I will give a brief introduction to the the emerging topic of Mean Field Games, introduced by J-M Lasry and P-L Lions some years ago as a model for the equilibrium of a population of agents each selecting his own optimal paths, according to a criterion which involves the density of the other agents, in the form of a congestion charge. This gives rise to a coupled system of PDEs, a continuity equation where the density moves according to the gradient of a value function, and a Hamilton-Jacobi equation solved by the value function, where the density also appears.
I will mainly deal with the case where this equilibrium problem may be seen as optimality conditions of a convex variational problem, and give the main results in this framework. In particular, I will present some easy but recent regularity results, as well as the connection with optimal transport theory (in particular the dynamical formulation given by J-D Benamou and Y Brenier for numerical purposes). At the end of the talk I will present an interesting variant, where the congestion cost is replaced by a capacity constraint.