I  will discuss the derivation of effective evolution equations for the
dynamics of interacting Fermi systems, at high density. I will start by
reviewing the dynamics of mean-field Fermi systems, and the derivation
of the time-dependent Hartree-Fock equation. I will then focus on
extended systems at high density, and I will present a recent result
establishing the validity of the Hartree equation on a local scale, for
fermions with pseudo-relativistic dispersion. The proof relies on
Bogoliubov transformations, semiclassical commutator estimates and their
propagation in time, control of the fluctuations between effective and
many-body dynamics. An important ingredient in the derivation of the
Hartree equation on a local scale is the proof of propagation of
locality for the dynamics of fluctuations, which allows to quantify the
deviation of the two dynamics on small spatial scales. Joint work with
Luca Fresta (University of Roma 3) and Benjamin Schlein (University of
Zurich).