Explore the foundations of many-body perturbation theory and Green's function methods in this comprehensive lecture by Michael Lindsey from New York University Mathematics. Delve into the world beyond single-particle density functional theory, examining widely used computational methods in quantum many-body physics. Learn about the GW method, dynamical mean-field theory (DMFT), and bold Feynman diagrams. Understand the Luttinger-Ward functional and its role in formalizing Green's function methods. Investigate the rigorous justification of these theories in the context of Euclidean lattice field theories and their application to interacting fermions. Discover open problems in the fermionic setting and gain insights into the mathematical structure of Green's function methods. Cover topics such as density-density form, self-energy, first and second-order diagrams, infinite resumation, large reward functions, renormalization of diagrams, impurity problems, and partition functions.