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Analysis on Groups [MA5064] - SS 15

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The theory of topological groups combines the local concept of 'nearness' (given by a topology) with the global concept of 'homomgeneity' (given by a group law). The interplay between these two structures results in a rich theory that finds applications in

This course covers the basic results and techniques of the following topics:

The theory will be accompanied by illustrating examples, and exercises.

News

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Timetable

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Lecture   Wednesday,    10:15 - 11:45,   03.08.011

Exercises    Weekday    Time    Place    Tutor  
Group 1   Friday    12:00 - 13:00    02.08.020    Dominik Jüstel     
Group 2   Tuesday    11:00 - 12:00    03.08.011    Dominik Jüstel     

Exam (30 minutes oral)  
21.07. - 23.07.   MI 03.08.022A    

Repeat Exam  
tba   tba   tba  

Course Material

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Week 1
Lecture 1: Introduction; Topological groups. Appendix_topology
Lecture 2: Locally compact Hausdorff groups.   Lectures1-2
Exercises: Examples of locally compact groups; semidirect product; Banach spaces; countable groups Sheet1  Sheet1_solution

Week 2
Lecture 3: Functions and measures on locally compact groups.
Lecture 4: The Haar measure.   Lectures3-4
Exercises: Examples of Haar measures; Point measures; Haar measure of the shearlet group Sheet2  Sheet2_solution

Week 3
No Lecture Fachschaftsvollversammlung  
No Lecture    
Exercises: p-adic numbers Sheet3  Sheet3_solution

Week 4
Lecture 5: Existence and uniqueness of the Haar measure.  
Lecture 6:   Lectures5-6
Exercises: p-adic numbers (continued)  

Week 5
Lecture 7: The modular function, unimodular groups.  
Lecture 8: The Weil formula.   Lectures7-8
Exercises: Haar measure on semidirect products; invariant measures on quotients; Haar measure on SO(3) Sheet4  Sheet4_solution

Week 6
Lecture 9: The convolution algebra L^1(G).  
Lecture 10: The algebra M^1(G) of complex measures.   Lectures9-10
Exercises: Convolution, involution, and Haar measure; C_0(G) Sheet5  Sheet5_solution

Week 7
Lecture 11: The dual group of a lca group.  
Lecture 12: The Fourier transform on L^1(G).   Lectures11-12
Exercises: Basic properties of the Fourier transform; the discrete Fourier transform Sheet6  Sheet6_solution

Week 8
Lecture 13: The inverse Fourier transform.  
Lecture 14: Positive-definite functions (functions of positive type).   Lectures13-14
Exercises: Fourier transform on \R_+; Mellin transform Sheet7  Sheet7_solution

Week 9
Lecture 15: Unitary representations of locally compact groups.  
Lecture 16: Unitary representations and positive-definite functions.   Lectures15-16
Exercises: Eigenfunctions of group actions Sheet8  Sheet8_solution

Week 10
Lecture 17: Bochner's theorem.  
Lecture 18: Fourier inversion theorem.   Lectures17-18
Exercises: Symmetries of differential operators Sheet9  Sheet9_solution

Week 11
Lecture 19: Pontryagin duality.  
Lecture 20: The Poisson summation formula.   Lectures19-20
Exercises: Discrete and compact groups; Shannon sampling on lca groups Sheet10  Sheet10_solution

Week 12
Lecture 21: The Fourier transform on L^2(G).  
Lecture 22: Schwartz-Bruhat functions and tempered distributions.   Lectures21-22
Exercises: Fourier transform of periodic functions; The Zak transform; A Bloch-Floquet theorem Sheet11  Sheet11_solution

Week 13
Lecture 23: Lie groups.  
Lecture 24: The Lie algebra of a Lie group.   Lectures23-24
Exercises: Left-invariant vector fields and the Lie bracket Sheet12  Sheet12_solution

Week 14
Lecture 25: The exponential mapping.  
Lecture 26: Lie algebras of matrix groups.   Lectures25-26
No Exercises    

Literature

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Jüstel   Analysis on Groups    lecture notes    2015       
Folland   A Course in Abstract Harmonic Analysis    CRC Press    1995       
Reiter, Stegeman   Classical Harmonic Analysis and Locally Compact Groups    Oxford University Press    2000       
Hewitt, Ross   Abstract Harmonic Analysis (Volume I and II)    Springer    1963, 1970       
Varadarajan   Lie Groups, Lie Algebras, and thier Representations    CRC Press    1995       
Hall   Lie Groups, Lie Algebras, and Representations: An Elementary Introduction    Springer    2004       

People

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  NameSorted ascending eMail Room Consultation
Lecturer   Dominik Jüstel  juestelma.tum.de   MI 03.06.021   tba