This will be an advanced workshop in four different themes in harmonic analysis. First set of lectures will be by ProfessorAngela Pasquale on the theory of Heckman-Opdam hypergeometric functions which is a natural extension of the spherical function theory by Harish-Chandra on semisimple Lie groups. Starting from the definition of these functions which are also natural generalization of Gauss’hypergeometric functions recent results characterizing the bounded ones (analogue of the Helgason-Jhonson theorem for non-compact symmetric spaces) will be explained. Second set of lectures by Professor J Faraut will focus 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. Horn’s theorem describing the projection of an orbit in the space of Hermitian matrices under the action of the unitary group and the orbital measure will be explained. Third set of lectures by Professor Linda Saal will deal with generalized Gelfand pairs associated to the Heisenberg group. Spherical analysis associated to these pairs will be discussed in detail. Fourth set of lectures by Professor Andrei Lerner, Professor Parasar Mohanty, Professor Saurabh Srivastava will focus on sparse operators and a proof of the A_2 conjecture.
Syllabus to be covered in terms of modules of 6 lectures each :
|Name of the Speakers with affiliation.||No. of Lectures||Detailed Syllabus|
|1.Swagato Ray Indian Statistical Institute, Kolkata||6||
Prerequisites for Fourier analysis on rank one Riemannian symmetric spaces of noncompact type, convolution and Kunze-Stein phenomenon, Hardy-Littlewood maximal function, Fourier restriction theorems.
|2. J. Faraut Université Pierre et Marie Curie, France||6||Orbital measures, action of unitary group on Hermitian matrices, determinantal formula of Olshanskii, spline functions, Horn’s theorem|
|3.Linda Saal University of Cordoba, Argentina||6||
Gelfand pairs associated to Heisenberg groups, spherical analysis, generalized Gelfand pairs, spherical distributions, spherical transform.
|4. Parasar Mohanty IIT Kanpur||6||Calderon-Zygmund analysis on Euclidean spaces. Weighted norm inequalities. A_2 conjecture and related topics.|
|5. Saurabh Srivastava IISER Bhopal||6||Introduction to sparse operators and applications in harmonic analysis. Proof of the A_2 conjecture.|
- E. M Opdam: Lecture notes on Dunkl operators for real and complex reflection groups, Mathematical Society of Japan, Tokyo 2010.
- B. Schapira: Contributions to the hypergeometric function theory of Heckman-Opdam : Sharp estimates, Schwartz space, heat kernel, Geom. Funct. Anal. 18 (1) (2008) 222-250
- E. K. Narayanan, A Pasquale and S. Pusti: Asymptotics of Harish-Chandra expansions, bounded hypergeometric functions associated to root systems and applications, Adv. Math. 252 (2014)227-259.
- G. van Dijk: Introduction to harmonic analysis and generalized Gelfand pairs. De Gruyter Studies in Mathematics, 36, Walter de Gruyter & Co, Berlin
- T. Godoy and L Saal: On the spectrum of the generalized Gelfand pair (U(p, q), H_n), Math. Scand.105 (2009), no.2, 171-187.
- F. Levstein and L. Saal: Spherical distributions of some generalized Gelfand pairs attached to the Heisenberg group. Contemp. Math., 537, Amer. Math. Soc. 2011
- J. Faraut: Rayleigh theorem, projection of orbital measures and spline functions. Adv. Pure Appl. Math. 6 (2015) , no.4 261-283
- A. Lerner: A simple proof of the A_2 conjecture: Int. Math. Res Not. (2013) no. 143159-3170.
Tentative time-table mentioning names of the speakers and tutors with their affiliation:
|Lecture 2 11:00-12:00||Lecture 3 12:00-1:00||Lunch
| Lecture 4
|Name of the speaker||Name of the speaker||Name of the speaker||Name of the speaker||Name of the speaker|
SR: Swagato Ray (Indian Statistical Institute, Kolkata)
JF: J. Faraut (Université Pierre et Marie Curie, France)
LS: Linda Saal (National University of Cordoba, Argentina)
SS: Saurabh Srivastava (IISER Bhopal, India)
PM: Parasar Mohanty (IIT Kanpur, India)
AL: Andrei Lerner (Bar-Ilan University, Israel)