Contact geometry in dimensions high and low

本提案的重点是解决几个基本问题，在低维和高维接触几何。在低维中，最基本的问题，即哪三个流形承认紧密接触结构，以及与各种接触流形手术的结果有关的问题仍然是开放的。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基础上，阐明双曲流形上紧密接触结构的本质，以及紧密接触流形上的Legendrian手术。此外，他将扩展Legendrian结理论的最新进展，不仅更好地理解这种结的结构，而且还将对一些三流形族上的接触结构进行分类，包括一些研究得很多且非常困难的小Seifert纤维空间。在更高的维度中，连接触结构的存在也不能完全理解。高维接触几何的最新进展使得对这些结构进行深入研究的时机成熟。也就是说，几年前，Niederkrueger引入了塑料材料的概念，希望在所有维度上找到著名的三维紧与超扭曲二分法的类比（其他提出的概念，如bLobs，最近浮出水面），最近首席研究员已经完全回答了五个流形上接触结构的存在问题（另一个研究小组也是如此）。项目的一部分将涉及解决所有奇维流形上接触结构的存在，以及研究高维中超扭曲的概念。项目负责人还将进一步发展高维的接触同调计算，并研究优雅的法向结构，以便将接触几何技术更广泛地应用于三维结理论和嵌入理论的研究。接触几何是一门古老的学科，它作为几何光学、热力学和经典力学的自然语言而兴起。人们每天在平行停车、滑冰或观看一杯水中的光线时都会遇到接触结构。数学家和物理学家长期以来一直在研究接触几何，但在过去十年左右的时间里，它已经发展成为一个非常丰富而美丽的理论，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和基本问题。此外，首席研究员将继续与一大群研究生合作，组织会议和研讨会，以帮助教育下一代研究人员，并创造有利于新思想和合作发展的肥沃环境。

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.