The derivation of the Vlasov-Poisson equation, that describes the evolution of a systems of N interacting non-collisional particles at macroscopic scale, is a classical problem in mathematical physics. In this talk we will review this topic in the quantum setting, by using the Hartree equation as a bridge between the N-body Schrödinger equation and the Vlasov-Poisson system. We will show that a solution of the Hartree equation converges strongly towards a solution of the Vlasov-Poisson equation in the semiclassical limit.
The results hold for singular interactions (included the Coulomb and gravitational potentials) and exhibit explicit bounds on the convergence rate.

I will show how, in the context of diffeology, one can find the prequatization framework for general diffeological spaces (finite or infinite dimensional, in presence of singularities or not) as a groupoid, quotient of the space of paths, equipped with a left-right invariant connectionlike-form. This construction suggests a variant of the program of geometric quantization which I will describe.