Cohomological integrality isomorphisms

To understand the topology of a manifold, one often considers its singular cohomology, which forms a finite-dimensional graded vector space. The dimensions of the graded components, called Betti numbers, encode enumerative information about the space. For example, the first Betti number of a Riemann surface counts (twice) the number of holes. In algebraic geometry, however, we encounter objects that are not manifolds but belong to a broader class of geometric objects called stacks. The cohomology of a stack is usually infinite-dimensional, and it is not immediately clear how to extract meaningful enumerative invariants. In my talk, illustrated with elementary but representative examples, I will explain how to define new enumerative invariants for stacks obtained as a quotient of a vector space by the action of a reductive groups. This definition involves parabolic induction morphisms, which allow us to break the infinite-dimensional cohomology into more manageable finite-dimensional pieces. I will give a conjectural interpretation of these new enumerative invariants in terms of intersection cohomology, a refinement of singular cohomology that takes singularities into accounts.