Riemann surfaces

Riemann surfaces are surfaces that locally look like the complex plane:

sphere

["Orange Torus" by Gelada - Own work. Licensed under CC BY-SA 3.0 via Wikimedia Commons]

Their geometry has rather strong implications for the existence and number of holomorphic or meromorphic functions defined on them. The basic principles which we will learn in this seminar have far reaching generalizations in geometry, topology, and algebra. Riemann surfaces provide a rich class of simple, but still interesting examples and thus serve as an ideal topic for a gentle introduction to more advanced techniques that will also appear in courses on differential geometry, algebraic topology, or algebraic geometry. Examples are coverings and the fundamental group, differential forms and de Rham cohomology, sheaves and sheaf cohomology.

Basic knowledge of analysis and function theory is required for the seminar.

If all participants agree, this seminar can be held in German. The written report can be in English or German.

Prüfungsbestandteile Vortrag, schriftliche Ausarbeitung; je nach zutreffendem Modulkatalog ist entweder der Vortrag oder die schriftliche Ausarbeitung eine Studienleistung. Giving a presentation, writing a detailed report; depending on the applicable Modulkatalog: either the presentation or the written report is a Studienleistung.

Anteile der Bestandteile an der Note Je nach zutreffendem Modulkatalog basiert die Note entweder auf der schriftlichen Ausarbeitung oder auf dem Vortag. Depending on the applicable Modulkatalog: the grade is based either only on the written report or only on the presentation.