*Programme leader: E.M. Opdam*

The central research themes are:

1- Harmonic analysis on groups, homogeneous spaces and
related structures, such as Hecke algebras. Interactions with
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- Approximation theory and its applications.

5- Analysis in several complex variables.

6- 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.

7- Analysis on Lie Groups. The main topics studied are: representations of Lie groups, harmonic analysis on homogeneous spaces and quantization. We make special study of canonical representations (in the sense of Gelfand, Graev, Vershik) and oscillator representations (in the sense of R. Howe). Close cooperation exists with research groups in Russia, in particular in Tambov (Molchanov) and with Pevzner in Paris.

This programme unites a number of themes which are mutually
connected and have a stimulating influence on each other. Some of
the themes have an algebraic or geometric flavor, while others
belong to classical analysis. We mention the following highlights
of research in 2002.

- Theme 1: There has been stimulating progress in the study of
canonical representations (in the sense of Gelfand, Graev,
Vershik) and oscillator representations (in the sense of R. Howe).
Another vein of research within (1) concerns applications of ideas
from noncommutative geometry to various completions of affine
Hecke algebras.

- Theme 2: There has been a lot of activity on quantum groups,
double affine Hecke algebras and the related q-special functions.
New ideas have led to involvement of elliptic functions. Another
result that fits in (2) is the construction of a quasi-hereditary
covering of the representation category of a Hecke algebra, by
means of the study of certain differential equations.

- Theme 3: Involvement with the production of a new and edition of
the famous *Handbook of mathematical functions'* (Abramowitz &
Stegun). This new edition will also become available as an
interactive database on the internet.

- Theme 4: A new result on the determinacy of the simultaneous
distribution of non-negative random variables has been obtained.

- Theme 5: There has been progress towards the Gleasons problem.
Questions around complete pluripolarity of maximal analytic
varieties have been studied. Korevaar is writing an extensive
monograph on Tauberian theorems. The manuscript is expected to be
completed by 2004.

- Theme 6: This theme concerns the interaction between operator
algebras, noncommutative geometry, quantization theory, and
quantum field theory. In 2002, the mutual relationship between all
these fields and both K-theory and the index theory of elliptic
operators has been added as an important research theme. The
central goal is the quantization of singular spaces, with
applications to physics. Also in (6) important results have been
obtained towards the classification of evolution equations by the
existence of infinitely many generalized symmetries.

- Theme 7: Van Dijk and Molchanov completed a study on so-called H-invariants in the irreducible representations of the universal covering group of SL(2,R).
Progress was made with a textbook on harmonic analysis on hyperboloids in 3-dimensional space, a joint project with Molchanov.
With Pevzner and Aparicio a first step was set in the study of the multiplicity free decomposition of Hilbert spaces invariant under the action of the oscillator representation.