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，拓扑学是一个对物理学有重要应用的数学领域。 从两个方面来看，这是一个相对简单的问题，在世纪早期就已得到解决。 在20世纪80年代和90年代，被称为表征论的数学领域在三维中应用于这一目标。 在20世纪90年代后期，包含数学各个领域的分类程序被引入到四维中来解决这个问题。这个由美国国家科学基金会资助的合作项目旨在进一步加深数学两个基本领域之间的联系。在20世纪80年代量子群被引入后不久，某些纽结不变量，如琼斯多项式，就从这些对象中构造出来。 将量子群的参数专门化为单位根导致了三维流形的不变量。 范畴化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，用范畴代替向量空间。 20世纪90年代发现的一个链接同源性将琼斯多项式归类。 这导致了对量子群的各个方面进行分类的研究激增，以及它们在量子参数的一般值上的表示。 为了对来自量子群的三流形不变量进行分类，人们必须试图在统一的根上理解已分类的量子群。 在2000年代初，hopfological代数的主题被概述为实现这一目标。 单位根上的量子群是定义在分圆多项式环上的结合代数。 这样的环被归为截断多项式代数上的模的稳定范畴。 研究人员将在这些稳定的范畴上寻找模范畴，以试图将量子群的表示论归类为统一的根，包括结和三流形不变量。 为了对相关的3维TQFT进行分类，必须在一个稍大的环上工作，其中某些整数被反转。 这个环最近已被归类，研究人员将试图将这种结构纳入范畴表征理论。这个奖项反映了NSF的法定使命，并已被认为是值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估的支持。

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.