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.

数学和物理学的主题始终紧密地交织在一起，每个人都在激励并告知另一个进步。该项目着重于当前相互作用的一个领域，在数学侧的拓扑（形状研究）和物理方面的弦理论之间。该项目的起点是一个有趣且出乎意料的联系，这是由双方的首席研究员和合作者最近发现的，在两个与太空中的结中相关的两个单独的代数结构之间：主要研究者开发的一种，而弦乐理论中的一种是过去几年中许多研究的重点。在本项目的过程中，主要研究人员将建立这种联系，目前仅在环境上得到支持；希望这项工作能够创造并加强数学和物理学之间的新沟通线，从而将每个学科的技术引入彼此之间。首席研究者还将使用该项目来培训各个级别的未来数学家，从为年度美国数学奥林匹亚贡献高中生的年度数学奥林匹克运动会，到监督本科生，研究生和博士后研究员的研究，到组织会议和研讨会的组织和研讨会，以供成熟的研究人员进行培养，以培训的人类介绍，并介绍了介绍的人，并研究了一个专家，并研究了一个专家，并研究了一个专家，并研究了一个专家，并研究了一个针对的人，并进行了研究，并进行了研究，并进行了研究，该专业人士曾是一个专业的研究员通过使用Gromov和Floer触发的符号几何形状中的一种方法来计算某些符号歧管中的全态曲线，以及最近在Eliashberg，givental和Hofer的符合性场理论中提出的。先前的工作表明，打结触点同源性是一种强大的不变性，可有效区分结并包含有关结的经典拓扑信息。 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.首席研究人员将使用拉格朗日填充物和格罗莫夫（Gromov-Witten）的潜力来解决这一猜想。这可能会在不同方向上产生重大影响：在结理论中，它将建立AJ猜想的变体；在《镜像对称性》中，它将通过同时田地理论产生一种新方法，以构建镜子calabi-yau 3倍。在拓扑弦理论中，它将为最近的结果提供数学基础。在相关工作中，主要研究者将在符号野外理论的某些方面发展并加强代数框架，包括结的接触同源性和符号同源性。在这种情况下，新的代数工具，例如差分级代数的表示理论，将使人们能够在符号几何形状中更有效地攻击问题，尤其是通过分析符号流形和legendrian的Weinstein结构和接触歧管中的横向结。