Abstract Harmonic Analysis 

Bachelor mathematics (In particular Analysis I+II, Topology, Symmetry). It is essential to have followed Introduction to Fourier Theory. Some background in Functional Analysis is helpful.

The student knows

• the basics of topology and measure theory for locally compact groups,
  
• their applications to the duality and structure theory for abelian groups and
  
• the representation theory of compact groups.
• (S)he has some understanding of the new phenomena and difficulties arising for non-abelian and non-compact groups.
• The student will be able to approach the literature on more recent aspects of the theory.
  

Abstract Harmonic Analysis is a relatively recent mathematical theory; it started in the late 1920s and is still growing rapidly. It has close connections with many subjects in analysis, topology and mathematical physics. It also is the starting point of generalizations like "quantum groups".

Abstract Harmonic Analysis starts from the classical Fourier theory (involving the abelian groups Z, S¹ and R) and from the representation theory of finite groups and combines them into a beautiful and powerful theory of harmonic analysis on and representation theory of locally compact groups. A crucial ingredient is the Haar measure, which allows to integrate over any locally compact group. In the abelian case, one obtains the duality theory of Pontrjagin which generalizes classical Fourier theory and has applications to the structure theory of such groups. On the other hand, one finds that the representation theory of compact groups has much in common with that of finite groups. For groups that are non-compact and non-abelian, very interesting new phenomena arise that are still the subject of much research.

• Basics of topological groups, in particular locally compact ones.
  
• Haar measure: Existence, uniqueness, properties.
• Pontrjagin duality of locally compact abelian groups and applications to structure theory of such groups.
• Representation theory of compact groups: Characters, Peter-Weyl theorem.
• Time permitting: Some aspects of non-compact non-abelian groups, based on some crucial examples.
  

Anton Deitmar & Siegfried Echterhoff: Principles of harmonic analysis.
Springer Universitext, 2009. ISBN 978-0-387-85468-7. (42 Euro.)