Monge-Ampère Equations on (Para-)Kähler Manifolds: from Characteristic Subspaces to Special Lagrangian Submanifolds

We present the basic notions and results of the geometric theory of second order PDEs in the framework of contact and symplectic manifolds including characteristics, formal integrability, existence and uniqueness of formal solutions of non-characteristic Cauchy problems. Then, we focus our attention to Monge-Ampere equations (MAEs) and discuss a natural class of MAEs arising in Kahler and para-Kahler geometry whose solutions are special Lagrangian submanifolds.