Lecture I. Introduction. (A general lecture that serves as an introduction to the subject. Samples of the applications to be discussed in the remaining lectures and an overview of what sorts of geometry one can reasonably hope to attach to a PDE.)
Lecture II. The Method of Equivalence. (An introduction to the method with several examples computed so that the audience can get a feel for how the computations are done. The rest of the talks will use this method.)
Lecture III. Control Theory and Pfaffian Systems. (Examples of problems from control theory and differential equations where the method of equivalence can be used to understand basic phenomena. Rigid curves, rolling bodies, parking problems, sub-Riemannian geometry.)
Lecture IV. The Geometry of Conservation Laws. (How the invariants of a PDE can be used to calculate the space of conservation laws, integrable extentions, and other important PDE constructions.)
Lecture V. Natural PDEs on 3-manifolds. (An investigation, using the above methods, of several geometrically natural PDE problems, such as prescribed curvature problems, both for metrics and connections, as well as the problem of describing special families of metrics in dimension 3 that are characterized by differential equations on the curvature tensors.)
All lectures will be given at the Salon Diego Bricio, Cimat, from 12:00-13:30.