NCMW Harmonic Analysis (2017)

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Venue: Indian Institute of Science,  Banglore, Karnataka
Date:  11th, Dec 2017 to 16th, Dec 2017

 


 

School Convener(s)

Name Prof. E.K. Narayanan Prof. P.K. Ratnakumar
Mailing Address Indian Institute of Science (IISc)
Bangalore
School of Mathematics
Harish-Chandra Research Institute
Allahabad 211019

 


 

Speakers and Syllabus 

This was an advanced workshop in four different themes in harmonic analysis. First set of lectures by Professor J Faraut focused on orbital measures and spline functions. The action of unitary group on the space of Hermitian matrices can be described in terms of the results of Olshanksii, Okunkov etc and thus involve harmonic analysis on compact Lie groups. Horns theorem describing the projection of an orbit in the space of Hermitian matrices under the action of the unitary group and the orbital measure was explained. Second set of lectures by Professor Linda Saal dealt with generalized Gelfand pairs associated to the Heisenberg group. Spherical analysis associated to these pairs was discussed in detail. Third set of lectures by Professor Shobha Madan and Professor Saurabh Srivastava focused on sparse operators and a proof of the A2 conjecture. Finally, the lectures by Professor Swagato Ray dealt with harmonic analysis on symmetric spaces. In particular, Kunze-Stein phenomenon and boundedness of convolution operators were established.

 Speakers and their affiliations with detailes of lecture

Speakers Affiliations Details of lectures delivered
J. Faraut, Professor, Universite Pierre et Marie Curie, France Introduction to orbital measures and statement of three problems related to orbital measures. Action of the unitary group on Hermitian matrices and Baryshiknov’s theorem on the joint distribution of eigenvalues. Olshanskii’s theorem on projection of orbital measures to the (1, 1)th co-ordinate of the matrices. Okunkov’s result on the distribution of X11. Action of orthogonal group on the space of real symmetric spaces. Discussion of the three problems and a formula for the projection of the orbital measure. Discussion of open problems in the case of U (p, q) action.
Shobha Madan Professor, IISER Mohali, India Introduction to Euclidean Fourier analysis. Convolutions. Plancherel theorem for the Euclidean Fourier transform. Schwartz space and Paley-Wiener theorems. Hardy-Littlewood maxi-
mal functions and its Lp− boundedness. Hilbert transform and its Lp boundedness properties. Introduction to singular integrals. Calderon -Zygmund singular integral operators. Calderon-Zygmund decompostions and covering lemmas. Weak type (1, 1) propertly of C–Z operators. Riesz transforms.
Swagato Ray Professor, ISI Kolkata, India  Iwasawa and Polar decomposition on SU (1, 1), relation with SL(2, R). Description of Haar measure in the coordinates KAN, AN K, N AK. Poincar ́e metric on the open unit disc, geodesics
and horocycles. Haar measure with respect to Polar decomposition.

Construction of eigenfunctions of the Laplace-Beltrami operator which are constant on horocycles. Definition of the Fourier transform. Fourier transform is well defined for Lp functions on the Poincar ́e disc. K−biinvariant functions and the spherical Fourier transform. The slice projection theorem for Radon, Fourier and the Abel transform.

The Plancherel theorem for L2 functions on G/K assuming the Plancherel theorem for K−biinvariant functions.The Abel inversion and the Fourier inversion for smooth, compactly supported K−biinvariant functions on G = SU (1, 1). Harish Chandra series for the elementary spherical functions. Harish Chandras c−function and asymptotics of the elementary spherical functions. Elementary cases of Kunze–Stein phenomena. Young’s convolution inequality for unimodular and non-unimodular groups. View a function on the hyperbolic space H 2 as a function on its isometry group which is unimodular and also as a function on the associated Iwasawa N A group which is non-unimodular and use this dichotomy to obtain a new convolution inequality known as Kunze–Stein phenomenon. Mapping properties of spectral projection, restriction and extension operators on the hyperbolic space H 2 and its connection to the convolution inequality mentioned above. Hardy- Littlewood maximal function (centered and uncentered), their mapping properties. Counter example to establish the sharpness of these properties and emphasize the difference between the centered and uncentered versions. Comments on the effect of polynomial versus exponential volume growth vis-a-vis the lack of ball doubling property of the metric-measure space H 2 on some results of analysis on this space. Discussion of some open questions.

Linda Saal Professor, University of Cordoba, Argentina Introduction to Gelfand pairs, class one representations and spherical functions. Examples, (M (n), SO(n)), (HM (n), U (n)) etc. General examples of the type (G K, K) where G is a two step nilpotent Lie group. Spherical functions in explicit form. Bessel functions and Laguerre functions. Generalized Gelfand pairs. U (p, q) action on the Heisenberg group. Associated generalized spherical distribution and spherical analysis associated to this pair. Open problems.
Saruabh Srivastava Associate Professor, IISER Bhopal, India Introduction to sparse operators. Calderon- Zygmund operators. Sparse bounds for C–Z operators with smooth kernels. A2 conjecture. Proof of the A2 conjecture due to A. K. Lerner.
Discussion of some open problems related to rough singluar integrals.

 


 
Time Table

Date Lecture 1
9:30-10:30
Tea
10:30 -11
Lecture 2
11-12
Lecture 3
12-1
Lunch
1- 2:30
Lecture 4
2:30-3:30
Tea
3:30 - 4
Lecture 5
  4-5
11-12-2017 JF   LS SM   SR   SS
12-12-2017 JF   LS SM   SR   SS
13-12-2017 JF   LS SM   SR   SS
14-12-2017 JF   LS SM   SR   SS
15-12-2017 JF   LS SM   SR   SS
16-12-2017 JF   LS SM   SR   SS

 


 

Actual Participants 
Sr.n  Name Affiliation Position held
1 Anoop V. P. NISER, Bhubaneswar Research scholar
2 Sayan Bagchi ISI, Kolkata Inspire faculty
3 Rijju Basak IISER Bhopal BS-MS student
4 Mithun Bhowmick  ISI, Kolkata Research scholar
5 Santanu Debnath Kolkata University Research scholar
6 Venku Naidu Dogga  IIT Hyderabad Assistant Professor
7 Abhishek Ghosh  IIT Kanpur Research scholar
8 Qaiser Jahan  IIT Mandi Assistant Professor
9  Jotsaroop Kaur  IIT Bombay Assistant Professor
10 Ashisha Kumar IIT Indore Assistant Professor
11 Shravan Kumar IIT Delhi Assistant Professor
12 Lakshmi Lavanya IISER Tirupati Assistant Professor
13 Arup Kumar Maity Harish-Chandra Rese-
arch Institute, Allahabad
 Research scholar
14 S. Pitchai Murugan RIASM, Chennai Research scholar
15 Muna Naik ISI, Kolkata Research scholar
16 Sanjay Parui NISER, Bhubaneswar Assistant Professor
17 Partha Sarathi Patra  IIT Hyderabad Research scholar
18  Savan Patel  St Xavier’s college, Ahemdabad  Assistant Professor
19 Sanjay Pusti IIT, Bombay Associate Professor
20 Senthil Raani ISI, Bangalore  Postdoctoral Fellow
21 Sivaramakrishna C  IIT Hyderabad Research scholar
22 Ratnakumar P. K. HCRI, Allahabad Professor
23 Kalachand Shuin IISER Bhopal Research scholar
24  Sumit Kumar Rano IIT Guwahati Research scholar
25  Samya Kumar Ray IIT Kanpur Research scholar
26  Sanjay P. K NIT, Kozhikode, Kerala Assistant Professor
27 Jayanta Sarkar ISI, Kolkata Research scholar
28 Rajesh Kumar Singh IIT Kanpur Research scholar
29  Devendra Tiwari  Delhi University Research scholar

 


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