Abstract: I will discuss recent rigorous progress on the problem of wave turbulence (WT) in two settings. First - ''classica'' - when for the Hamiltonian cubic NLS equation we take the WT limit (this will be explained) for random initial data and small time. Second - due to Zakharov-Lvov - when we take same equation, damp it by viscosity, drive be random force, and for zero initial data examine solutions under the WT limit, for all values of time.

Bio Sketch: Sergei Kuksin is a professor of mathematics at the Université Paris Cité and Sorbonne Université and at Steklov Mathematical Institute of RAS, a head of laboratory in the RUDN University (Moscow) and a professor at the Institute for Financial Studies, Shandong University. His research is the broad area of analysis and probability, including KAM theory, PDEs with randomness, turbulence and statistical hydrodynamics, and elliptic PDEs for functions between compact manifolds. In 1992 he was a plenary speaker with talk "KAM theory for partial differential equations" at the European Congress of European Mathematicians in Paris. In 1998 he was an invited speaker at International Congress of Mathematicians in Berlin. In 2016 he received the Lyapunov Prize of the Russian Academy of Sciences.