Knots, Sheaves, and Mirrors

该项目旨在连接与现代数学物理相关的几个数学领域。 结的理论--三维空间中普通的、缠结的弦环--可以追溯到开尔文勋爵，他提出元素周期表是由以太的不同打结形式组成的。 从那时起，纽结理论成为数学中一个成熟的学科，即使开尔文的提议在物理学上不受欢迎。 纽结理论在数学中的主要挑战是识别两个纽结何时相同或不同。 最近，纽结又重新出现在与超对称规范理论和弦论有关的数学物理中。通过它们在数学物理中的再现，结以复杂的形式重新出现在数学中。这个项目将阐明数学和物理学中这些领域之间的相互作用，从而更全面地理解这个主题。此外，首席研究员将培训研究生，支持研究研讨会，运行一个数学圈，并执行旨在培养年轻数学家的其他活动。“结、层和镜”首先是指勒让德结和可构造层之间的联系，其次是对标识所涉及的模空间的研究。勒让德纽结可以作为拉格朗日子流形的无穷远处边界出现，因此，考虑到与层和福谷范畴相关的微局部化定理（由主要研究者和纳德勒证明），勒让德纽结有一个纯粹的层理论描述也许并不奇怪。 事实上，确切的关系是有些微妙的，但最终可以等同于Chekanov-Eliashberg微分分次代数的增广与包含前图的基平面上的可构造层。 有了这种等价性在手，它是自然的，遵循镜像对称的思想，研究“秩一”对象的模空间-这些被证明是有趣的空间，与Bott-Samelson决议有关，从中可以恢复分类HOMFLY不变量的结。 该项目还将研究拓扑结（沿着Aganagic-Ekholm-Ng-Vafa的线）的Legendrian conormalals的层理论描述，对应于以结为链接的代数奇点的野生希钦系统。

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.