Explore the intricacies of the regularity problem for the Landau equation in this 57-minute lecture delivered by François Golse from Ecole Polytechnique at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the mathematical complexities surrounding this fundamental equation in plasma physics, examining its properties, challenges, and potential solutions. Gain insights into the latest research and methodologies employed to tackle the regularity issues associated with the Landau equation, and understand its significance in the broader context of kinetic theory and partial differential equations.