Cyclotomic rational Cherednik algebras and categorifications (I)

This will be part one of a joint talk with M. Varagnolo. We will present a proof of Varagnolo–Vasserot's conjecture on the equivalence between the category $O$ of rational Cherednik algebras and a parabolic category $O$ of affine Lie algberas and some applications of this result (joint work with R. Rouquier and E. Vasserot).
In this talk, we will present the conjecture, discuss its relationship with Schur–Weyl duality and explain one of the main ingredients of our proof – an extended version of Rouquier's theory on unicity of highest weight covers.