Operator Methods in Fractal Analysis, Wavelets and Dynamical Systems (06w5027)

Arriving in Banff, Alberta Saturday, December 2 and departing Thursday December 7, 2006


(University of Oslo)

(The University of Iowa)

David Kribs (University of Guelph)

(Louisiana State University)

(Lund University)


This workshop is aimed at developing new approaches and mathematical foundations for wavelet analysis, dynamical and iterated function systems, spectral and tiling duality, fractal iteration processes and non-commutative dynamical systems. The basic methods involved in this work derive from a combination of operator theory, harmonic analysis and representation theory.

The proposed workshop program is in mathematics. However, the issues to be studied in the program originate both from within mathematics and from observed natural phenomena and the engineering practice. The interplay and unified approaches to these significantly interrelated areas of mathematics is of great significance both for mathematics itself and its connections to other subjects and applications.

Wavelet theory stands on the interface between signal processing, operator theory, and harmonic analysis. It is concerned with the mathematical tools involved in digitising continuous data with a view to storage and compression, and with the synthesis process, recreating the desired picture (or time signal) from the stored data. The algorithms involved go under the name of filter banks, and their spectacular efficiency derives in part from the use of hidden self-similarity in the data or images that are analysed. Self-similarity is built into wavelets too as they are intrinsically defined using dynamical and iterated function systems. This makes wavelets also closely related to fractals and fractal processes. Investigation of this relation has huge theoretical and practical potential and thus it is becoming a subject of growing interest both in and outside mathematics. It has been recently shown (by Palle Jorgensen, Ola Bratteli, David Larson, X. Dai and others) that a unifying approach to wavelets, dynamical systems, iterated function systems, self-similarity and fractals may be based on the systematic use of operator analysis and representation theory.

The research community with interests in aspects of the workshop is large. Among the suggested participants there are a substantial number of distinguished senior and junior researchers both with theoretical and applied backgrounds. This mix will ensure a long-term impact of the program on the area, and on the mathematics and its applications as a whole. The research agenda is full of interesting problems of common interest to most of the participants. We expect therefore, as an important outcome of the workshop, the appearance of a number of joint publications and the creation of several new collaborative research projects and initiatives involving different groups of participants.