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，它已经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维度为自然数的向量空间构建的高阶结构，而向量空间本身则提升为类别。整数总是被提升到向量空间的复合体。连杆同调理论构成了这样一个量子连杆不变量的提升。后者与量子场论、统计力学和许多数学领域有着深刻的关系，而分类链同调扩展了这些关系，并在更高的结构水平上产生了新的关系。该提案将在统一的根源上研究分类，这将导致chen - simons理论和量子3流形不变量的结构提升，最终，部分共形场论。该项目还旨在加强和统一最近发现的上同操作的链接同源。这些操作使链路同调群刚性化，并展示了代数拓扑的连接；在后者中，一些最深刻的结果不仅需要研究同调群和它们的特殊对应物，还需要研究它们上同调运算的整个系统。很可能，理解链同调上同调运算的全代数将极大地推进这一主题以及与之相关的许多数学和数学物理领域。该项目将研究范畴和链接同调，找到刚性链接同调理论的上同调运算，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范畴。已知的上同操作可能只是作用于各种连杆同调理论，特别是Khovanov-Rozansky和Webster同调群的上同操作的更大代数的一小部分。最近关于单位根上量子群的分类的研究指出了一个深刻但尚未发展的理论的存在，包括一个类似的同调代数，其中复合体的作用由p复合体扮演。该项目的目标之一是通过在p复合体和hopological代数的框架内进一步发展量子群及其表示的分类，对琼斯多项式和其他量子环不变量在单位素根上进行分类。

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.