Abstract: In this talk we focus on the time dynamics of solutions of periodic nonlinear Schrödinger with random initial data and the key underlying difficulty of understanding how randomness propagates under the nonlinear flow. We discuss the theory of random tensors, a powerful new framework that we developed with Yu Deng and Haitian Yue, which allows us to unravel the propagation of randomness beyond the linear evolution of random data and probe the underlying random structure that lives on high frequencies/fine scales. This enables us to show the existence and uniqueness of solutions to the NLS in an optimal range relative to the probabilistic scaling. A beautiful feature of the solution we find is its explicit expansion in terms of multilinear Gaussians with adapted random tensor coefficients.