Abstract: This talk incorporates three aspects:

i) a review of the general theory of characteristics of PDEs with a particular emphasis on its physical
ramifications,

ii) an introduction to a recent study of 2nd order Monge-Ampère equations (MAEs) with the tools of contact/symplectic geometry, and

iii) a presentation of an ongoing investigation of 3rd order MAEs, with special attention to the peculiarities of this nearly unexplored case. These topics were freely inspired, respectively, by
arXiv:1311.3477, arXiv:1003.5177, and arXiv:1403.3521.

Throughout the talk, there will be a gradual transition from an initial physical perspective on PDEs and their characteristics in general, and MAEs in particular, towards a concluding entirely geometric picture, involving contact manifolds, their prolongations, and special sub-bundles of some Grassmannian-like bundles, improperly called here "Lagrangian". The reason behind this denomination is that they stem from a higher-order analog of the symplectic form, known as "meta-symplectic".

Surprisingly enough, the so-obtained framework, in spite of its abstractedness, allows to give an immediate answer to some non-equivalence problems and, more generally, to begin to understand the structure of the "space of all 3rd order MAEs".