Seminar talk, 17 May 2023
Speaker: Boris Doubrov
Title: Overdetermined systems of PDEs related to representations of semi-simple Lie algebras
We explore a class of finite-type systems of PDEs whose symbol is determined by an (arbitrary) irreducible representation of a graded semisimple Lie algebra.
We show that trivial equations with such symbol correspond to rational homogeneous varieties, non-trivial linear equations define symbol-preserving deformations of such varieties. In particular, we determine when such deformations exist. In terms of the corresponding PDE system this corresponds to the question when compatibility conditions imply that the system is equivalent to trivial. The answer to this question is given in terms of certain Lie algebra cohomology, which can be effectively computed using the results for the theory of semisimple Lie algebras.
We solve local equivalence problem for such systems under fiber+symbol preserving transformations and show how this is related to the projective geometry of submanifolds. Finally, we discuss the case of non-linear systems with the same symbol and show that under certain additional conditions their solution spaces admit remarkable geometric structures.