Collaborative research: New structures in link homology and categorification

确定两个形状是否相同是拓扑学的一个基本目标，拓扑学是数学的一个领域，在物理学中有着重要的应用。在二维空间中，这是一个相对简单的问题，在20世纪早期就已经被解决了。在20世纪80年代和90年代，被称为表征理论的数学领域在三维空间中应用于这一目标。在20世纪90年代末，包含数学各个领域的分类程序被引入来解决第四维的这个问题。这个由美国国家科学基金会资助的合作项目旨在进一步深化这两个数学基础领域之间已经很深的联系。20世纪80年代量子群被引入后不久，某些结不变量，如琼斯多项式，就从这些对象中构造出来了。将量子群的参数专门化到一个单位根，得到三维流形的不变量。分类是一个跨学科的研究领域，它试图“升级”数学中的某些结构，例如用向量空间代替数字，用类别代替向量空间。在20世纪90年代发现的一个链同调对琼斯多项式进行了分类。这导致了对量子群的各个方面及其在量子参数的通用值上的表示进行分类的研究激增。为了对来自量子群的三流形不变量进行分类，必须尝试从统一的根来理解分类的量子群。为了实现这一目标，在21世纪初概述了hopfological algebra这一主题。单位根处的量子群是在环形多项式环上定义的结合代数。这样的环被截断多项式代数上模的稳定范畴所分类。研究人员将在这些稳定范畴上寻找模范畴，以便尝试对包括结和三流形不变量在内的统一根上的量子群的表示理论进行分类。为了对相关的三维TQFT进行分类，必须在一个稍大的环上工作，其中某些整数是倒置的。这个环最近已经被分类，研究者将尝试将这个结构纳入范畴表征理论。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。

Determining whether or not two shapes are the same is a fundamental goal in Topology, an area of mathematics which has important applications to physics. In two dimensions, this is a relatively simple question which was addressed in the early part of the 20th century. In the 1980s and 1990s, the area of mathematics known as Representation Theory had applications to this goal in dimension three. In the late 1990s the program of categorification, which encompasses various areas of mathematics, was introduced to tackle this question in dimension four. This National Science Foundation funded collaborative project aims to further this already deep connection between the two fundamental fields of mathematics. Soon after quantum groups were introduced in the 1980s, certain knot invariants, such as the Jones polynomial, were constructed from these objects. Specializing the parameter of a quantum group to a root of unity led to invariants of three-dimensional manifolds. Categorification is an interdisciplinary area of research which seeks to "upgrade" certain structures in mathematics such as replacing numbers with vector spaces and vector spaces with categories. A link homology discovered in the 1990s categorified the Jones polynomial. This led a surge in research in categorifying various aspects of quantum groups and their representations at a generic value of the quantum parameter. In order to categorify three-manifold invariants coming from quantum groups, one must try to understand categorified quantum groups at a root of unity. The subject of hopfological algebra was outlined in the early 2000s to accomplish this goal. Quantum groups at roots of unity are associative algebras defined over rings of cyclotomic polynomials. Such rings are categorified by stable categories of modules over truncated polynomial algebras. The investigators will look for module categories over these stable categories in order to try to categorify the representation theory of quantum groups at roots of unity including knot and three-manifold invariants. In order to categorify the associated 3-dimensional TQFT, one must work over a slightly larger ring where certain integers are inverted. This ring has been categorified recently and the investigators will attempt to incorporate this structure into categorical representation theory.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.