Geometry of differential equations and integrable systems
The study of integrable systems (both finite and infinite dimensional) is crucial in the analysis of various problems arising in differential geometry, such as the characterization of metrics with projective symmetries, the integrability of the geodesic flow, the classification of Monge-Ampère equations up to a prescribed group of transformations and the variational analysis of particular functionals. A useful approach to the study of the above problems is to interpret a certain system of differential equations as a submanifold of an appropriate space of momenta (space of jets or reduction of a configuration space) equipped with an external differential system. The prolongation of this system permits to find its differential consequences and allows both the study of the existence of integral submanifolds and the analysis of the symmetries. Through the method of "moving frames" one can reduce the problem of finding a hierarchy of solutions (starting from a known one) to that of solving a system of linear partial differential equations. This requires the knowledge and the application of techniques coming from topology, analysis, representation theory of Lie groups as well as tools of algebraic geometry and homological algebra.