Randomness is ubiquitous in nature. It is exhibited in mathematics in a wide range of models and problems from different areas. From the point of view of PDEs, there are three main phenomena we are interested in: statistical description of the system, propagation of randomness, and stochastic regularization. I will talk about a series of recent works in nonlinear dispersive equations related to these three phenomena. Using the nonlinear Schrodinger equation as a model, I will discuss the dynamics of Gibbs measure (equilibrium statistical mechanics), mathematical treatment of wave turbulence (non-equilibrium statistical mechanics), and uniqueness of rough solutions. These are joint works with Zaher Hani, Andrea Nahmod, and Haitian Yue. Speaker(s): Yu Deng (University of Southern California)