In the past 20 years, the optimal transport problem introduced by Monge in 1781 has attracted a lot of attention due to its range of applications in various areas: PDEs, Statistics, Statistical Physics, Quantum Chemistry, Metric Geometry, and more. One striking result is the interpretation of certain parabolic PDEs as gradient flows in a metric space of measure.
In this talk, I will present how to derive this interpretation for simple PDEs such as the heat and porous medium equation. I will then discuss how these ideas give birth to a discretization in time algorithm for those equations, known as the JKO scheme (Jordan-Kinderlehrer-Otto)
