Introduction to Exterior Differential Systems

Mercredi 19 septembre 2018 14:00-17:00 - Benjamin McKay - University College Cork (Irlande)

Résumé : \’Elie Cartan described arbitrary systems of partial differential equations in terms of differential forms and submanifolds rather than equations and functions.
Many examples of such systems arise in differential geometry, sometimes in a manner that does not even require coordinates to state.
Cartan used this point of view to prove a theorem of existence and generality of solutions, further developed by Kaehler.
Their theorem applies only in the real analytic category, and only describes local solutions, so from a modern PDE perspective it seems a very poor theorem.
Nonetheless, because the theorem is coordinate independent, it can sometimes apply easily to problems which are easier to describe in differential forms than in coordinates.
For some problems, there is no other analytical tool to prove local existence of solutions, so the Cartan—Kaehler theorem is surprisingly rich.
The theorem has an algebraic flavour, as it demands no estimates, so it appeals to algebraic geometers, while analysts tend to find it unattractive for the same reason.
However, the theorem demands strong nondegeneracy and smoothness hypotheses, which make it frustrating to everyone.
I want to explain how to state and employ the Cartan Kaehler theorem, with simple examples, and give an overview of some points of the proof.
This talk will explain very old ideas, with no new ideas.

Lieu : Salle 3L8

Introduction to Exterior Differential Systems  Version PDF