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

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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.