Oberseminar Funktionentheorie, The dark side of the unit disk I: Introduction

In this series of lectures we introduce a unified framework for studying conformally invariant differential operators on the unit disk and the Riemann sphere.This approach is based on viewing the unit disk as well as the Riemann sphere as subsets of a two-dimensional complex manifold Ω, which can be seen as the complement of the complexified unit circle.  We unfold the structure of the Fréchet space H(Ω) of holomorphic functions living on Ω and the invariant differential operators acting on  H(Ω). We also discuss various applications such as

• Star products of Wick-type in strict deformation quantization;
• Peschl-Minda differential operators on the unit disk;
• the invariant eigenspaces of the hyperbolic Laplacian on the unit disk studied by Helgason, Rudin and many others;
• spherical harmonics.