In this talk, we will discuss wave turbulence theory and its recent progress. Wave turbulence theory describes the statistical evolution of weakly nonlinear dispersive waves on long time scales. In this talk, I will introduce the basic kinetic picture, and review recent rigorous derivations of the wave kinetic equations which are wave analogues of the Boltzmann equation. I will briefly investigate the diagrammatic expansions and the role of resonance relations. Moreover, I will describe the molecule approach, which is a central concept of Deng-Hani machinery. At the end of the talk, I will present ongoing work on a derivation of the wave kinetic equation of the Klein-Gordon equation, arising from crystal. This talk is based on joint work with Zaher Hani (U of Michigan) and Katja Vassilev (U of Chicago).
