Mean field dynamics of interacting fermionic systems

We consider the dynamics of interacting fermionic systems, in the mean-field regime. As the number of particles goes to infinity, the evolution of the system is expected to be well approximated by the time-dependent Hartree-Fock equation, a well-known example of effective evolution equation. We review some rigorous results about the validity of this approximation. We start by discussing the case of systems of particles interacting via bounded interaction potentials, at zero and at positive temperature. Under the assumption that a suitable semiclassical structure is propagated in time along the flow of the Hartree-Fock equation, the result can be extended to the case of Coulomb interactions.