The problem of describing solutions to a particular PDE is — at a first glance — totally unrelated to the problem of describing all the PDEs sharing a common feature. Such PDEs are usually collected into a class. For instance, we usually deploy the toolbox of functional analysis within the class of linear PDEs, and such a possibility is precisely the feature making this class so important. But there are other classes, whose definition (and existence) is not so obvious (and clear) as that of the class of the linear ones. For instance, everybody is able to provide a rigorous definition of a linear PDE, whereas defining the class of Monge—Ampère equations requires a deeper thinking. There exist, however, other classes of PDEs which do not possess an actual definition, but are more similar to collections of examples: these usually arise in Physics, where people tend to group PDEs according to the similitudes in their solving methods. The chief examples are provided by the various notions of integrable PDEs.
In this seminar I will focus on the class of completely exceptional (nonlinear) PDEs, introduced in 1954 by P. Lax, by imposing a certain „linear behavior" on their (potential) solutions. Postponing the discussion of the mathematical rigor of Lax's definition, it is truly remarkable that he was able to single out a certain class of PDEs by requiring their solutions to behave in a certain way, without the necessity of computing the solutions themselves. I will show that, by using an appropriate geometric framework, Lax's condition appears to be natural and, as such, absolutely rigorous. More precisely, I will construct a differential operator whose solutions are precisely the operators defining all the completely exceptional PDEs, thus showing that the two problems mentioned at the beginning are in fact one and the same. The existence of such an operator should not come as a surprise: for instance, linear operators are the zeros of the operator ''which takes second-order derivatives'', though nobody ever thinks of linear PDEs in this cumbersome way.
This seminar is based on the recent publication ''Completely exceptional 2nd order PDEs via conformal geometry and BGG resolution'', by Jan Gutt, Gianni Manno and myself.