Recent progress on SPDEs beyond subcriticality

Effective models of interacting particle systems or turbulent fluids are often given by singular stochastic partial differential equations (SPDEs). While research on subcritical SPDEs continues to flourish, building on the foundations of Hairer's regularity structures, recent years have seen major breakthroughs in tackling critical and supercritical models. In this survey talk I will explore this new frontier, covering foundational models like the 2D stochastic heat equation and KPZ/Burgers equation, as well as complex systems from fluid dynamics like stochastic Navier-Stokes or surface geostrophic equations. I will discuss the origin of these equations, including the challenging Dean-Kawasaki equation, and survey some novel techniques that now allow for their rigorous analysis.