Mean field control and games. Some prospects

I will first give a brief overview of the notion of mean-field control and games. In particular, I will focus on the Eulerian formulation, through the notion of master equation, which is a nonlinear PDE on the space of probability measures. This PDE is notoriously difficult to solve, at least in a classical sense, except in cases presenting a form of convexity or monotonicity in the argument of the measure. I will then present two problems outside the convex/monotonic framework: (i) non-convex mean-field control problems and their particle approximation, (ii) mean-field games subject to certain non-standard forms of common noise.