Collaborative Research: New Structures in Link Homology and Categorification

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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.