NCMW - Harmonic Analysis (2019)

Venue: IISER Bhopal, Bhopal

Dates: 10 Dec 2019 to 14 Dec 2019

Name: Dr. Saurabh Shrivastava Dr. Rahul Garg
Mailing Address: Office-213, Academic Building 1,
IISER Bhopal,
Bhauri, Bhopal Bypass Road,
Bhopal, Madhya Pradesh 462 066
Office-209, Academic Building 1,
IISER Bhopal,
Bhauri, Bhopal Bypass Road,
Bhopal, Madhya Pradesh 462 066.
Email: saurabhk at rahulgarg at

Please Note:  Participants have to arrange for their own travel.

This will be an advanced workshop in four different themes in harmonic analysis. First set of lectures will be by Professor Luca Fanelli on "Hardy inequalities and applications" , introducing a unified point of view through Quantum Mechanics, Functional Analysis and Fourier Analysis to describe Hardy inequalities as mathematical manifestations of the Uncertainty Principle. Later on, some useful applications to Spectral Theory and the study of Dispersive PDE’s will be discussed.

Second set of lectures will be by Professor Bent Orsted on "Branching problems for unitary representations and Poisson transforms". An interesting problem studied in the literature is to investigate how certain unitary representations of semi-simple Lie groups decompose when restricted to subgroups. This is known as Branching problem and it leads to structures that shed new light on certain elliptic boundary value problems. For example, for the groups O(1,n+1) and U(1,n+1) the branching theory leads to the well known Poisson transforms associated to symmetric spaces. Studying this problem in the general context one is led to Symmetric Breaking operators of Kobayashi and Speh providing new examples of Poisson transforms.

Third set of lectures will be on "Bochner-Riesz conjecture and related problems" by Prof. Sanghyuk Lee. The Bochner-Riesz conjecture is an attempt to understand the convergence of multidimensional Fourier series and integral, which is one of the most fundamental questions in the classical harmonic analysis. Though the conjecture is settled in 2-dimensions, it is still open in higher dimensions. As is well known to experts in this field, the progresses in the Bochner-Riesz conjecture are closely tied to those of Fourier restriction problems, where we have witnessed rapid developments in recent years. Bilinear and multilinear generalizations of linearestimates turned out to be the most efficient tools. In this series of lectureswe will cover developments in the Bochner-Riesz conjecture and its related problems in connection with the Fourier restriction problem.

The Fourth set of lectures will on the topic of "Noncommutative Harmonic Analysis" by Prof. Parasar Mohanty. This course will begin with an introduction to the noncommutative operator space structure of L^pspaces. We shall discuss several important examples and the main results in this direction. This allows us to formulate problems fromclassical harmonic analysis in the context of group von Neumann algebras. In particular, we shall develop the theory of Calderon-Zygmundoperators and completely bounded multipliers in generalon noncommutativeL^pspaces.