Module 24-M-GT-TOPG Topological Groups

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master advanced content and methods of the theory of Topological Groups, in particular they can independently carry out very complex proofs in this area requiring a high level of mathematical expertise. They can explain and correctly apply central concepts of the theory (e.g. Haar measure, lattice, continuous action) and apply them correctly. Students can derive formulas for the Haar measure for explicitly given Lie groups, e.g., by using integration by substitution. They know leading examples of the theory (matrix groups over real or p-adic numbers, isometry groups) and can use them to illustrate concepts and theorems. For example, students can derive bounds for the covolume of some lattice.

Furthermore, students recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of Topological Groups.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

The following advanced content of teaching from the field of Topological Groups is compulsory:

• Important definitions from topology:
  • local compactness
  • separability
  • metrizability
  • separability
  • fundamental group
• Continuous actions and subgroups
• Commutativity of the fundamental group
• Existence of the Haar measure and uniqueness up to scaling
• Concrete formulas for the Haar measure in examples (like GL_n)
• Unimodularity
• Lattices
• Important examples:
  • real Lie groups
  • p-adic Lie groups
  • isometry groups
• Totally disconnected groups

In addition, the following content of teaching can be covered, for example:

• Hilbert's fifth problem
• duality theory of Pontryagin
• structure theory of compact groups

This module prepares the content of a master's thesis.

Measure and Integration Theory, Algebra, Geometry and Topology

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