Delve into the world of quantum embedding theory through this comprehensive lecture by Lin Lin from the University of California, Berkeley. Explore the mathematical foundations of quantum embedding methods, including dynamical mean-field theory (DMFT) and density matrix embedding theory (DMET), and their applications in studying strongly-correlated quantum materials. Examine the computational challenges faced in these methods, particularly at the single-particle level, such as the bath fitting problem in DMFT and the correlation potential fitting problem in DMET. Gain insights into recent approaches using convex optimization to tackle these issues. Cover topics including second quantization, operators, quantum antibody problems, noninteracting systems, quantum impurity, and orbital partitioning. Discover the high-level ideas behind various embedding theories and their solvers, as well as the similarities and goals of different approaches.