In Complex Analysis II (called Funktionentheorie II in German) we develop the theory of Riemann surfaces. These are manifolds that are locally pieces of the complex plane (1 complex-dimensional). To these spaces we generalise the results of Funktionentheorie I, in the same way that the results of Analysis I and II are generalised to manifolds in Analysis III.
Important results include the uniformisation theorem, the classification of Riemann surfaces, cohomology theory, Serre duality, and the Riemann-Roch theorem. To close we will discuss Riemann surfaces with singularities.
Method of Instruction
This is a guided reading course. That means that there will be no lectures or tutorials (although, there are some videos from 2020). Instead each week students are expected to read an assigned portion of the script and meet with me to discuss any questions and the exercises. There is also the flexibility, if you would like to go deeper into a particular topic, to supplement the script with a number of textbooks. If you are taking this course, please email me (r.ogilvie at uni-mannheim.de) to arrange a meeting time.
If you would like to take this course, but are missing prerequisites (such as Funktionentheorie I or Analysis III), it may still be possible to take this course. Please get in contact to see if something can be arranged.
(The script is in a mix of German and English. If this is a problem, I can complete the translation)