The goal of this set of lectures is to provide an introduction to some current themes of research in quantum many-body theory in condensed matter physics. The lectures will be primarily aimed at a non-specialist audience and I will try to emphasize the main results and open questions, while minimizing the technical derivations.

The tentative plan is :

1. Introduction to strongly correlated systems.
   • From Fermi liquids to Mott insulators.
   • Examples of Mott insulators in solid state and atomic physics.
   • Doping a Mott insulator : the example of high-temperature superconductors.
2. Quantum impurity models in equilibrium.
   • Kondo and Anderson models. Kondo effect.
   • Methods of solutions of quantum impurity models.
   • Introduction to Dynamical Mean Field Theory and its applications.
3. Quantum impurity models out-of-equilibrium.
   • Schwinger-Keldysh formalism.
   • Illustration on simple examples.
   • Wingreen-Meir formula. Kondo effect in quantum dots.