We'll examine a branch of algebra which lies at the intersection of Lie
theory, Poisson geometry and category theory.
W-algebras are related to universal enveloping algebras, and like them carry a lot of information about the representation theory of the corresponding Lie algebras. In particular, the representation theory of W-algebras is closely related to the block decomposition of the BGG
category O. We'll discuss how the geometry of the nilpotent cone and a
quantum version of Hamiltonian reduction by stages can be used to relate different W-algebras.