CAREER: Link homology -- in type A and beyond

职业：链接同源性——A 型及以上

• 批准号: 2144463
• 负责人: David Rose
• 金额: $ 42.5万
• 依托单位: University of North Carolina at Chapel Hill
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2144463&HistoricalAwards=false
• 关键词: CAREER; Link; homology; type; beyond

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.

该奖项的全部或部分资金根据《2021 年美国救援计划法案》（公法 117-2）提供。链接是一种数学对象，它形式化了一根或多根打结的绳子的概念。链接的数学理论（结理论）是对此类物体连续变形的研究，也就是说，我们想象我们的链接是由柔性材料制成的，如果我们可以将一个链接变形为另一个链接，我们就认为两个链接是相同的。尽管结理论看似专业，但它已被证明可应用于 3 维和 4 维空间理论、理论物理学，并且已应用于 DNA 重组和蛋白质折叠的研究。区分两个链接的问题具有挑战性，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；其次，鉴于我们怀疑有两个不同的链接，很难直接而严格地证明一个链接事实上不能变形为另一个链接。解决后一个问题的一种技术是通过链接不变量：将更简单的数学对象分配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理解强大的现代链接不变量（称为链接同源理论）。伴随的教育活动有一个共同的主题，即增加各个级别对数学科学的参与，包括课程开发、学生研究和 K-12 级别的外展活动。该项目的更广泛影响旨在通过研究指导工作支持数学领域通常代表性不足的群体的持久存在，并通过北卡罗来纳大学科学博览会促进与公众的联系。链接同源性是链接的现代不变量，它概括（并且分类）量子不变量，例如琼斯多项式。除了提供第三维和第四维的深层拓扑信息之外，它们还具有与现代表示理论的联系所产生的丰富代数结构。因此，它们是表示论、拓扑学和理论物理相关考虑研究的重要纽带。到目前为止，链接同调的代数发展集中在与李代数 sl(n) 相关的 Khovanov-Rozansky 理论。 PI 将开发超出此（非超级）A 类案例的链接同源性。这些进展对于在量子和经典拓扑结构之间提供长期以来备受追捧的桥梁以及研究与简单复杂李代数相关的链接同源性至关重要，这些简单复杂李代数与 sl(n) 不同，而 sl(n) 基本上对此一无所知。在一项工作中，PI 将构建 gl(m|n) 连接同调性（超类型 A），该同调性已通过理论物理学的考虑预测存在，并且推测性地提供了 Khovanov-Rozansky 理论和纽特弗洛尔同调性之间的联系。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且可计算的链接同调。在此过程中，他还将解决寻找非 A 型简单复李代数的量子群表示类别的生成器和关系描述这一长期存在的问题。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和知识进行评估，被认为值得支持。 更广泛的影响审查标准。