Ich halte eine der Übungsgruppen zur Vorlesung von Stefan Haller. Die Standards für Abhaltung und Bewertung der Übungen sind für alle Gruppen einheitlich und finden sich im Vorlesungsverzeichnis. Die Anmeldung erfolgt wie üblich online, die Abwicklung der Kreuzerlisten wird über Moodle erfolgen. Mit der Anmeldung zur Lehrveranstaltung werden Sie automatisch auch in die entsprechende Moodle-Gruppe aufgenommen, wo auch die Übungsbeispiele zur Verfügung gestellt werden.

This is a topics course from the area "geometry and topology" which can be used for studentst of other areas within the modules "Mathematische Verbreiterung" and "Mathematisches Wahlfach". There are close relations to algebra. The course may also be of interest for physics students, e.g. via the connections to homogeneous (pseudo-)Riemannian manifolds as used in GR and via the theory of bundles that is used in several parts of physics.

The course is situated at the intersection of classical geometry, differential geometry and Lie theory. Since Felix Klein's "Erlangen Program" classical geometries (e.g. Euclidean, affine, projective, hyperbolic) are described in terms of transitive actions of Lie groups. In modern language, one studies various types of objects on a homogeneous space G/H which are invariant under the natural action of the group G.

The main topic of the course is that determining G-invariant geometric objects on G/H can be systematically reduced to problems of finite diemensional linear algebra or representation theory. On the way to these results, we will devleop a substantial part of the modern concepts and language of differential geometry. In particular, we will discuss various type of fiber bundles (vector bundles, principal bundles, associated bundles) and connections (linear connections, principal connections, induced connections). Several examples of homogeneous geometric structures, for example on spheres of different dimensions, will be discussed in detail.

To follow the course, a good background in analysis on manifolds and a basic background on Lie groups (in particular the Lie group/Lie algebra correspondence and the basic results on homogeneous spaces) will be needed.

A set of rough lecture notes and some related materials will be posted online here in due time.

I am organizing this seminar jointly with Roland Donninger, who is the main organizer. The main topic will be the study of several well known explicit solutions (like Schwarschild and Kerr) of the Einstein equations of general relativity. Depending on the background of the participants we will also discuss fundemantal material on pseudo-Riemannian geometry. Students are expected to prepare and present a 90 minutes talk and participate in the discussion of other presentations in order to complete the seminar. We will follow the book "The Geometry of Kerr Black Holes" by O'Neill, background material will be discussed as neccessary in the seminar.