Abstract: The Riemann mapping theorem says that every simply connected domain in the Riemann sphere which does not contain two points is biholomorphic to the unit disc. Carathéodory prime ends theory is an effective tool to study the boundary behavior of Riemann mappings. The aim of this course is to present the Carathéodory prime ends topology in a sistematic way.
Program (4 hours)
First hour: No Koebe arcs theorem, Lehto-Virtanen theorem. Slits.
Second hour: Prime ends. Circular null chains. Impression of a prime end.
Third hour: The Carathéodory topology. Carathéodory extension theorem.
Fourth hour: Radial limit, non-tangential limits and unrestricted limits versus principal parts and impressions.
Due to time limitation, most proofs will be only sketched and the focus will be on the main ideas underlying the theory.
Prerequisites: basic knowledge of complex analysis.
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