Explore recent advancements in Density Functional Theory through this 45-minute lecture presented by Mathieu Lewin of CNRS and Université Paris-Dauphine at IPAM's Model Reduction in Quantum Mechanics Workshop. Delve into key topics such as the Lieb-Oxford bound and the Local Density Approximation, drawing from joint works with Elliott Lieb and Robert Seiringer. Gain insights into Schrödinger's equation for electrons, Lieb's convex formulation of DFT, low and high density limits, kinetic energy, and the uniform electron gas energy. Recorded on April 12, 2022, this comprehensive presentation offers a valuable overview of cutting-edge research in quantum mechanics and its applications.