Apparaît dans la collection : Riemannian Geometry Past, Present and Future: an homage to Marcel Berger

The point of this talk is to introduce a polynomial differential operator which is an analogue of the classical real and complex Monge-Ampère equations. This operator makes sense on any symplectic manifold with a Gromov metric, and its solutions are exactly the functions obtained as upper envelopes of Lagrangian plurisubharmonic functions. Both the homogeneous and inhomogeneous Dirichlet problems for this operator are solved on Lagrangian convex domains, and the homogeneous result also holds for all other branches of the equation. In C^n fundamental solution is established, where the inhomogeneous term is a delta function. There are many interesting open questions. (Talk presented by P. PANSU)