Knots and contact topology through holomorphic curves

数学和物理的主题一直紧密交织在一起，每一个都激励和通知对方的进步。这个项目的重点是当前相互作用的一个领域，在数学方面的拓扑学（形状的研究）和物理方面的弦理论之间。该项目的出发点是一个有趣的和意想不到的连接，最近由首席研究员和双方的合作者发现，在两个独立的代数结构之间与空间中的节点相关：一个是由首席研究员开发的拓扑结构，另一个是弦理论，在过去几年中一直是许多研究的焦点。在这个项目的过程中，首席研究员将建立这种联系，这是目前只支持情况下，希望这项工作将创建和加强数学和物理学之间的新的通信线路，从每一个学科的技术引入到其他。首席研究员还将利用该项目培养各级未来的数学家，从为高中生的年度美国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，再到为既定研究人员组织会议和研讨会。主要研究者介绍并研究了一组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，使用由格罗莫夫和弗洛尔开创的辛几何中的方法，最近在埃利阿什伯格、吉文塔尔和霍费尔的辛场论中达到顶峰。以前的工作表明，结接触的同源性是一个强大的不变量，是有效的区分结，并包含经典的拓扑信息的结。2012年，Aganagic、Ekholm、Vafa和主要研究者发现结接触同调与弦理论和镜像对称有着意想不到的潜在强大关系：增广多项式，一个从结接触同调导出的结不变量，被证明等于Aganagic和Vafa的Q-变形A-多项式，它出现在拓扑弦的上下文中。主要研究者将使用拉格朗日填充和Gromov-Witten势来处理这个猜想。这可能在不同的方向上产生重要的分支：在纽结理论中，它将建立AJ猜想的变体;在镜像对称中，它将产生一种通过辛场论来构造镜像卡-丘3-折叠的新方法;在拓扑弦理论中，它将为最近的结果提供数学基础。在相关的工作中，首席研究员将发展和加强辛场论某些方面下的代数框架，包括结接触同调和辛同调。新的代数工具在这方面，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温斯坦结构和接触流形上的勒让德和横结。

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.