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Hello everybody, I would like to know about the work of Élie Cartan of PDE's that relate to the theory of foliations and differential forms.

I am interested in the subject and will be happy to receive basic references on the subject (articles) as well as explanations on the importance of the subject in mathematics today.

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  • $\begingroup$ I'm wondering if this question is a bit overbroad. (See the FAQ.) Also I am not sure why the OP wanted a dynamical-systems tag on this question. For the time being I'll leave it there. $\endgroup$ – Willie Wong Aug 27 '10 at 16:01
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    $\begingroup$ A starting point may be section 2 of projecteuclid.org/euclid.bams/1183516693 $\endgroup$ – Willie Wong Aug 27 '10 at 16:05
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    $\begingroup$ I was going to suggest the book by Bryant, Chern, ... on Exterior Differential Systems, which is available for free from the MSRI: msri.org/communications/books/Book18/… $\endgroup$ – José Figueroa-O'Farrill Aug 27 '10 at 16:13
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    $\begingroup$ Ivey and Landsberg, cartan for beginners $\endgroup$ – Deane Yang Aug 27 '10 at 16:46
  • $\begingroup$ Thanks Wong, Yang and Figueroa! It is a good start for my bibliographic research $\endgroup$ – Klein Aug 27 '10 at 18:43
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Robert Bryant is the reigning expert on this. An excellent book on the subject (later than the one mentioned) is: Exterior Differential Systems and Euler-Lagrange Partial Differential Equations, Chicago Lectures in Mathematics (2003), University of Chicago Press (vii+213 pages, ISBN: 0-226-07794-2.) by R. Bryant, Phillip Griffiths and Dan Grossmann.

I just recalled, Bryant has a very nice set of nine introductory lectures on the subject. It may be just what you are looking for! They are available online here:

https://services.math.duke.edu/~bryant/Introduction_to_EDS.pdf

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    $\begingroup$ A preliminary version of the book is available on the arXiv: arxiv.org/abs/math.DG/0207039 $\endgroup$ – Willie Wong Aug 27 '10 at 22:05
  • $\begingroup$ Hi, I made the address of the nine lectures a genuine link. Will. $\endgroup$ – Will Jagy Aug 28 '10 at 4:53
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Another highly readable reference for this material is the book Equivalence, Invariants and Symmetry by Peter Olver.

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