The porous medium equation (PME) is a nonlinear degenerate diffusion equation that describes a wide variety of physical phenomena, including fluid flow through porous media, heat conduction in insulators, plasma physics, and population growth. In this course, the student will learn the well-posedness of weak solutions, regularity of weak solutions, asymptotic behavior and long-time behavior, and other qualitative properties including the maximum principle, comparison principle, and the behavior of solutions near the free boundaries. Students should seek approval from the course instructor to take this course.