CAREER: Link homology -- in type A and beyond

该奖项全部或部分由《2021年美国救援计划法案》（公法117-2）资助。链接是一个数学对象，它形式化了一个或多个打结的字符串的概念。连杆的数学理论（结理论）是研究这些物体直到连续变形的，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连杆变形成另一个连杆，我们就认为两个连杆是相同的。尽管结理论似乎是专门的，但它已被证明可以应用于三维和四维空间理论，理论物理，并已应用于DNA重组和蛋白质折叠的研究。区分两个链接的问题是具有挑战性的，原因有两个：首先，很难表现出识别两个看起来不同但实际上相同的链接的变形；第二，鉴于我们怀疑有两种不同的联系，很难直接和严格地证明，事实上，其中一种不能变形为另一种。解决后一个问题的一种技术是通过链路不变量：将一个更简单的数学对象赋值给变形时不变的链路。如果两个连杆有不同的不变量，我们就知道它们确实是不同的。该项目的主要研究目标是进一步加深我们对强大的现代连杆不变量的理解，即连杆同调理论。伴随的教育活动有一个共同的主题，即在各个层次上增加对数学科学的参与，包括课程开发、学生研究和K-12层次的推广活动。该项目的更广泛影响旨在通过研究指导努力支持数学领域代表性不足的群体的坚持，并通过北卡罗来纳大学科学博览会促进与公众的联系。链接同调是链接的现代不变量，推广（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度拓扑信息外，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，它们是表征理论、拓扑学和理论物理相关研究的重要纽带。到目前为止，连杆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将在此（非超级）A型病例之外发展链接同源性。这些发展对于在量子和经典拓扑结构之间提供长期寻求的桥梁以及研究与简单复李代数相关的链路同调至关重要，而这些代数与sl(n)截然不同，基本上对sl(n)一无所知。在一项工作中，PI将构建gl（m|n）链接同调（超A型），该链接同调是根据理论物理学的考虑预测存在的，并且在推测上提供了Khovanov-Rozansky理论和结花同调之间的联系。在第二项工作中，PI将构建与a型之外的简单复李代数相关的显式和可计算的链路同调。在此过程中，他还将解决长期存在的问题，即寻找非a型简单复李代数量子群表示类别的生成器和关系描述。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.