Knots, Sheaves, and Mirrors

这个项目旨在把与现代数学物理有关的几个数学领域联系起来。结理论——三维空间中普通的、纠结的弦环——可以追溯到开尔文勋爵，他提出元素周期表是由不同形式的以太结构成的。从那时起，即使开尔文提出的物理理论不受欢迎，结理论在数学中也成为了一个公认的学科。数学中结理论的主要挑战是识别两个结是相同的还是不同的。最近，结在与超对称规范理论和弦理论相关的数学物理中重新出现。通过它们在数学物理中的再现，结以复杂的形式再次出现在数学中。该项目将阐明数学和物理中这些领域之间的相互作用，从而对该主题有更全面的了解。此外，首席研究员将培训研究生，支持研究研讨会，运行数学圈，并开展旨在培养年轻数学家的其他活动。“结、束和镜”首先是指勒让德结和可构造的束之间的联系，其次是对识别下涉及的模空间的研究。一个Legendrian结可以作为拉格朗日子流形的无穷边界出现，所以给定关于束和Fukaya范畴的微局部化定理（由主要研究者和Nadler证明），也许并不奇怪Legendrian结有一个纯粹的束理论描述。事实上，确切的关系有些微妙，但最终人们可以将Chekanov-Eliashberg微分梯度代数的增广等同于包含前图的基面上的可构造轴。有了这个等价，遵循镜像对称的思想，研究“排名第一”对象的模空间是很自然的——这些被证明是有趣的空间，与bot - samelson分辨率相关，从中可以恢复结的分类HOMFLY不变量。该项目还将研究拓扑结（沿着Aganagic-Ekholm-Ng-Vafa的路线）的Legendrian正规的束理论描述，与代数奇点相对应的野生Hitchin系统，其链接是结。

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.