Exterior Differential Systems, from Elementary to Advanced

Abraham D. Smith

Published 2017-01-18Version 1

This monograph was developed to support a series of lectures at the Institute of Mathematics at the Polish Academy of Sciences in September 2016, as part of a Workshop on the Geometry of Lagrangian Grassmannians and Nonlinear PDEs. The goal is to cut the shortest-possible expository path from the common, elementary concepts of geometry (linear algebra, vector bundles, and algebraic ideals) to the advanced theorems about the characteristic variety. Hopefully, these lectures lower the barrier to advanced topics in exterior differential systems by exposing the audience to elementary versions of several key results regarding the characteristic variety, and to outline how these results could be used to push the frontiers of the subject. These key results are: 1) The incidence correspondence of the characteristic variety, 2) Guillemin normal form and Quillen's thesis, 3) The Integrability of Characteristics (Guillemin, Quillen, Sternberg, Gabber), 4) Yang's Hyperbolicity Criterion. \end{enumerate} To accomplish this, the subject of exterior differential systems is reinterpreted as the study of smooth sub-bundles of the Grassmann bundle over a smooth manifold. These notes intentionally obscure the role of exterior differential forms in computation, instead focusing on tableaux as subspaces of homomorphisms and on symbols as varieties of endomorphisms. Techniques involving differential ideals or PDE analysis are hardly mentioned. Instead, Guillemin normal form (a generalization of Jordan decomposition from a single endomorphism to a variety of endomorphisms) is the primary computational tool.