COURSE DESCRIPTION: The lectures will focus on the behavior of numerical invariants of smooth projective varieties under derived equivalences, and on the structure of the total cohomology of bundles of holomorphic forms as modules over the exterior algebra via the BGG correspondence. I will explain recent results with Ch. Schnell on the behavior of the Picard variety under derived equivalence, in particular showing the invariance of Hodge numbers for derived equivalent threefolds. I will also describe work with R. Lazarsfeld, which uses the BGG correspondence and generic vanishing in order to describe the complexity of the canonical cohomology module, and give bounds for basic numerical invariants. Ample background will be provided.