MATH 848I: Exterior Differential Systems
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 4-7 (first edition)/Chapters 1 and 5-8 (second edition), which includes the Cartan-Kähler algorithm yielding a criterion for existence of solutions with given initial conditions, as well as applications to particular classes of PDEs such as Monge-Ampère systems.
- 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 Euler-Lagrange 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).
Class meets Tu-Th 11-12:15 in Math 1311.
- 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: Maurer-Cartan form (1.6 in IL, see also App. A.4 for Lie algebras)
- 2/6: Fundamental Theorem for the Maurer-Cartan 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)