The Study of Wave Interactions: Where Beautiful Mathematical Ideas Come Together
Phenomena involving interactions of waves happen at different scales and in different media: from gravitational waves to the waves on the surface of the ocean, from our milk and coffee in the morning to infinitesimal particles that behave like wave packets in quantum physics. These phenomena are difficult to study in a rigorous mathematical manner, but maybe because of this challenge mathematicians have developed interdisciplinary approaches that are powerful and beautiful. In the first lecture, which will be colloquium style, I will describe some of these approaches and show for example how the need to understand certain multilinear and periodic wave interactions provided also the tools to prove a famous conjecture in number theory, or how classical tools in probability gave the right framework to still have viable theories behind certain deterministic counterexamples. In the second and third lecture I will open a small window into the concept of weak wave turbulence. I will start with the deterministic approach of Bourgain, involving the study of long time asymptotic of higher Sobolev norms of solutions of dispersive equations, and I will end with the rigorous derivation of a 3-wave kinetic equation.