ABSTRACT PART I: Multidimensional Monge-Ampère equations are, in a sense, the simplest nonlinear PDEs of order two, and to explain this point of view, I will briefly outline the ideas and the results contained in the paper "Contact geometry of multidimensional Monge-Ampère equations: characteristics, intermediate integrals and solutions" by D. Alekseevsky et al. (Ann. Inst. Fourier, 2012). In particular, I will stress the role of characteristics in the description of these equations: a characteristic is a direction in the manifold of independent variables along which the Cauchy-Kowalevskaya theorem fails in uniqueness, and for (non-elliptic, two-dimensional) Monge-Ampère equations, the knowledge of all the characteristics corresponds to the knowledge of the equation itself. This easy feature, which is usually overlooked, plays a key role here, and it can properly formulated in terms of the geometry of the three-dimensional Lagrangian (or "symplectic") Grassmannian L(2,4).

ABSTRACT PART II: In this second part, I will provide a solid mathematical foundation to the statement that "the simplest nonlinear PDEs of order three (in two independent variables) are of Monge-Ampère type". Basically, I will mimic all the steps, explained in the first part, which allowed to "reconstruct" a classical (non-elliptic) Monge-Ampère equation out of its characteristics. As it will turn out, everything goes rather smoothly, except for the definition of the "third-order analog" of the Lagrangian Grassmannian, which I denote by L(2,5) and refer to as the "meta-symplectic Grassmannian". I will explain in detail how to define the four-dimensional space L(2,5), how to frame it in the jet-theoretic framework for nonlinear PDEs, and how to recognize in its hyperplane sections the natural third-order analogues of the Monge-Ampère equations. Finally, I will show how such a perspective on third-order Monge-Ampère equations can help in solving equivalence problems and in finding exact solutions.

The second part of this seminar is based on a joint work with G. Manno, available at http://arxiv.org/abs/1403.3521