CAREER: Link homology -- in type A and beyond

该奖项的全部或部分资金来自《2021年美国救援计划法案》(公法117-2)。链接是一个数学对象，它将一个或多个打结的字符串的概念形式化。连杆的数学理论(纽结理论)是对这些对象直到连续变形的研究，也就是说，我们想象我们的连杆是由柔性材料制成的，如果我们可以将一个变形为另一个，我们认为两个连杆是相同的。尽管纽结理论看似专门化，但它已被证明在三维和四维空间理论、理论物理中有应用，并已被应用于DNA重组和蛋白质折叠的研究。区分两个链接的问题具有挑战性，原因有两个：第一，很难显示出识别两个看起来不同但实际上相同的链接的变形；第二，考虑到我们怀疑是不同的两个链接，很难直接和严格地表明一个实际上不能变形到另一个。解决后一个问题的一种技术是通过链接不变量：将一个较简单的数学对象赋给在变形下不变的链接。如果两个链接具有不同的不变量，我们知道它们确实是不同的。这个项目的主要研究目标是加深我们对强大的现代链接不变量的理解，这些链接不变量称为链接同调理论。随之而来的教育活动有一个共同的主题，那就是在不同级别增加对数学科学的参与，包括课程开发、学生研究和K-12级别的推广活动。这个项目的更广泛的影响旨在通过研究指导努力支持在数学中通常代表不足的群体的持久性，并通过北卡罗来纳大学科学博览会促进与公众的联系。链接同源是链接的现代不变量，它概括(而且分类)量子不变量，如琼斯多项式。除了提供三维和四维的深层拓扑信息外，它们还具有丰富的代数结构，这些结构源于与现代表示理论的联系。因此，它们是理论物理中表示理论、拓扑学和相关考虑的研究的重要纽带。到目前为止，环同调的代数发展主要集中在与李代数sl(N)相关的Khovanov-Rozansky理论上。PI将在这个(非超级)A型情况下发展链接同源性。这些发展对于提供量子和经典拓扑结构之间的长期桥梁以及研究与简单复李代数不同的与sl(N)本质上一无所知的简单复李代数相关联的链接同调是至关重要的。在一项工作中，PI将构建gl(m|n)链同调(超型A)，这些同调已被理论物理中的考虑因素预测存在，并推测在Khovanov-Rozansky理论和纽结Floer同调之间提供了联系。在第二项工作中，PI将构建与A型以外的简单复李代数相关的显式和可计算的链接同调。在此过程中，他还将解决长期存在的问题，即寻找非A型简单复李代数的量子群表示范畴的生成元和关系描述。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This award is funded in whole or in part under the American Rescue Plan Act of 2021 (Public Law 117-2). A link is a mathematical object that formalizes the notion of one or more knotted pieces of string. The mathematical theory of links (knot theory) is the study of such objects up to continuous deformations, that is we imagine that our link is made from a flexible material, and we consider two links to be the same if we can deform one into the other. Despite the seemingly specialized nature of knot theory, it has been shown to have applications to the theory of 3- and 4-dimensional spaces, to theoretical physics, and it has been applied to the study of DNA recombination and protein folding. The problem of distinguishing two links is challenging for two reasons: first, it can be difficult to exhibit the deformations that identify two links that look different but are indeed the same; second, given two links that we suspect to be distinct, it is difficult to directly and rigorously show that in fact one cannot be deformed to the other. One technique for solving the latter problem is via a link invariant: an assignment of a simpler mathematical object to a link that is unchanged under deformation. If two links have different invariants, we know that they are indeed distinct. The main research goal of this project is to further our understanding of powerful modern link invariants called link homology theories. The accompanying educational activities share a common theme of increasing participation in the mathematical sciences at a variety of levels, including course-development, student research, and outreach activities at the K-12 level. The broader impacts of this project aim to support the persistence of groups typically underrepresented in mathematics via research mentoring efforts, and to foster connections with the public through the UNC Science Expo.Link homologies are modern invariants of links that generalize (and moreover, categorify) quantum invariants such as the Jones polynomial. In addition to providing deep topological information in dimensions three and four, they enjoy a rich algebraic structure arising from connections to modern representation theory. Consequently, they are an important nexus for research in representation theory, topology, and related considerations in theoretical physics. Thus far, algebraic developments in link homology have focused on the Khovanov-Rozansky theory, which is associated with the Lie algebra sl(n). The PI will develop link homology beyond this (non-super) type A case. These developments are crucial for providing long sought-after bridges between quantum and classical topological structures and for studying link homology associated to simple complex Lie algebras distinct from sl(n) about which essentially nothing is known. In one line of work, the PI will construct the gl(m|n) link homologies (super type A) that have been predicted to exist by considerations in theoretical physics, and which conjecturally provide a connection between the Khovanov-Rozansky theory and knot Floer homology. In a second line of work, the PI will construct explicit and computable link homologies associated to simple complex Lie algebras outside type A. Along the way, he will also resolve the long-standing problem of finding generators-and-relations descriptions of categories of quantum group representations for non-type A simple complex Lie algebras.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.