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 noncommutative 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, PeterWeyl 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 
PeterWeyl Theory of compact groups Examples: Unitary groups, Orthogonal groups 
Swagato Roy 
6 
Abstract Theory of Gelfand Paris, Spherical Fourier Transforms, PlancherelGodement 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 PlancherelGodment 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.
TimeTable
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 (3.353.55) 
Tutorial (4.005.00) 
Snacks 5.055.30 
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 