Course: Manifolds and group actions (14722.0039).
Lecture: Tuesdays and Thursdays 10:00 - 11:30.
IMPORTANT: There is a change of the room. Tuesdays lecture will be taking place in Seminar room 3, Mathematischen Institut (Raum 314), but Thursdays lecture will be taking place in Übungsraum 1, Mathematischen Institut (basement), with the EXCEPTION OF Thursday April 27, when the lecture is in Übungsraum 2.
Office Hours: Wednesdays 2-3 pm, room 05a.
It is mandatory to register also for the exercise session for Manifolds and group actions (14722.0040).
Exercise Sessions with Thomas Rot: Tuesdays 16-17:30, Seminar room 2, Mathematischen Institut, Raum 204. Requested change of time was not possible as there is no room avaialbe at 12.
Homeworks:
Course description:
This course is devoted to manifolds and proper Lie group actions on them.
In the first part of the course we study differential geometry, mostly following [W] and [L]:
- abstract manifolds and smooth maps between them,
- vector fields and
- differential forms.
Then we move to the study of Lie groups, i.e. groups which are also smooth manifolds, and their Lie algebras.
The second part is devoted to proper group actions.
We analyse some classical examples (adjoint and coadjoint actions, the action of the fundamental group of a manifold, polygon spaces, etc) and discuss the following topics, following [A] and [DK]:
- Bochner linearization theorem,
- slice theorem,
- principal and associated bundles,
- orbit type stratification,
- averaging method producing invariant structures (like invariant metric, symplectic or contact structure),
- bi-invariant metrics on Lie groups,
- Seifert fibered spaces,
- homogeneous spaces.
If time permits, we can also discuss some of the following topics: classification of simple Lie groups, Mostov-Palais embedding theorem, classifying spaces and equivariant cohomology, torus actions and their relation to combinatorics.
Literature:
[A] M. Audin, "The topology of torus actions on symplectic manifolds",
[L] J.M. Lee "Introduction to Smooth Manifolds,"
[DK] J.J. Duistermaat and J.A.C. Kolk,"Lie groups",
[W] Frank Warner, "Foundations of differentiable manifolds and Lie groups",