Harmonic analysis on locally compact groups 1000-1S25AH
The plan of the seminar is to familiarize participants with a wide range of issues related to harmonic analysis on locally compact groups. There is a natural division between the abelian and non-abelian case.
1 Harmonic analysis on locally compact abelian groups.
- Basic knowledge of the Fourier transform, Fourier-Stieltjes and Pontryagin duality.
- Theorems on the structure of locally compact abelian groups.
- Idempotent measures.
- Special sets (primarily Sidon's and Helson's).
- Ideals in L1.
2. Harmonic analysis on locally compact (non-abelian) groups.
- Existence of Haar measure on locally compact groups.
- C∗-algebras of groups.
- Fourier and Fourier-Stieltjes algebras.
- Multipliers on groups.
- Relationships to properties of groups, such as medializability or Haagerup's property.
The above list is indicative, and the detailed selection of topics will depend on the interests and abilities of the participants.
Type of course
Prerequisites (description)
Course coordinators
Learning outcomes
Seminar participants will learn the basics of harmonic analysis, which will broaden their mathematical horizons and allow them to define interesting research problems.
Assessment criteria
Credit for the seminar is given by delivering a paper and participating in class.
Bibliography
C. C. Graham and K. E. Hare, Interpolation and Sidon sets for compact groups, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, Springer, New York, 2013;
MR3025283
P. Eymard, L’algèbre de Fourier d’un groupe localement compact, Bull. Soc. Math. France 92 (1964), 181–236; MR0228628
G. B. Folland, A course in abstract harmonic analysis, second edition, Textbooks in Mathematics, CRC Press, Boca Raton, FL, 2016; MR3444405
C. C. Graham and O. C. McGehee, Essays in commutative harmonic analysis, Grundlehren der Mathematischen Wissenschaften, 238, Springer, New York-Berlin, 1979; MR0550606
E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. I, second edition, Grundlehren der Mathematischen Wissenschaften, 115, Springer, Berlin-New York, 1979; MR0551496
E. Hewitt and K. A. Ross, Abstract harmonic analysis. Vol. II: Structure and analysis for compact groups. Analysis on locally compact Abelian groups, Die Grundlehren der mathematischen Wissenschaften, Band 152, Springer, New York-Berlin, 1970; MR0262773
L. H. Loomis, An introduction to abstract harmonic analysis, D. Van Nostrand Co., Inc., Toronto-New York-London, 1953; MR0054173
W. Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962; MR0152834
Additional information
Information on level of this course, year of study and semester when the course unit is delivered, types and amount of class hours - can be found in course structure diagrams of apropriate study programmes. This course is related to the following study programmes:
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