Lie systems and their geometry

Lie systems are systems of differential equations whose dynamics can be interpreted as a curve taking values in a finite dimensional Lie algebra of vector fields, aka a Vessiot Guldberg Lie algebra. This underlying algebra brings out interesting properties of the system: the general solution can be described in terms of a nonlinear superposition rule of particular solutions, the solutions can be interpreted as a certain type of projective foliation over an appropriate bundle, they also serve for the analysis of flat g valued connections, among other properties… Furthermore, there exist different geometric structures that are compatible with the Vessiot Guldberg Lie algebra, as for example, Poisson and Dirac structures. We will depict how the different geometric backgrounds imply important physical applications.