Functional probability inequalities with their applications to RMT and Samplings

There have been an intensive research activities around functional probability inequalities in the past decades. They do not only play an important role in characterizing probability distributions, but also have a wide range of applications in apparently distinct fields. In this talk I shall mainly focus on three recent examples: (1) Quantitative normal approximations of linear eigenvalue statistics, (2) A dynamical approach to Dyson-Gaudin-Mehta-Wigner conjecture for Wigner matrices, (3) Rates of convergence in Metropolis-adjusted-Langevin algorithm in a local non-convex target distribution sampling. I shall also present some problems of interest from my personal taste.