Lecture

Lokally Compact Groups

Winter term 2024/25

Course Locally Compact Groups
Locally compact groups are topological groups, i.e. groups which carry a topology such that multiplication and inversion are continuous maps. They play a role in different areas of mathematics such as geometry, geometric group theory, Lie theory, operator theory or harmonic analysis. We first study general topics in topological groups such as subgroups, quotients, connectedness, and actions. Then we consider profinite groups and van Dantzig's theorem. After this we turn to the more advanced structure theory. We will introduce the Haar integral and use it to prove the Peter-Weyl theorem about the structure of compact groups. In the last part of the course we will consider Pontrjagin-Van Kampen duality of compact and locally compact abelian groups.

Audience: The course is aimed at advanced BSC students and MSC students. The topic is well suitable for a master's thesis (or a bachlor's thesis).

Prerequisites: Basic algebra (groups, rings, vector spaces and modules) and a solid background in point-set topology (as covered in the course Grundlagen der Analysis, Topologie und Geometrie). Knowledge about Lie groups or functional analysis is certainly helpful, but not required for this course.

The class takes place on Tuesday and Friday 8:15 - 10:00 Uhr in lecture room M4. We begin on Tuesday Di 8.10.2024 at 8:15. We have set up a course in the Learnweb for the class and the tutorial. The Learnweb key will be anounced in the class.

There will be a weekly tutorial for the class. It is essential that you participate actively in the tutorial and that you do the homework problems. The date for the tutorial will be fixed during the first week of classes.

Literature

Außenhofer, Dikranjan, Giordano Bruno, Topological groups and the Pontryagin-van Kampen duality
Deitmar, Echterhoff, Principles of harmonic analysis
Dugundji, Topology
Hewitt, Ross, Abstract harmonic analysis I
Hofmann, Morris, The structure of compact groups
Hofmann, Morris, The Lie theory of connected pro-Lie groups
Stroppel, Locally compact groups
Wilson, Profinite groups