Darboux Integrability and the Quotient Theory of Differential Equations.

The goal of this talk is to discuss finding closed form solutions to partial differential equations, with emphasis on second order PDE in the plane. In particular the classical method of Jean-Gaston Darboux was a theory of compatibility of systems of differential equations (which was the motivation behind Elie Cartan's five variable paper) which lead to  the pioneering work of Ernest Vessiot that provided a group theoretical basis behind the construction of the closed form formulas obtained by Darboux.  Underlying Darboux integrability is the quotient theory of differential equations (or differential systems in general)  by a Lie group. I will demonstrate this theory with examples while highlighting the difference between hyperbolic equations and elliptic.