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
 signal and image analysis (where topological groups appear as structure of the signal space or as group of transformations of images),
 physics (where topological groups appear as symmetry groups of equations or theories),
 pure maths (where topological groups play a central role in number theory, as well as algebraic geometry, and topology).
This course covers the basic results and techniques of the following topics:
 integration on groups: existence and uniqueness of a canonical measure (Haar measure), and its integration theory,
 Fourier analysis on abelian groups: the Fourier transform on abelian groups, its basic properties, and some classic theorems, e.g. inversion theorem, Plancherel, Pontryagin duality, Poisson summation formula,
 Lie groups: groups with differentiable (manifold) structure, the Lie algebra of a Lie group, the exponential map, 'Taylor series' on Lie groups.
The theory will be accompanied by illustrating examples, and exercises.
News
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 15.07.2015: The exam will take place in MI 03.08.022A (Glaskasten).
 08.07.2015: The dates for the exam have been fixed (you should have gotten an eMail). Note that the last two lectures (on Lie groups) are note relevant for the exam.
 01.07.2015: The oral exams will take place from 21.07.  23.07. You can choose your favourite slot in this Doodle ^{}
 18.06.2015: In yesterday's lecture, we decided to change from 60 minutes written to 30 minutes oral exam. Application is possible till 30.06.
 27.05.2015: I added the lecture notes (currently Lecture 112) to the Literature section below. I will update the file after each lecture.
 22.05.2015: On Tuesday, 26.05., there will be no exercises (Pentecost).
 06.05.2015: On Tuesday, 12.05., there will be no exercises (stud. Vollversammlung). Unfortunately, we will thus not be able to finish Sheet 3.
 22.04.2015: One of you noted that my symbol for the semidirect product is the wrong way round  I corrected this in the notes and the sheets.
 22.04.2015: There will be no exercises on Friday, 24.04., and no lecture on Wednesday, 29.04., (Fachschaftsvollversammlung).
 16.04.2015: The material for the first week has been uploaded.
 15.04.2015: Note that the dates for the exercises have been fixed. There will be two groups: Friday, 1213, 02.08.020, and Tuesday, 1112, 03.08.011. The Friday group will start on 17.04.2015, the Tuesday group on 21.04.2015.
 13.04.2015: Please, choose your preferred time slots for the exercises in this Doodle ^{}.
 10.04.2015: You might be interested in the course Representations of compact groups (MA5054) by Prof. Robert König. It perfectly complements this course.
 03.03.2015: Registration for the course is now open in TUMonline.
Timetable
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Lecture 
Wednesday, 
10:15  11:45, 
03.08.011 
Course Material
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Week 1 

Lecture 1: 
Introduction; Topological groups. 
Appendix_topology 
Lecture 2: 
Locally compact Hausdorff groups. 
Lectures12 
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. 
Lectures34 
Exercises: 
Examples of Haar measures; Point measures; Haar measure of the shearlet group 
Sheet2 Sheet2_solution 
Week 4 

Lecture 5: 
Existence and uniqueness of the Haar measure. 

Lecture 6: 

Lectures56 
Exercises: 
padic numbers (continued) 

Week 5 

Lecture 7: 
The modular function, unimodular groups. 

Lecture 8: 
The Weil formula. 
Lectures78 
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. 
Lectures910 
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). 
Lectures1112 
Exercises: 
Basic properties of the Fourier transform; the discrete Fourier transform 
Sheet6 Sheet6_solution 
Week 8 

Lecture 13: 
The inverse Fourier transform. 

Lecture 14: 
Positivedefinite functions (functions of positive type). 
Lectures1314 
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 positivedefinite functions. 
Lectures1516 
Exercises: 
Eigenfunctions of group actions 
Sheet8 Sheet8_solution 
Week 11 

Lecture 19: 
Pontryagin duality. 

Lecture 20: 
The Poisson summation formula. 
Lectures1920 
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: 
SchwartzBruhat functions and tempered distributions. 
Lectures2122 
Exercises: 
Fourier transform of periodic functions; The Zak transform; A BlochFloquet theorem 
Sheet11 Sheet11_solution 
Week 14 

Lecture 25: 
The exponential mapping. 

Lecture 26: 
Lie algebras of matrix groups. 
Lectures2526 
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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