Most phenomena in Nature are nonlinear, and their modeling requires nonlinear partial differential equations (PDE). These are elusive objects, and solving them explicitly is almost always out of reach. We are then left to prove that solutions exist, are unique, satisfy specific properties that somehow characterize them, and then implement numerical methods to approximate those solutions. Existence shows we are modeling a real problem and uniqueness that it is meaningful. However, for the model to be useful and the numerical methods to converge, we need to understand the behavior of the solutions even if unable to see their face. The connection between a PDE’s structure and its solutions’ properties is the essence of modern PDE theory.

This masterclass highlights different topics, ranging from fundamental questions on regularity theory to applications in reinforcement learning. It will include talks on nonlinear wave propagation, hyperbolic conservation laws, applications of Fourier analysis to music, and a stunning visual guide to PDE models.