Dates: From January 10 to July 10, 2011                                      Link to the Weekly Seminar

Place: Centre de Recerca Matemàtica

Xavier Massaneda, Universitat de Barcelona

Joaquim Ortega-Cerdà, Universitat de Barcelona

Background

Spectral complex analysis was created in the classical works by Carleman and Wiener, and then developed by Beurling, Krein, Levinson and many other prominent analysts of the 20th century. The unifying theme of these works was the complex Fourier transform, which translates various problems of harmonic analysis in the real domain into the language of complex analysis. Originally this circle of ideas and problems included sampling, interpolation and uniqueness in Paley-Wiener spaces of entire functions and related properties of exponential systems in L^2-spaces; later it expanded to the uncertainty principle, to various notions of spectrum of a function and to related questions of spectral analysis and synthesis.

Around 1960, fundamental results on multipliers and completeness were obtained by Beurling and Malliavin in two profound works. At the same time, a beautiful unifying theory of Hilbert spaces of entire functions was developed by de Branges. Since then, de Branges spaces are ubiquitous in the spectral theory of self-adjoint 2nd order canonical systems and Krein's strings, in harmonic analysis, and in theory of Gaussian processes.

Since the 60ies very powerful new methods and ideas have been developed: estimates of Hilbert transform and Cauchy integrals, harmonic measure estimates, asymptotically holomorphic functions, subharmonic techniques in theory of entire functions, sampling and interpolation in the Fock-Bargmann-type spaces in the disk and in the plane, functional model of Nagy-Foias and Lax-Phillips, Aleksandrov-Clark measures, to mention a few. Many classical spectral problems that were out of reach 30-40 years ago might be accessible nowadays. Last, but not least, in the recent years complex analysis has regained its central role in different areas of mathematics. Methods of complex analysis have been crucial in achievements in direct and inverse problems of the spectral theory of self-adjoint differential operators, Gabor analysis of signals, random matrix theory, critical two-dimensional lattice models of statistical mechanics, conformal field theory. This has boosted new developments of complex analysis, in particular, of its spectral aspects.

Objectives

The programme will be focused on Hilbert spaces of entire and analytic functions with emphasis on their applications, both emerging and classical.
This is a broad area of research with many developments in recent years.
From our today’s perspective, the objectives of the programme should include at least the following closely related to each other themes:
1) De Branges spaces of entire functions. The chief goal is to better understand the structure of de Branges spaces and their relations with other areas of analysis and mathematical physics.
2)Methods of complex analysis in spectral theory of self-adjoint differential operators. The aim of these topic is to reveal new and clarify known relations between problems and methods of complex analysis and those of direct and inverse spectral problems and of scattering theory for one-dimensional Schrödinger type operators
3) Fock-Bargmann spaces. This theme includes applications of these spaces in mathematical physics (2D Coulomb gas, conformal field theory), in electrical engineering (time-frequency analysis of signals), and in probability theory (determinantal point processes and zero point processes of Gaussian analytic and entire functions).

Perspectives of the programme

This programme would provide an excellent opportunity to bring together mathematicians working in different areas of complex and harmonic analysis and mathematical physics. The aim of the programme is to learn about major developments in spectral complex analysis, to introduce researchers working in complex analysis to related problems that appear in other areas of mathematics and to explore new interdisciplinary directions and perspectives.