Invariant Gibbs measures, propagation of randomness and the theory of random tensors for NLS

In this talk, we will review recent joint work with Yu Deng and Haitian Yue on the solution of the invariance of the Gibbs measure under the 2D nonlinear Schrödinger flow (NLS) flow via the method of random averaging operators and the development of the random tensors theory. The latter yielded the resolution of the random data Cauchy problem for NLS in its full probabilistic subcritical regime. In particular,  we will explain the fundamental shift in paradigm that arises from the notion of probabilistic scaling for random data Cauchy problems and how these ideas opened the door to unveil the random structures of nonlinear waves that live on high frequencies and fine scales as they propagate. We will end the talk with a short discussion of some open challenges.