Teaching

• Turnus: Jedes Sommersemester
• Voraussetzungen:  Inhalte der Vorlesungen Analysis 1 und 2 (Bachelor).
• Inhalt: Der kanonische Stoff einer einführenden Vorlesung zur Funktionentheorie: Cauchy-Theorie, Fundamentaleigenschaften holomorpher und meromorpher Funktionen, konforme Abbildungen und der Riemannsche Abbildungssatz, konstruktive Funktionentheorie.

• When: Every winter term
• Prerequisites: A basic Complex Analysis course at the Bachelor's level
• Syllabus: Advanced  topics in Complex Analysis such as Potential Theory and Geometric Function Theory

Applications of methods from Complex Analysis to  Harmonic Analysis, Operator Theory, Spectral Theory, Banachalgebras and Mathematical Physics, e.g.

• Spectralprojections using Cauchy's Integral Formula
• Von Neumann's Inequality
• Spectral Theorem for unbounded operator
  Direct proof based on  the Herglotz Formula
• Hardy- and Bergman Spaces of holomorphic functions
• Invariant subspaces

Prerequisites: Introductory Course on Complex Analysis (Bachelor's level), a little functional analysis

Constructive methods in Complex Analysis based on the dbar equation and methods of Harmonic Analysis with applications to Potential Theory, Topology, the inverse problem of Galois Theory (Grothendieck's dessins d'enfants) and uniformization of Riemann Surfaces.

Basics of  Loewner  Theory,  a particularly modern topic in geometric function theory with Fields-Medals for W. Werner (2006) and St. Smirnow (2010) with applications in Mathematical Physics.

Introduction to the mathematics of Julia sets and the Mandelbrot set.

Uniformisation  and Deformation of complex and conformal structures.

Prerequisites: Complex Analysis (Master) and Differential Geometry (Master)