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Fourier analysis and representation theory Semester
WiSe 2023 / 24
Lecturer
Lukas Prader
Type of course (Veranstaltungsart)
Seminar
German title
Fourieranalysis und Darstellungstheorie
Contents
This seminar carries the subtitle “Analysis meets group theory”, which already describes quite well what you may expect.
The idea is that certain concepts from analysis (like integration or Fourier series) can be better understood if one studies them from a group theoretic (or more precisely, representation theoretic) point of view. This leads to an exciting theory, which was initiated in 1927 by a publication by Peter and Weyl.
In the preliminary meeting, I will try to illustrate this principle in the easiest possible case: Having the strong machinery of representation theory at hand, the fact that continuous (or more generally, square integrable) functions on the circle may be expanded into Fourier series simply becomes an incarnation of the circle group begin compact and abelian. (Don't worry if you should miss the preliminary meeting; we will revisit this example in even more detail during the seminar.)
In the first part of the seminar, we will introduce topological groups and study their properties. For instance, we will see that on topological groups which are locally compact (this is a notion from topology), one has a well-behaved integral (called “Haar measure”), which enables us to perform analysis on them. Depending on the number of participants, we will further cover (some of) the following topics: the Plancherel Theorem, Pontryagin Duality, the Peter-Weyl Theorem, applications (e.g. to number theory).
Finally, here are further (and more differentiated) reasons for attending this seminar:
If you enjoyed the lectures “Linear Algebra I+II” and/or “Analysis I+II”, then I am convinced that you will also like this seminar. Indeed, we will basically transfer certain topics from the analysis lectures (like integration) into a more general setting, and representation theory is (roughly speaking) the study of groups through linear algebra. In particular, you will learn a quite general theory, which you may then apply to various special cases (like real analysis, complex analysis, …).
If you like differential geometry or mathematical physics, then this seminar will certainly be of use for you, since Lie groups (which are of great importance in these fields) are special cases of topological groups. The main difference is that, in contrast to Lie groups, there is no notion of “differentiation” on a general topological group.
And if you are a number theorist, then I very much recommend to attend this seminar. (I understand if this sounds strange to you, but believe me: I am a number theorist.) Nowadays methods from abstract harmonic analysis (which is a common term to describe the contents of this seminar) are indispensable for number theory and led to fantastic results (e.g., to “Tate’s thesis”, or to the theory of automorphic representations). The idea is that for any prime number p, one may construct the so-called field of p-adic numbers (which, in particular, is a topological group with respect to addition), which turns out to be the “right place” to study questions that only pay respect to one single prime number p.
Literature
We will cover selected parts of the books
A. Deitmar, S. Echterhoff: Principles of Harmonic Analysis. 2nd edition, Springer (2014)
G.B. Folland: A course in abstract harmonic analysis. 2nd edition, Productivity Press (2015)
Recommended previous knowledge
Analysis I+II, Linear Algebra I+II
Time/Date
Mo 16-18
Location
M101
Course homepage
https://elearning.uni-regensburg.de/course/view.php?id=62329
(Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)
Registration- Organisational meeting/distribution of topics: The preliminary meeting takes place on
Monday,
July 17th, at 6:30-7:30 pm via Zoom. Meeting-ID: 637 9871 3793;
Password: 235711 - If you would like to participate in the seminar, then please register for the seminar in
GRIPS
(for the link, see "Course homepage"). If you have questions (or if you are
unable to
attend the preliminary meeting), feel free to write me an e-mail: lukas.prader
"at"
ur.de - Registration for course work/examination/ECTS: FlexNow
Course work (Studienleistungen)- Presentation: Giving a seminar talk of roughly 90 minutes
Examination (Prüfungsleistungen)- Detailed written report of the seminar talk
Modules
BSem, MV, MSem, LA-GySem, Nebenfach Master
ECTS
BSem, MSem: 4,5 ECTS. LA-GySem: 6 ECTS.
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