Automorphism groups and elliptic complex geometry : doctoral dissertation

This dissertation presents the results obtained by the author during the course of his doctoral studies. The first two chapters provide a detailed introduction to the Andersen-Lempert theory which can be used as an introduction to the theory of holomorphic automorphisms of complex Euclidean spaces and more in general of Stein manifolds with the density property. The latter are complex manifolds admitting a large group of holomorphic automorphisms. Afterwards, we move to the original results concerning parametric jet interpolation by automorphisms. We first provide a complete picture in complex Euclidean spaces and then move to the more technical result for Stein manifolds with the density property. It is at this point that we focus on the topic of tame sets, recalling some of the classical theory developed by Rosay and Rudin and proceeding with the new results. Here, we will focus mostly on linear algebraic Lie groups and the action of holomorphic vector fields. Of particular importance will be the study of the ring of invariant functions for such vector fields. In the last chapter, we present questions that arose during the described research and suggest future lines of exploration in this field.