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Knots and contact topology through holomorphic curves

通过全纯曲线的结和接触拓扑

基本信息

批准号：
1406371
负责人：
Lenhard Ng
金额：
$ 43.67万
依托单位：
Duke University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-09-01 至 2018-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406371&HistoricalAwards=false
关键词：
Knots contact topology through holomorphic

项目摘要

The subjects of mathematics and physics have always been closely intertwined, with each one motivating and informing progress in the other. This project focuses on one current area of interplay, between topology (the study of shapes) on the mathematical side and string theory on the physics side. The jumping-off point for the project is an intriguing and unexpected connection, recently discovered by the Principal Investigator and collaborators on both sides, between two separate algebraic structures associated to knots in space: one in topology developed by the Principal Investigator, and one in string theory that has been the focus of much research in the past few years. In the course of this project, the Principal Investigator will establish this connection, which is currently only supported circumstantially; it is hoped that this work will create and strengthen new lines of communication between mathematics and physics, introducing techniques from each discipline into the other. The Principal Investigator will also use this project to train future mathematicians at all levels, from contributing to the annual USA Mathematical Olympiad for high school students, to supervising the research of undergraduates, graduate students, and postdoctoral fellows, to organizing conferences and seminars for established researchers.In the past decade, the Principal Investigator has introduced and studied a package of knot invariants called knot contact homology, which arises by counting holomorphic curves in certain symplectic manifolds, using a method in symplectic geometry pioneered by Gromov and Floer and more recently culminating in the Symplectic Field Theory of Eliashberg, Givental, and Hofer. Previous work has shown that knot contact homology is a robust invariant that is effective at distinguishing knots and contains classical topological information about the knot. In 2012, it was discovered by Aganagic, Ekholm, Vafa, and the Principal Investigator that knot contact homology has an unexpected and potentially powerful relation to string theory and mirror symmetry: the augmentation polynomial, a knot invariant derived from knot contact homology, is conjectured to be equal to Aganagic and Vafa's Q-deformed A-polynomial, which arises in the context of topological strings. The Principal Investigator will approach this conjecture using Lagrangian fillings and Gromov-Witten potentials. This could have significant ramifications in different directions: in knot theory, it would establish a variant of the AJ conjecture; in mirror symmetry, it would produce a new approach via Symplectic Field Theory to constructing mirror Calabi-Yau 3-folds; and in topological string theory, it would provide a mathematical foundation for recent results. In related work, the Principal Investigator will develop and strengthen the algebraic framework underneath certain aspects of Symplectic Field Theory, including knot contact homology and symplectic homology. New algebraic tools in this context, such as representation theory for differential graded algebras, would enable one to more effectively attack problems in symplectic geometry, in particular by analyzing Weinstein structures on symplectic manifolds and Legendrian and transverse knots in contact manifolds.

数学和物理学科一直紧密地交织在一起，每一门学科都激励着另一门学科的进步。这个项目集中在一个当前的相互作用领域，数学方面的拓扑学(形状研究)和物理方面的弦理论之间的相互作用。该项目的起点是两个独立的代数结构之间的一种有趣和意想不到的联系，这是首席研究人员和双方的合作者最近发现的，两个独立的代数结构与空间中的节点有关：一个是由首席研究人员开发的拓扑学，另一个是弦理论，这是过去几年来许多研究的焦点。在这个项目的过程中，首席研究员将建立这种联系，目前这种联系只得到间接的支持；希望这项工作将建立和加强数学和物理之间的新的交流渠道，将每个学科的技术引入到另一个学科。首席研究人员还将利用这个项目来培养所有级别的未来数学家，从为一年一度的美国高中生数学奥林匹克竞赛做出贡献，到监督本科生、研究生和博士后研究员的研究，再到为知名研究人员组织会议和研讨会。在过去的十年里，首席研究人员引入并研究了一套称为纽结接触同调的纽结不变量，它是通过计算某些辛流形中的全纯曲线而产生的，使用了格罗莫夫和弗洛尔开创的辛几何方法，最近发展到了埃拉什伯格、吉夫勒和霍弗的辛场理论。前人的工作表明，纽结接触同调是一种稳健的不变量，它在区分纽结方面是有效的，并且包含了关于纽结的经典拓扑信息。2012年，由Aganial，Ekholm，Vafa和主要研究者发现，纽结接触同调与弦理论和镜像对称性有一种意想不到的潜在的强大联系：由纽结接触同调导出的纽结不变量--增广多项式被猜想等同于Aganial和Vafa的q-变形A-多项式，后者产生于拓扑弦的背景下。首席调查员将使用拉格朗日填充和Gromov-Witten势来处理这个猜想。这可能会在不同的方向产生重大的影响：在纽结理论中，它将建立AJ猜想的一个变体；在镜像对称中，它将产生一种通过辛场理论构造镜像Calabi-Yau三折叠的新方法；在拓扑弦论中，它将为最近的结果提供数学基础。在相关工作中，首席研究员将在辛场理论的某些方面下发展和加强代数框架，包括纽结接触同调和辛同调。在这一背景下，新的代数工具，如微分分次代数的表示理论，将使人们能够更有效地攻击辛几何中的问题，特别是通过分析辛流形上的Weinstein结构和接触流形中的Legendrian和横向纽结。