Resolvent method for holomorphic mappings and semigroups

In the hundred and fifty year period since their discovery, numerous extensions and generalizations of the classical Denjoy-Wolff Theorem, Schwarz and Schwarz–Pick Lemmas, Julia’s Lemma, Julia-Wolff Carathéodory Theorem have appeared in their connections with complex analysis, complex dynamical systems and geometric function theory. Ideas generated by these now classical results continue to attract attention of mathematicians to this day. Our main goal in this talk is to present some links and interactions between generation theory for semigroups of holomorphic mappings and geometric function theory in the complex spaces of finite and infinite dimensions. In particular, we present basics of the reolvent method theory for finite and infinite dimensional cases and point out some new trends and problems related to this issue. A special attention we pay to the geometric properties of the nonlinear resolvent of holomorphic generator in the spirit of geometric function theory.