CATEGORIFICATIONS IN TOPOLOGY AND REPRESENTATION THEORY – CATORE

Recently, new methods have appeared in representation theory and quantum topology, based on the notion of categorical representations of Kac-Moody algebras. They have already had remarkable applications. In representation theory, categorifications give a proof of important particular cases of Broué’s conjecture on modular representations of finite reductive groups, they give a better understanding of the Lusztig conjecture on modular representations of algebraic groups, they yield a proof of a character formula for simple modules in the category O of cyclotomic rational double affine Hecke algebras. In topology, categorifications give new knots invariants which are richer than the Reshetikhin-Turaev invariants, such as Khovanov homology, which permit, in particular, to detect the trivial knot. New types of categorical representations, such as categorical representations of Heisenberg algebras, have also appeared but are not well understood yet. Inspired by physicists’ works on gauge theory, a new approach to knots invariants which is based on the representation theory of double affine Hecke algebras has been proposed. On the other hand, representation theory of quantum groups has taken a proeminent role in computing the equivariant quantum cohomology or K-theory of some standard varieties such as the flag manifolds or the Nakajima’s quiver varieties. Although the relation between quantum cohomology of flag manifolds and Toda lattices appeared at the very begining of the theory in Givental’s works, the relations between Bethe algebras, double affine Hecke algebras, symplectic duality and equivariant quantum cohomology or K-theory is still unclear. The aim of the project is to put together specialists of categorical methods in different domains of representation theory with topologists and geometers so that all the members of the team will profit of mutual interactions. This project will gather 10 mathematicians from 9 universities and it will give them the possibility to organize meetings, collaborations and invitations in order to progress in the understanding of these new structures. In order to promote this research area among young mathematicians, we’ll propose a PhD position. We’ll investigate potential applications of categorifications to finite dimensional representations of quantum affine algebras or to fusion data associated with finite reductive groups, as well as to invariants of knots and 3-manifolds in quantum topology or to equivariant quantum cohomology and K-theory of flag varieties and quiver varieties in geometry.