Explore the Wannier localization problem in quantum mechanics through this 53-minute lecture by Cornell University's Anil Damle at IPAM's Model Reduction in Quantum Mechanics Workshop. Delve into the mathematical foundations of localization methods, examining their role in Hartree-Fock and Kohn-Sham density functional theory calculations. Discover new perspectives on deriving objective functions for condensed phase systems, and understand their impact on practical computations of localized functions. Gain insights into recent advancements in initialization schemes, software developments, and approaches for unoccupied orbital localization. Follow the lecture's progression from theoretical foundations to practical applications, including a simple 1D example, periodic approximations, and the MLWF approach, culminating in an exploration of the truncated density convolution method and its application to BaTiO3.