Geometry and algebra of the McKay correspondence

I will give an overview of my main research results starting with the McKay correspondence, which relates the representation theory of Lie algebras of ADE type and finite subgroups of SL_2(C) to the geometry of the minimal resolutions of Kleinian singularities. I will then explore the connections with homological invariants such as the derived category and their symmetries, and with categorification of Heisenberg algebras.