Link homology, cohomological operations, and categorification at roots of unity

这个研究项目旨在研究分类，这是一个相对较新的数学分支，它成功地将以前已知的对象提升到新的结构水平。这包括将涉及自然数的组合系统转换为由向量空间构建的高阶结构，向量空间的维度是这些数字，而向量空间本身提升到类别。整数一直被提升到向量空间的复体。链同调理论构成了量子链不变量的提升。后者与量子场论、统计力学和数学的许多领域有着深刻的关系，而将链接同调分类扩展了这些关系，并在更高的结构层次上产生了新的关系。该提案将研究统一根的分类，这将导致陈-西蒙斯理论和量子3-流形不变量的结构提升，并最终导致共形场论的一部分。该项目还旨在增强和统一最近发现的链接同源上的同调操作。这些操作使同调群刚性化，并表现出与代数拓扑的联系;在代数拓扑中，一些最深刻的结果不仅需要研究同调群及其非凡的对应物，而且需要研究它们上的上同调操作的整个系统。 很可能，了解完整的代数上同调操作链接同源将显着推进这一主题，以及许多领域的数学和数学物理在其附近。该项目将研究分类和链接同源性，找到使链接同源性理论僵化的上同调运算，并在量子群，它们的表示，量子链接和3-流形不变量的统一根上推进分类。 已知的上同调运算很可能只是作用于各种链同调理论的更大的上同调运算代数的一小部分，特别是作用于Khovanov-Rozansky和韦伯斯特同调群的上同调运算代数。最近关于量子群在单位根上的分类的工作指出了一个深刻但尚未发展的理论的存在，包括同调代数的类似物，其中复合物的作用由p-复合物发挥。该项目的目标之一是通过进一步发展量子群的分类及其在p-复体和hopfological代数框架内的表示，将琼斯多项式和其他量子链不变量分类在单位的素根上。

This research project aims to study categorification, a relatively new branch of mathematics that has been successful at lifting previously known objects to new structural levels. This includes transforming combinatorial systems that involve natural numbers into higher-order structures built out of vector spaces whose dimensions are those numbers, while vector spaces themselves lift to categories. Integers are consistently lifted to complexes of vector spaces. Link homology theories constitute such a lift of quantum link invariants. The latter boast deep relations to quantum field theory, statistical mechanics, and many areas of mathematics, and categorifying link homology extends these relations and produces new ones at a higher structural level. The proposal will investigate categorification at roots of unity, which should lead to such structural lifting of the Chern-Simons theory and quantum 3-manifold invariants and, eventually, parts of conformal field theory. The project also aims to enhance and unite recent discoveries of cohomological operations on link homologies. These operations rigidify link homology groups and exhibit connections to algebraic topology; in the latter some of most profound result required studying not just homology groups and their extraordinary counterparts, but entire systems of cohomological operations on them. It is likely that understanding the full algebra of cohomological operations on link homology will significanly advance this subject as well as many areas of mathematics and mathematical physics in its proximity. The project will investigate categorification and link homology, find cohomological operations that rigidify link homology theories, and advance categorification at roots of unity for quantum groups, their representations, and quantum link and 3-manifold invariants. Known cohomological operations are likely just a small part of much bigger algebras of cohomological operations acting on various link homology theories, in particular, on Khovanov-Rozansky and Webster homology groups. Recent work on categorification of quantum groups at roots of unity points to existence of a deep but undeveloped theory, including an analogue of homological algebra where the role of complexes is played by p-complexes. One of the project's goals is to categorify the Jones polynomial and other quantum link invariants at prime roots of unity by further developing categorification of quantum groups and their representations within the framework of p-complexes and hopfological algebra.