Topological Groups (Summer Semester 2020)
- Lecturer: Dr. Rafael Dahmen
- Classes: Lecture (0155300), Problem class (0155310)
- Weekly hours: 2+2
Due to the current pandemic this course will be hold only online.
More information will be added in the coming weeks to the following ILIAS-course:
If you have any questions, do not hesitate to ask per e-mail: firstname.lastname@example.org
|Lecture:||Thursday 15:45-17:15||20.30 SR 3.068|
|Problem class:||Wednesday 14:00-15:30||20.30 SR 2.058|
|Lecturer, Problem classes||Dr. Rafael Dahmen|
|Room 1.024 Kollegiengebäude Mathematik (20.30)|
Content of the course
Groups are omnipresent in mathematics. They appear naturally when describing symmetries of various mathematical objects. Many of those groups come equipped with a natural topology which is compatible with the algebraic structure. Therefore, it makes sense to use this additional information when discussing the group (and its actions on other objects).
The goal of this lecture is to provide an overview in the theory of topological groups and to give many interesting examples. Besides other topics, we explore the interaction of topological properties (like connectedness, compactness and metrizability) with the group structure. Furthermore, we will explore when a continuous group homomorphism is automatically open (Open Mapping Theorems for topological groups). Differential structures on groups (Lie groups) will not be a topic of this course, although there are plenty of connections between the corresponding theories.
Besides only elementary point set topology and group theory, no other prerequisites are needed.
This lecture will follow lecture notes which will be provided in the course.
Further useful information concerning topological groups can be found in:
- D. DIKRANJAN, Introduction to topological groups (2007; online available at https://users.dimi.uniud.it/~dikran.dikranjan/ITG.pdf)
- T. B. SINGH, Introduction to topology, Singapore: Springer (2019; relatively new overview over (algebraic and point-set) topology including a long chapter about topological groups; available from the KIT at https://link.springer.com/book/10.1007%2F978-981-13-6954-4)
- P. J. HIGGINS, An introduction to topological groups, Cambridge University Press (1974)