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，已经成功地将先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这些数字的向量空间构建的高阶结构，而向量空间本身则提升为范畴。整数被一致地提升为向量空间的复形。链同调理论构成了量子链不变量的这样一种提升。后者与量子场论、统计力学和许多数学领域有着深刻的联系，分类链接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。该提案将从统一性的根源进行分类，这应该会导致对Chern-Simons理论和量子3-流形不变量的结构性提升，并最终导致部分共形场论。该项目还旨在加强和统一关于链接同调的上同调运算的最新发现。这些运算使链同调群刚性，并显示出与代数拓扑学的联系；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，而且需要研究它们上的上同调运算的整个系统。理解环同调上同调运算的全代数将极大地推进这一学科，以及数学和数学物理的许多领域。该项目将研究分类和链接同调，找到使链接同调理论僵化的上同调运算，并推进量子群、它们的表示以及量子链接和3-流形不变量的单位根分类。已知的上同调运算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Webster同调群的上同调运算。最近关于单位根处量子群分类的工作表明，存在一个深刻但尚未发展的理论，包括一种类似于同调代数的理论，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-复形和Hopfology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位于单位素根的量子链接不变量进行分类。

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.