The subject Harmonic analysis forms one of the mainstreams of mathematics, whose relevance in mathematics and other branches like Physics, ElectricalEngineering and Economics is well known. It had found its origins in the study of Fourier series. In modern day mathematics, by harmonic analysis we generally mean analysis over homogeneous spaces using the study of Groups and their representations. This branch of Mathematics has pervaded into manyother branches of mathematics, for instance,differential geometry, functional analysis, operator algebras, discrete mathematics.
The modest aim of this school is to introduce and study Spherical harmonic analysis, an important and well understood aspect of non-commutative harmonic analysis in general, and apply it to understand Heisenberg motion group and compact groups. We begin from the basics and make the course self- contained. The topics to be studied are for instance Structure theory of locally compact groups,Fourier analysis on Euclidean spaces, Representation Theory of groups, Peter-Weyl Theory, Abstract spherical harmonic analysis, Compact Gelfand Pairs, Heisenberg Motion group.
Syllabus
|
Name of the Speaker with affiliation |
No. of Lectures |
Detailed Syllabus |
|
Rudra Sarkar I.S.I. Kolkatta |
6 |
Groups and Homogenous spaces, linear Lie groups, Computation of Haar measures on some known examples, Convolution Various function spaces. |
|
Parashar Mohanty I.I.T Kanpur |
6 |
Harmonic Analysis over Torus and Euclidean spaces. Generalities about locally compact abelian groups, which includes Fourier Inversion Formula, Bochner’s theorem. |
|
V. Muruganandam NISER, Bhubaneswar |
6 |
Basic Representation Theory, Induced representations, Positive Definite functions, Schur’s lemma, Naimark Theorem |
|
Sanjay Parui NISER, Bhubaneswar |
6 |
Peter-Weyl Theory of compact groups- Examples: Unitary groups, Orthogonal groups |
|
Swagato Roy |
6 |
Abstract Theory of Gelfand Paris, Spherical Fourier Transforms, Plancherel-Godement Theorem, Inverse Sperical Fourier Transforms, Compact Gelfand Pair |
|
S. Thangavelu |
6 |
Representation Theory of Heisenberg group, Gelfand pair consisting of Heisenberg group and Unitary group. Associated Plancherel-Godment Theorem, Special functions. |
References:
- J. Faraut. Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques. In J.L. Clerc et al., editor, Analyse harmonique. C.I.M.P.A., 1982.
- J. Faraut. Analysis on Lie Groups: An Introduction. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2008.
- J. Faraut and K. Harzallah. Analyse Harmonique: Fonctions Speciales Et Distributions Invariantes. Progress in Mathematics. Birkh¨auser, 1987
- G. Folland, A course in abstarct harmonic analysis, CRC Press, 1994.
- Y. Katznelson: Introduction to Harmonic Analysis J. Wiley and Sons, 1968
- Lecture notes from the Speakers.
Time-Table
|
Day |
Date |
Lecture 1 (9.30–11.00) |
Tea (11.05–11.25) |
Lecture 2 (11.30–1.00) |
Lunch (1.05–2.25) |
Tutorial (2.30–3.30) |
Tea (3.35-3.55) |
Tutorial (4.00-5.00) |
Snacks 5.05-5.30 |
|
Mon |
17.12.2018 |
R. Sarkar |
|
P. Mohanty |
|
R. Sarkar |
|
P. Mohanty |
|
|
Tues |
18.12.2018 |
P. Mohanty |
|
R. Sarkar |
|
P. Mohanty |
|
R. Sarkar |
|
|
Wed |
19.12.2018 |
R. Sarkar |
|
P. Mohanty |
|
R. Sarkar |
|
P. Mohanty |
|
|
Thu |
20.12.2018 |
P. Mohanty |
|
R. Sarkar |
|
P. Mohanty |
|
R. Sarkar |
|
|
Fri |
21.12.2018 |
R. Sarkar |
|
P. Mohanty |
|
R. Sarkar |
|
P. Mohanty |
zz |
|
Sat |
22.12.2018 |
P. Mohanty |
|
R. Sarkar |
|
P. Mohanty |
|
R. Sarkar |
|
|
SUNDAY : OFF |
|||||||||
|
Mon |
24.12.2018 |
V.Muruganandam |
|
S. Parui |
|
V.Muruganandam |
|
S. Parui |
|
|
Tues |
25.12.2018 |
S. Parui |
|
V.Muruganandam |
|
S. Parui |
|
V.Muruganandam |
|
|
Wed |
26.12.2018 |
V.Muruganandam |
|
S. Parui |
|
V.Muruganandam |
|
S. Parui |
|
|
Thu |
27.12.2018 |
S. Parui |
|
V.Muruganandam |
|
S. Parui |
|
V.Muruganandam |
|
|
Fri |
28.12.2018 |
V.Muruganandam |
|
S. Parui |
|
V.Muruganandam |
|
S. Parui |
|
|
Sat |
29.12.2018 |
S. Parui |
|
V.Muruganandam |
|
S. Parui |
|
V.Muruganandam |
|
|
SUNDAY : OFF |
|||||||||
|
Mon |
31.12.2018 |
S.Roy |
|
S. Thangavelu |
|
S.Roy |
|
S. Thangavelu |
|
|
Tues |
1.1.2019 |
S. Thangavelu |
|
S.Roy |
|
S. Thangavelu |
|
S.Roy |
|
|
Wed |
2.1.2019 |
S.Roy |
|
S. Thangavelu |
|
S.Roy |
|
S. Thangavelu |
|
|
Thu |
3.1.2019 |
S. Thangavelu |
|
S.Roy |
|
S. Thangavelu |
|
S.Roy |
|
|
Fri |
4.1.2019 |
S.Roy |
|
S. Thangavelu |
|
S.Roy |
|
S. Thangavelu |
|
|
Sat |
5.1.2019 |
S. Thangavelu |
|
S.Roy |
|
S. Thangavelu |
|
S.Roy |
|
Tutorial Assistants:
|
S. No. |
Name |
Affiliation |
|
1 |
Dr. Senthil Raani |
IISER, Beehembpur |
|
2 |
Dr. Saswatha Adhikari |
NISER, Bhubaneswar |