Wave turbulence theory seeks to describe the nonequilibrium statistical mechanics of nonlinear wave systems. At its core lies wave kinetic theory, a kinetic formalism that captures the mesoscopic evolution of microscopic ensembles of waves through a collisional paradigm.
In this talk, I will first give a brief introduction to wave turbulence theory, including its history and motivation. I will then discuss recent progress in wave kinetic theory, focusing on breakthroughs by Yu Deng and Zaher Hani as well as results from my joint work with Zaher Hani and Jalal Shatah. I will conclude by presenting open problems and conjectures arising from my joint work with Pierre Germain.