，这一联系非常有影响力，也是拟议研究的起点。PI建议更好地理解叶状结构和接触拓扑之间的关系。这包括扩展逼近定理的适用性，以及加强这类定理的结论。这也包括对存在和唯一性问题的调查，对于紧凑的叶状结构和紧密的接触结构。

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.