Inverse Hamiltonian reduction refers to a series of conjectural relations between W-algebras corresponding to distinct nilpotent orbits in a Lie algebra. I will outline a proof of this conjecture in type A that relies on novel geometric methods. Along the way, we shall encounter a technique for localising vertex algebras and, time permitting, speak briefly on the deformation theory thereof. To build intuition, I shall focus on the finite type analogue of this story, where such techniques are more commonplace. This talk is based on joint work with Dylan Butson.