# Eleftherios Theodosiadis: The Loewner equation

**Time: **
Fri 2019-09-20 13.00 - 14.00

**Lecturer: **
Eleftherios Theodosiadis

**Location: ** rum 306, hus 6, Kräftriket

### Abstract

We are going to discuss Loewner’s method (1923) for the representation of single slit mappings of the unit disk. This method was used by Loewner to prove the Bieberbach Conjecture for the third coefficient and later by L. de Branges (1984) to prove the Bieberbach Conjecture in general. We start with the basics, for domains and simply connected domains on the Riemann sphere. We give the Riemann’s mapping theorem and some cases of continuous extensions. In more depth, we have Caratheodory’s convergence theorem and its application, on the density of single slit mappings in the class of univalent functions. Finally, we conclude to Loewner’s equation, giving the basic structure of his proof.