The positions have been filled!

General information

The PhD-students will be based at the Research Institute for Mathematics, Astrophysics and Particle Physics IMAPP, Radboud Universiteit, Nijmegen.
The goal is to write a PhD-thesis (in 4 years), which is to be defended at the Radboud Universiteit. A position is for 1+3 years. After 8 months a decision will be made on the extension of the 1 year contract to a total 4 year contract. In case the applicant joins in the VOHO (Voortgezet Onderwijs Hoger Onderwijs) programme an appropriate extension is possible. There is a teaching duty for at most 25%. For more information on the conditions of employment see the P&O-website.

Short research description for project 1: Noncommutative integration

The position is part of the Geometry and Quantum Theory (GQT) cluster. The general goal of the research project is to understand integration theory in quantum group theory and its interrelations with integration in non-commutative geometry and non-commutative integration theory.
Integration in quantum group theory is given by a normal semi-finite faithful (nsf) weight on a suitable von Neumann algebra equipped with additional structures such as a comultiplication. Contrary to the classical case this weight is in general not tracial, but it has a modular group measuring its deviation from traciality. A first research goal is to obtain a (co-)representation theoretic interpretation of the modular group and the associated operators in terms of Duflo-Moore operators. Here we have to distinguish between compact and non-compact quantum groups, and a study of specific special examples from both sets seems necessary.
A second research goal is to understand the relation between integration in quantum group theory and in noncommutative geometry, especially using the results from the first research theme. Already a great deal of examples have been worked out, and for some of these examples there is more than one suggestion (especially for the Dirac operator on the quantum SU(2) group), and we naturally expect that the results of the first part can lead to such a natural relation. A secondary theme in the second research goal is to establish a link with non-commutative integration theory.
The research project is related to research interests of other members, in particular Walter van Suijlekom, of the Mathematical Physics group of IMAPP.

Short research description for project 2: Matrix-valued spherical functions

Scalar valued spherical functions play an important role in representation theory of Lie groups and quantum groups, and such functions can often be realized as matrix elements of an irreducible unitary spherical representation of a Gelfand pair (G,K) of a Lie group G and a compact subgroup K. Examples of such spherical functions are subclasses of Jacobi polynomials and Jacobi functions.
For matrix-valued spherical functions there is a general theory available, starting with Godement in the 1950ies, and with a generalized notion of Gelfand pair. Amongst others, the paper by F.A.Grunbaum, I.Pacharoni and J.Tirao, Matrix valued spherical functions associated to the complex projective plane, (J. Funct. Anal. 188(2002), 350--441) indicates that there is an interesting relation to matrix-valued orthogonal polynomials, a subject which is also studied in its own right. The purpose of the research proposal is to develop a theoretical framework for matrix-valued spherical function using Banach space representations, where the Banach space has the additional structure of a Hilbert C*-algebra. Hilbert C*-algebras behave similarly to Hilbert spaces, but there are essential differences. For such representations one can develop analogues of induced representations and Frobenius reciprocity, and the goal is to identify matrix-valued spherical functions as generalized matrix elements in this context.
Apart from developing a general theory, there are two important related subprojects. The first is to study specific examples in this context, and to relate this interpretation to specific results, e.g. orthogonality, recurrence, addition theorems, etc, for the matrix-valued special functions involved. Secondly, to generalize this to the level of quantum groups using either the notion of quantum subgroups or the notion of co-ideals. This will require new ideas for Banach space valued corepresentations of quantum groups.
The research project is related to research interests of other members, in particular Gert Heckman, of the Mathematical Physics group of IMAPP. Erik van den Ban (U Utrecht, Lie theory) and Walter Van Assche (KU Leuven, special functions) are also members of the research team.

More information

In case you are interested, or if you want to learn of a more detailed research description, for either project, please contact: 
Erik Koelink
e_dot_koelink@math_dot_ru_dot_nl
(+31) (0)24 3652597

Deadline for applications

Please send your application (including your resume and references (at least one)) before February 11, 2008, to Erik Koelink at the email address mentioned above. In case you have not yet obtained a MSc-degree, but expect to do so shortly, you can also apply. Please indicate clearly in which project you are interested.