  This paper describes recent results obtained in collaboration with M.
Huesmann and F. Otto on the regularity of optimal transport maps. The main
result is a quantitative version of the well-known fact that the linearization
of the Monge-Amp{\`e}re equation around the identity is the Poisson equation.
We present two applications of this result. The first one is a variational
proof of the partial regularity theorem of Figalli and Kim and the second is
the rigorous validation of some predictions made by Carraciolo and al. on the
structure of the optimal transport maps in matching problems.
