Filippo Bracci: What is a Loewner chain?

A (classical) Loewner chain can be interpreted in terms of a bi-parametric intertwining family of maps for a suitable family of holomorphic self-maps of the unit disc. Starting from this consideration, after recalling some facts of dynamical systems theory, we will define abstractly Loewner chains for complete hyperbolic complex manifolds of any dimension and we will show how the classical theory can be recovered from a categorial construction and the uniformization theorem.