Convergence of the Cahn-Hilliard equation

I will discuss some variational methods to guess and prove convergence results for some singular PDEs and stochastic dynamics. As a toy example, one can study limits of heat equations (or Brownian motion in a fast oscillating potential). The main focus of the talk will be on a singular limit of 1-dimensional Cahn-Hilliard equation. As time allows, possible extensions to non-reversible Markov processes and convergence of stochastic particles systems will be discussed.