Explore quantum impurity and quantum embedding theory in this 51-minute lecture presented by Lin Lin from the University of California, Berkeley. Delve into the quantum many-body problem, starting with non-interacting systems and progressing to quantum impurity problems. Examine various static quantum embedding theories, including a concise derivation of Density Matrix Embedding Theory (DMET). Learn about bath construction, basis rotation, interacting impurity Hamiltonians, and matching conditions. Investigate least squares and convex optimization approaches in DMET, and address challenges with gapless problems. Discover the basic structure of quantum impurity and the sparsity of self-energy. Explore dynamical quantum embedding theories, focusing on enhancing the robustness of Dynamical Mean-Field Theory (DMFT) using semidefinite relaxation. Gain insights into the Luttinger-Ward functional, Green's function domains, and Lindsey's conjecture.