Knots, Sheaves, and Mirrors

这个项目旨在将与现代数学物理相关的几个数学领域联系起来。打结理论可以追溯到开尔文勋爵，他提出元素周期表是由不同打结形式的乙醚组成的。从那时起，纽结理论就成了数学中一门久负盛名的学科，尽管开尔文提议的物理学并不受欢迎。数学中纽结理论的主要挑战是识别两个纽结何时相同或不同。最近，纽结又出现在与超对称规范理论和弦理论相关的数学物理中。通过它们在数学物理中的再现，纽结以复杂的形式在数学中重新出现。这个项目将阐明数学和物理中这些领域之间的相互作用，从而导致对这一主题的更全面的理解。此外，首席研究员将培训研究生，支持一个研究研讨会，运行一个数学圈，并开展其他旨在培养年轻数学家的活动。“结、垂和镜”首先指的是传奇结和可构造的滑轮之间的联系，其次指的是在标识下涉及的模空间的研究。Legendrian纽结可以作为拉格朗日子流形的无穷边界出现，因此，给出了与Sheets和Fukaya范畴相关的微局部化定理(由首席研究者和Nadler证明)，Legendrian有一个纯粹的层论描述也许就不足为奇了。事实上，确切的关系有些微妙，但最终人们可以等同于Chekanov-Eliashberg微分分次代数在包含前图的基平面上的增广具有可构造的层。有了这个等价性，就很自然地按照镜像对称的思想来研究“排名第一”对象的模空间-这些被证明是有趣的空间，与Bott-Samelson分解有关，从中可以恢复纽结的分类HOMFLY不变量。该项目还将研究拓扑纽结的Legendrian余法(沿着Agan-EkholmNg-Vafa的线)的层论描述，即对应于其链接为纽结的代数奇点的野生Hitchin系统。

This project aims to connect several areas of mathematics that relate to modern mathematical physics. The theory of knots -- ordinary, tangled loops of string in three-dimensional space -- goes back to Lord Kelvin, who proposed that the periodic table was made from different knotted forms of the ether. Since that time, knot theory became a well-established subject in mathematics, even as the physics of Kelvin's proposal came into disfavor. The main challenge of knot theory in mathematics is to recognize when two knots are the same or different. More recently, knots have reappeared in the mathematical physics relating to supersymmetric gauge theory and string theory. Through their reappearance in mathematical physics, knots have re-emerged in mathematics in sophisticated form. This project will elucidate the interplay among these fields within math and physics, leading to a more comprehensive understanding of the subject. In addition, the principal investigator will train graduate students, support a research seminar, run a math circle, and perform other activities aimed at developing young mathematicians."Knots, Sheaves, and Mirrors" refers first to a connection between Legendrian knots and constructible sheaves, and second to a study of the moduli spaces involved under the identification. A Legendrian knot can arise as a boundary-at-infinity of a Lagrangian submanifold, so given the micro localization theorem (proved by the principal investigator and Nadler) relating sheaves and the Fukaya category, it is perhaps not surprising that Legendrians have a purely sheaf-theoretic description. In fact, the exact relationship is somewhat subtle, but in the end one can equate augmentations of the Chekanov-Eliashberg differential graded algebra with constructible sheaves on the base plane containing the front diagram. With this equivalence in hand, it is then natural, following the ideas of mirror symmetry, to study the moduli spaces of "rank-one" objects --- and these prove to be interesting spaces, related to Bott-Samelson resolutions, from which one can recover categorized HOMFLY invariants of the knot. The project will also study the sheaf-theoretic description of Legendrian conormals of topological knots (along the lines of Aganagic-Ekholm-Ng-Vafa), wild Hitchin systems corresponding to algebraic singularities whose link is the knot.