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接触几何形状中的几个基本问​​题。在低维度中，最基本的问题询问哪些三种流形承认紧密的接触结构仍然是开放的，与与接触歧管的各种手术操作有关的问题有关。作为该提案的一部分，首席研究人员将以过去的工作为基础，研究这些问题，以阐明双曲线歧管上紧密接触结构的性质和在紧密接触歧管上的legendrian手术。此外，他将扩展Legendrian结理论的最新进展，不仅可以更好地了解此类结的结构，还可以对三个歧管的一些家庭进行分类，包括一些研究和臭名昭著的小型​​Seifert纤维空间。在较高的维度中，甚至没有完全理解接触结构的存在。较高维度接触几何形状的最新进展使得对这些结构进行严格研究的时间成熟。就是说，几年前，尼德克鲁格（Niederkrueger）介绍了plastikstufe的概念，希望能在各个方面找到一个著名的三维紧密与二分法的二维二分法的类似物（其他提议的概念，例如Blobs，例如Blobs，甚至最近有更多的研究者），以及主要的研究者在五个方面的研究者已经完全回答了五个宣言，该公司已经完全解决了五个宣言，这是五个宣言的五分之际，五个宣言（五个人都已经在五个方面都在建立了五分之际。该项目的一部分将涉及解决所有奇数歧管上的接触结构的存在，并研究更高维度中明确的概念。主要研究者还将在更高维度上进一步开发接触同源性计算，并研究优雅的综合构建，以便将接触几何技术应用于“尺寸”中的结理论的研究，并更普遍地嵌入理论。接触几何学是一个古老的主题，它是几何光学，热力学和经典力学的自然语言。当平行停车，滑冰或观看一杯水中的光线时，每天都会遇到接触结构。长期以来，数学家一直在物理学家研究接触几何学，但在过去的十年左右的时间里，它已经成长为一种非常丰富而美丽的理论，与歧管的拓扑（这是空间和时空的结构），现代物理学中的弦理论，riemannian的几何形状和流体动力学。主要研究者将阐明接触几何与黎曼几何形状与歧管拓扑之间的联系。他还将探讨有关在高维度上的接触结构及其submanifolds的存在和独特性的基本和基本问题。此外，首席研究人员将继续与大量的研究生合作，并组织会议和研讨会，以帮助教育下一代研究人员并创造肥沃的环境，在这些环境中，新的想法和合作可以增长。