Programme leader: E.M. Opdam
Central research themes are:
1- Analysis on groups and homogeneous spaces. This includes themes from operator algebras, noncommutative geometry and quantization.
2- Special functions associated with root systems and their interpretation in relation to the above mentioned structures.
3- Special functions and orthogonal polynomials in one variable: analysis, asymptotics, approximation theoretic properties and algorithmic aspects.
4- Wavelet theory and its applications to imaging.
5- Approximation theory and its applications.
6- Analysis in several complex variables.
7- Analysis aspects of mathematical physics. This includes topics from operator algebras, noncommutative geometry, quantization theory and quantum field theory. This theme also includes the study of integrable systems and evolution equations.
This programme unites a number of themes which are mutually connected and have a stimulating influence on each other. Some of these themes have a quite algebraic setting, while others belong to classical analysis. A stimulating line within theme (1) is the study of canonical representations for Hermitian and para-Hermitian symmetric spaces, and its relation with Berezin quantization. A new development within (1) is the application of ideas from noncommutative geometry to various completions of Hecke algebras. Within theme (3) there is involvement with the production of a new edition of the famous Handbook of mathematical functions (Abramowitz & Stegun). This new edition will also become available as an interactive database on the internet. In (5) a radically new and general approach has been developed to describe the closure of modules over the polynomials or trigonometric functions in a wide range of practical situations. Among others this gives rise to new criteria for the determinacy of moment problems in one and in higher dimensional situations. Theme (7) unites many disciplines of pure mathematics. Apart from the topics that were already mentioned, techniques from Lie groupoids and category theory play an important role here as well. A wide range of applications within mathematical physics is being studied.