Contact Geometry, Contact Homology and Open Book Decompositions

接触几何、接触同调和开卷分解

• 批准号: 0804820
• 负责人: John Etnyre
• 金额: $ 42.33万
• 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: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0804820&HistoricalAwards=false
• 关键词: Contact; Geometry; Homology; Open; Book

AbstractAward: DMS-0804820Principal Investigator: John B. EtnyreThe focus of this proposal is to better understand the intriguingnature of contact structures in all (odd) dimensions, withspecial attention given to dimension three and to apply contactgeometric techniques to questions in topology. The first maintheme of the proposed research is open book decompositions. Theconnection of contact geometry and these specially fibered linkshave provided a fount of information concerning contactstructures and low-dimensional topology. As part of this proposalthis fundamental connection will be exploited to define and studynew invariants of contact structures and special knots in them -specifically, Legendrian and transversal knots. From prior workit is clear these invariants have subtle connections to the famedtight vs. overtwisted dichotomy in dimension three and propertiesof symplectic fillings of contact structures, which in turn arecrucial to applications to topology. Some of the major outcomesexpected from this part of the project are various classificationresults for Legendrian and transversal knots in tight andovertwisted contact structures and a better understanding of thefundamental process of contact surgery. The project will alsostudy the connection between contact structures and open books inhigh dimensions with the goal of illuminating existence questionsin high dimensions and trying to generalize the notion ofovertwistedness here as well. Similar to the connection with openbooks, contact structures are also related to foliations. Thisconnection has been instrumental in applications of contactgeometry to low-dimensional topology. Many of the outstandingquestions concerning this connection will be explored. A secondmain theme of the proposal is the development and analysis ofLegendrian contact homology in higher dimensions. There is agreat deal of beautiful analytic and algebraic structure in thistheory, much of which is yet to be discovered. Moreover, throughthe elegant conormal construction one can then use this to defineinvariants of manifolds and of their embeddings in Euclideanspace. These invariants appear to be extremely powerful and willbe thoroughly investigated.Contact structures on manifolds are very natural objects, bornover two centuries ago, as a natural language for geometricoptics, thermodynamics and classical mechanics. Everyday, oneencounters contact structures when parallel parking a car,skating, or watching the prismatic play of light in a glass ofwater. Contact structures have been studied by manymathematicians and seem to touch on diverse areas of mathematicsand physics, but only recently have they moved into theforeground of mathematics. This is due to the many remarkablebreakthroughs in contact topology, resulting in a rich andbeautiful theory with many far reaching applications to areassuch as three and four dimensional topology (that is thestructure of space and space- time), string theory and large Ndualities in modern physics, and fluid dynamics. The PrincipalInvestigator will study the relation between contact geometry andvarious topological objects (such as open book decompositions andfoliations). In addition he will develop analytic and algebraictools for studying contact structures in high dimensions. As awhole it is expected that at the conclusion of this project wewill have a much better understanding of contact structures inall dimensions and see many more surprising implications for low-dimensional manifolds.

摘要：项目负责人：John B. etnyre本研究的重点是更好地理解所有（奇数）维度接触结构的有趣性质，特别关注三维空间，并将接触几何技术应用于拓扑学问题。本研究的第一个主题是开卷分解。接触几何和这些特殊纤维链路的联系提供了有关接触结构和低维拓扑的大量信息。作为这个提议的一部分，这个基本联系将被用来定义和研究接触结构的新不变量和其中的特殊节-特别是，Legendrian节和横向节。从先前的工作中可以清楚地看出，这些不变量与著名的三维紧与过扭二分法以及接触结构的辛填充的性质有微妙的联系，这反过来又对拓扑的应用至关重要。这部分项目的一些主要成果是对紧密和过度扭曲接触结构中的Legendrian节和横向节的各种分类结果，以及对接触手术的基本过程的更好理解。该项目还将研究接触结构与高维开放书籍之间的联系，目的是阐明高维存在问题，并试图在这里推广过度扭曲的概念。与开放书的连接类似，接触结构也与叶状结构有关。这种联系有助于接触几何在低维拓扑中的应用。关于这一联系的许多悬而未决的问题将被探讨。该建议的第二个主题是高维的legendrian接触同调的发展和分析。在这一理论中有大量美丽的解析和代数结构，其中许多尚未被发现。此外，通过优雅的正规构造，我们可以用它来定义流形及其在欧几里得空间中的嵌入的不变量。这些不变量似乎非常强大，将被彻底研究。流形上的接触结构是非常自然的物体，在两个多世纪以前，作为几何光学、热力学和经典力学的自然语言。每天，当平行停车、滑冰或观看一杯水中光线的棱镜游戏时，人们都会遇到接触结构。接触结构已经被许多数学家研究过，似乎涉及数学和物理的不同领域，但直到最近它们才进入数学的前景。这是由于接触拓扑学的许多显著突破，产生了丰富而美丽的理论，在三维和四维拓扑学（即空间和时空的结构）、现代物理学中的弦理论和大偶性以及流体动力学等领域具有许多深远的应用。首席研究员将研究接触几何和各种拓扑对象之间的关系（如打开的书分解和叶状）。此外，他将开发用于研究高维接触结构的分析和代数工具。总的来说，预计在这个项目结束时，我们将对所有维度的接触结构有更好的理解，并看到低维流形的许多令人惊讶的含义。