The book is based on the lectures given at the CIME school Quantum many body systems held in
the summer of 2010. It provides a tutorial introduction to recent advances in the mathematics
of interacting systems written by four leading experts in the field: V. Rivasseau illustrates
the applications of constructive Quantum Field Theory to 2D interacting electrons and their
relation to quantum gravity R. Seiringer describes a proof of Bose-Einstein condensation in
the Gross-Pitaevski limit and explains the effects of rotating traps and the emergence of
lattices of quantized vortices J.-P. Solovej gives an introduction to the theory of quantum
Coulomb systems and to the functional analytic methods used to prove their thermodynamic
stability finally T. Spencer explains the supersymmetric approach to Anderson localization
and its relation to the theory of random matrices. All the lectures are characterized by their
mathematical rigor combined with physical insights.
