EDS Seminar

From March to May 2018 I organised a reading seminar on Exterior Differential Systems, based on the book and notes by Robert Bryant (see precise references below).

The main goal of this seminar was to understand the general theory of EDS, i.e. a geometric way to study partial differential equations on manifolds, focusing in particular on the Cartan-Kähler theorem (a tool to produce integral manifolds, i.e. solutions, to analytic equations) and the Cartan-Kuranishi theorem (a condition to test involutivity by means of a finite number of prolongations).

Possible sequels of this seminar (which unfortunately never happened) may have included a few applications to relevant PDEs coming from geometric problems and to calculus of variations.

The seminars were run weekly during the Friday Fish, in room HFG 610 (occasionally on other days because of conflicts in the schedule).

Plan of the seminars (together with the notes of the talks - read at your own risk and peril):

• 02/03/18, 11-13 (HFG 610): Introduction to EDS and Cartan-Kähler theorem (Francesco Cattafi)

In this first talk, after a brief historical overview and a couple of motivating examples (the Frobenius and Darboux theorems), we will give a quick introduction to the main objects of our study, namely EDS, integral manifolds and integral elements. The purpose is to provide all the ingredients in order to understand the statement (of one version) of the Cartan-Kähler, whose proof will be given in a later talk.

• 09/03/18: no talk (Higher/Lower workshop)

• 15/03/18, 13-15 (HFG 610): Integral elements of EDS: fundamental properties and Cartan test (Luca Accornero)

The aim of this talk is twofold: understanding better the structure and regularity properties of integral elements and stating a more direct version of the Cartan-Kähler theorem. We first review the definitions of integral element and Kähler regularity, providing a couple of explicit examples; then we discuss some fundamental properties and define Cartan flags and ordinary integral elements. This enables us to state an explicit version of the Cartan-Kähler theorem. Finally, we state and prove the Cartan test, a fundamental tool for discussing whether or not an integral element is ordinary.

• 16/03/18: no talk (symplectic geometry workshop)

• 23/03/18, 11-13 (HFG 610): Proof and consequences of Cartan-Kähler theorem (Francesco Cattafi)

We will give a complete proof of the Cartan-Kähler theorem, relying on the definitions and lemmas given in the previous two talks. This is done in two steps, first constructing the analytic submanifold, and then showing that it is integral; both steps will use the Cauchy-Kovalevski theorem, which we will state without proof. Moreover, we will also state and prove an important corollary of the Cartan-Kähler theorem, which is basically an iterative application of the main theorem (sometimes called Cartan-Kähler theorem itself): it says that we can find an integral manifold of an analytic EDS starting from a "nice" (i.e. ordinary) integral element.

• 30/03/18: no talk (Easter holiday)

• 06/04/18: no talk (Easter holiday)

• 13/04/18, 11-13 (HFG 610): Cartan-Kähler theorem and Cartan test (Luca Accornero)

After a short review we are going to conclude the proof of the existence of integral manifolds tangent to an ordinary integral element, using the result proved in the last talk (i.e. the extension of integral manifolds). We will sketch a discussion about the "generality" of the space of integral manifolds. Then we will state and discuss (time permitting, in full details) the Cartan test for ordinariness of an integral element.

• 20/04/18, 11-13 (HFG 610): EDSs and PDEs (Francesco Cattafi)

In this talk we will explore three important classes of EDSs: the aim is on one side to highlight their connections with PDEs and on the other to lay the groundwork for the study of prolongations.

An independence conditions for an EDS consists of a class of differential forms not belonging to the ideal; every PDE can be written as an EDS with a natural independence condition, corresponding to the choice of the independent variables.

Involutivity is the existence of an ordinary integral element at any point: for real-analytic EDSs, this is precisely the hypothesis to apply Cartan-Kähler and get integral manifolds through every point. We will give the examples of an involutive EDS (the one given by an involutive distribution) and a non-involutive one (having an extra "hidden" integrability condition).

Last, all EDSs induced by PDEs turn out to be examples of Pfaffian systems with a certain "linearity" property (with an unfortunate choice of name); indeed, we conclude reviewing the definition of Pfaffian bundle (the "geometric structure" encoding a PDE on a jet bundle) and showing that it also induces a linear Pfaffian system with an independence condition.

• 27/04/18: no talk (Koningsdag)

• 30/04/18, 11-13 (HFG 610): Overview of prolongation theory (Luca Accornero)

In this talk we will give a (sketchy) introduction to prolongation theory. After a motivating example, we will define prolongations of EDS and we will describe their local structure. The theorems of Matsushima (also known as persistence of involutivity) and of Cartan-Kuranishi will be stated and discussed.

• 04/05/18: no talk (Masterclass on topological data analysis)

• 11/05/18: no talk (Hemelvaartsdag)

• 18/05/18: no talk (schedule problems)

• 25/05/18: no talk (schedule problems)

• 28/05/18, 11-13 (HFG 610): An Application of the theory of EDS to G-Structures (Ori Yudilevich)

Following a paper by Singer & Sternberg, I will present an application of the framework of Exterior Differential Systems (and the Cartan-Kähler Theorem, in particular) to the theory of G-Structures. I will prove that, in the real-analytic category, if two G-structure have constant intrinsic torsion and the two intrinsic torsions are equal, then the G-structures are locally equivalent (=isomorphic). As a corollary, we will conclude that the so-called transitive G-structures are precisely those with constant intrinsic torsion.

Main references:

• Robert Bryant, Nine Lectures on Exterior Differential Systems, 2015 (incomplete)

• R. Bryant, S. Chern, R. Gardner, H. Goldschmidt, P. Griffiths, Exterior differential systems, Springer-Verlag, New York, 1991

Extra/partial references:

• Robert Bryant, Notes on Exterior Differential Systems, 2014

• Y. Choquet-Bruhat, Yvonne, C. DeWitt-Morette, M. Dillard-Bleick, Analysis, manifolds and physics, North-Holland, 1977

• T. Ivey, J. Landsberg, Cartan for beginners: differential geometry via moving frames and exterior differential systems, American Mathematical Society, 2003