Contact geometry in dimensions high and low

高尺寸和低尺寸的接触几何形状

• 批准号: 1309073
• 负责人: John Etnyre
• 金额: $ 28.8万
• 依托单位: Georgia Tech Research Corporation
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309073&HistoricalAwards=false
• 关键词: Contact; geometry; dimensions; low

The focus of this proposal is to address several fundamental questions in low and high dimensional contact geometry. In low dimensions, the most basic question asking which three manifolds admit tight contact structures is still open as are questions related to the result of various surgery operations on contact manifolds. As part of this proposal the Principal Investigator will build on past work studying these questions to, among other things, illuminate the nature of tight contact structures on hyperbolic manifolds and Legendrian surgery on tight contact manifolds. In addition, he will extend recent advances in Legendrian knot theory to not only better understand the structure of such knots but also to classify contact structures on some families of three manifolds including some of the much studied and notoriously difficult small Seifert fibered spaces. In higher dimensions even the existence of contact structures is not completely understood. Recent progress in higher dimensional contact geometry makes the time ripe for an intense investigation of these structures. Namely, a few years ago Niederkrueger introduced the notion of a plastikstufe in hopes of finding an analog of the famed three dimensional tight vs. overtwisted dichotomy in all dimensions (other proposed notions, such as bLobs, have even more recently surfaced) and most recently the Principal Investigator has completely answered the existence question for contact structures on five manifolds (as has another team of researchers). Part of the project will involve addressing the existence of contact structures on all odd dimensional manifolds as well as investigating notions of overtwistedness in higher dimensions. The Principal Investigator will also further develop contact homology computations in higher dimensions and study the elegant conormal construction in order to apply contact geometric techniques to the study of knot theory in dimension three and embedding theory more generally. Contact geometry is a venerable subject that arose as a natural language for geometric optics, thermodynamics and classical mechanics. One encounters contact structures everyday when parallel parking a car, skating, or watching the play of light in a glass of water. Contact geometry has long been studied by mathematicians in physicists but in the last decade or so it has blossomed into a remarkably rich and beautiful theory with close ties to the topology of manifolds (that is the structure of space and space-time), string theory in modern physics, Riemannian geometry, and fluid dynamics. The Principal Investigator will illuminate the connection between contact geometry and Riemannian geometry and the topology of manifolds. He will also explore basic and fundamental questions concerning the existence and uniqueness of contact structures and their submanifolds in high dimensions. In addition the Principal Investigator will continue working with a large group of graduate students and organize conferences and seminars to help educate the next generation of researchers and create fertile environments in which new ideas and collaborations can grow.

该提案的重点是解决低维和高维接触几何中的几个基本问​​题。在低维中，最基本的问题（即哪三个流形允许紧密接触结构）仍然悬而未决，以及与接触流形上的各种手术操作的结果相关的问题。作为该提案的一部分，首席研究员将在过去研究这些问题的工作的基础上，除其他外，阐明双曲流形上的紧密接触结构和紧密接触流形上的勒让德手术的性质。此外，他还将扩展勒让德结理论的最新进展，不仅可以更好地理解此类结的结构，还可以对某些三流形族上的接触结构进行分类，包括一些经过深入研究且众所周知困难的小 Seifert 纤维空间。在更高的维度中，甚至接触结构的存在也没有被完全理解。高维接触几何学的最新进展使得对这些结构进行深入研究的时机已经成熟。也就是说，几年前，Niederkrueger 引入了塑料的概念，希望找到所有维度中著名的三维紧密与过度扭曲二分法的类似物（其他提出的概念，例如 bLobs，最近才出现），最近，首席研究员完全回答了五个流形上接触结构的存在问题（另一组研究人员也是如此）。该项目的一部分将涉及解决所有奇数维流形上接触结构的存在以及研究更高维度中过度扭曲的概念。首席研究员还将进一步发展更高维度的接触同调计算，并研究优雅的共正规构造，以便将接触几何技术应用到三维纽结理论和更广泛的嵌入理论的研究中。接触几何是一门古老的学科，作为几何光学、热力学和经典力学的自然语言而出现。人们每天在平行停车、滑冰或观看一杯水中的光的变化时都会遇到接触结构。接触几何长期以来一直被数学家和物理学家研究，但在过去十年左右的时间里，它已经发展成为一种非常丰富和美丽的理论，与流形拓扑（即空间和时空的结构）、现代物理学中的弦理论、黎曼几何和流体动力学密切相关。首席研究员将阐明接触几何和黎曼几何以及流形拓扑之间的联系。他还将探讨有关高维接触结构及其子流形的存在性和唯一性的基本问题。此外，首席研究员将继续与一大批研究生合作，组织会议和研讨会，以帮助教育下一代研究人员，并创造新想法和合作得以发展的肥沃环境。