Geometry of differential equations

This project is situated in pure mathematics, more specifically in differential geometry. The goal of the project is to extend several fundamental theoretical tools from a class of geometric structures known as parabolic geometries in order to apply them to questions coming from the geometric theory of differential equations. In particular, we seek a characterization of the so- called C-class, which was introduced by E. Cartan. Differential equations in this class can be solved without integration because of special properties of their invariants. Technically, our investigations will require a study of Cartan geometries modelled on certain homogeneous spaces for non-semisimple Lie groups. While these do not belong to the class of parabolic geometries, they still exhibit many of the structural features available in the parabolic case.

In this project, modern geometric tools were applied to the study and characterization of a class of differential equations introduced by Élie Cartan in 1938. Any equation in this class satisfies the fascinating property that all its differential invariants are first integrals, i.e. are constant along solutions. Consequently, generic equations admit sufficiently many first integrals and can be in principle solved algebraically, so without any integration. This is in stark contrast to the well-known symmetry-based methods for solving differential equations.