MATH 848I: Exterior Differential Systems
Spring 2020
This course will be an introduction to Cartan's method of moving frames and EDS. The latter is a method for encoding systems of partial differential equations on manifolds as differential ideals on associated manifolds, which is particularly adapted to the intrinsic geometry of the PDEs. The core of the course will be based on Ivey and Landsberg's Cartan for Beginners Chapters 1 and 47 (first edition)/Chapters 1 and 58 (second edition), which includes the CartanKähler algorithm yielding a criterion for existence of solutions with given initial conditions, as well as applications to particular classes of PDEs such as MongeAmpère systems.
Suggested Texts:
 T.A. Ivey and J.M. Landsberg: Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems (2nd ed.), AMS Graduate Studies in Mathematics 175, Providence, RI (2016).
 R. Bryant, P. Griffiths, D. Grossman: Exterior Differential Systems and EulerLagrange Partial Differential Equations, Chicago Lectures in Mathematics, Chicago (2003).
 R. Bryant, S.S. Chern, R.B. Gardiner, H. Goldschmidt, P. Griffiths: Exterior Differential Systems, Springer (1990).
Announcements

Class meets TuTh 1112:15 in Math 1311.
Lectures
 1/28: Examples of overdetermined PDEs (1.3 in IL, see also App. B1, B2 for a review of differential forms)
 1/30: Differential ideals and the Frobenius Theorem (1.10 in IL)
 2/4: MaurerCartan form (1.6 in IL, see also App. A.4 for Lie algebras)
 2/6: Fundamental Theorem for the MaurerCartan form (1.6 in IL; see handout for proof of last point of theorem)
 2/11: Examples of moving frames: curves in the Euclidean plane,
maps between complex domains (1.4, 1.7 in IL)
 2/13: Tableaux and prolongations (5.1 in IL)
 2/18: Solutions of tableaux in the form of the "first example" (5.2 in IL)