The course will discuss the mathematics of many-body quantum mechanics, with a focus on the rigorous derivation of effective theories for complex quantum systems. Topics to be covered include:

-) Introduction to quantum mechanics. The hydrogenic atom. Uncertainty principles, stability of matter of the first kind.

-) Bosons and fermions, density matrices. Introduction to large Coulomb systems, as models for atoms and molecules.

-) Lieb-Thirring inequalities, semiclassical approximations.

-) Thomas-Fermi theory. The Thomas-Fermi energy functional. Existence and uniqueness of the minimizer. The no-binding theorem.

-) Proof of stability of matter via Thomas-Fermi theory.

-) Derivation of Thomas-Fermi theory for large quantum systems. -) Quasi-free states, Hartree-Fock theory, Bogoliubov tranformations.

-) Many-body quantum dynamics. Derivation of nonlinear effective evolution equations, the case of fermionic systems. Mean field regime, time-depentent Hartree-Fock equation, Vlasov equation.

-) The correlation energy of interacting Fermi gases via rigorous bosonization.