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These are expository notes from the 2008 Srni Winter School. They have twopurposes: (1) to give a quick introduction to exterior differential systems(EDS), which is a collection of techniques for determining local existence tosystems of partial differential equations, and (2) to give an exposition ofrecent work (joint with C. Robles) on the study of the Fubini-Griffiths-Harrisrigidity of rational homogeneous varieties, which also involves an advance inthe EDS technology.

Exterior differential systems, Lie algebra cohomology, and the rigidity of homogenous varieties