Seminarium "The Trans-Carpathian Seminar on Geometry & Physics"

This talk incorporates three aspects: i) a review of the general theory ofcharacteristics of PDEs with a particular emphasis on its physicalramifications, ii) an introduction to a recent study of 2nd orderMonge-Ampère equations (MAEs) with the tools of contact/symplecticgeometry, and iii) a presentation of an ongoing investigation of 3rd orderMAEs, with special attention to the peculiarities of this nearlyunexplored 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 initialphysical perspective on PDEs and their characteristics in general, andMAEs in particular, towards a concluding entirely geometric picture,involving contact manifolds, their prolongations, and special sub-bundlesof some Grassmannian-like bundles, improperly called here "Lagrangian".The reason behind this denomination is that they stem from a higher-orderanalog of the symplectic form, known as "meta-symplectic". Surprisinglyenough, the so-obtained framework, in spite of its abstractedness, allowsto give an immediate answer to some non-equivalence problems and, moregenerally, to begin to understand the structure of the "space of all 3rdorder MAEs".