Quantum link invariants, including the Jones, Kauffman, and other link polynomials, can be recovered from quantum groups, which are Hopf algebra deformations of the universal enveloping algebras of simple Lie algebras. These invariants can be extended to functors from the category of tangles to categories of quantum group representations. The invariant of a tangle is a homomorphism of representations. Link homology lifts quantum invariants one dimension up and can be viewed as functors from the category of link cobordisms to the category of multi-graded abelian groups. The invariant of a tangle becomes a functor between triangulated categories assigned to the boundaries of the tangle. Categorification leads to invariants of tangle cobordisms which take values in natural transformations between these functors. The categories that appear in this way categorify tensor products of quantum group representations and appear thoughout representation theory, symplectic topology, and algebraic geometry. The proposal aims to further elucidate the structure of link homology, discover new cohomological operations in them, further tie them up with Hochschild homology, generalize the Rassmussen invariant, and relate link homology with the categorification of quantum groups. Categorification of quantum groups, discovered less that two years ago, realizes them as Grothendieck groups of categories of projective modules over certain diagrammatically defined rings, which also appear throughout representation theory. The PI believes that categorified quantum groups will prove ubiquitous in several areas of mathematics and will continue studying them for the next few years.
Mathematicians discovered in the past 30 years deep relations between the theory of knots and 3-manifolds (objects that locally look like our space but have a different global behaviour) and a plethora of sophisticated structures in algebra and geometry. Many of these structures have to do with the elucidation of the notion of symmetry. All symmetries of a given mathematical or physical object constitute what is known as a group - a collection of symmetries that can be composed and reversed. Various developments in the past decades led to a far-fetched generalizations of the notion of a group, including the discovery of quantum groups by Drinfeld and Jimbo. More recent further progress in the direction, in which the PI was involved, resulted in the discovery of so-called categorified quantum groups, which are even more sophisticated objects, from which quantum groups can be recovered by forgetting most of the information. These categorified quantum groups are expected to be intimately related to the topology in four dimensions. Four-dimensional topology studies objects that locally look like our space plus the time direction, but may have complicated global structure. This line of research is expected to further tie together many areas of mathematics, including representation theory, homological algebra, topology and geometry.
