Dynamics, fractals and wavelets

In 2004, Associate Professor Sergei Silvestrov, Lund University received a STINT Institutional Grant for development of cooperation, with groups at the Departments of Mathematics in Oslo University, Norway, University of Iowa, USA, Keio University, Japan and Copenhagen University, Denmark, in the area of operator non-commutative analysis of dynamics, fractals and wavelets. The project has received an annual funding of SEK 300 000.

This four-year institutional exchange program was launched in 2004 with the main objective to develop long term collaboration between Centre for Mathematical Sciences, Lund University and research groups at Departments of Mathematics in Oslo University, Norway (repr. by Prof. Ola Bratteli) University of Iowa, USA (repr. by Prof. Palle Jorgensen), Keio University, Japan (repr. by Professor Yoshiaki Maeda) and Copenhagen University, Denmark in the area of operator non-commutative analysis of dynamics, fractals and wavelets.

The cooperation and exchange 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. Wavelet theory stands on the interface between signal processing, operator theory, and harmonic analysis. It is concerned with the mathematical tools involved in digitizing 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 analyzed. 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. In the program the interplay and unified approaches are being developed to wavelet analysis, dynamical and iterated function systems, spectral and tiling duality, fractal iteration processes and commutative and non-commutative dynamical systems, based on novel methods from and interplay with operator theory and operator algebras, harmonic analysis and representation theory.

The international research community with interests in various aspects of the project is large including substantial number of distinguished senior, junior researchers and doctoral and master students both with theoretical and applied backgrounds. This mix ensures broad interest in various aspects and activities of the program, a long-term impact of the program activities on the area, and on the mathematics and its applications as a whole.

The activities and exchange within the project include

• Personal exchange of students, post-docs and senior researchers
• joint research and educational seminars
• advanced mini-courses and intensive programs and schools
• international workshops and conferences
• joint supervision and exchange of PhD students and post-docs
• preparation of joint publications
• preparation of joint applications for research funding

Among recent international events organized in connection to the project could be specially highlighted two big international conferences that took place in autumn 2006:

BIRS international conference “Operator methods in fractal analysis, wavelets and dynamical systems”, Banff international research station, Banff, Canada, with 40 participants from Sweden, other Scandinavian and European countries, Japan, USA, Australia and Canada with 30 lectures; Baltic-Nordic AGMF network Workshop “Algebra, Geometry and Mathematical Physics”, October 12-14 in Lund University, Sweden attracting 50 participants and 40 lectures from more then 12 countries. Proceedings of both conferences are in preparation for publication with international publisher. Doctoral and Master theses and joint publications have been completed in connection to this STINT project, and further joint publications and Doctoral and Master thesis are in preparation. Support from STINT provides with possibilities for significantly expanding network of international contacts and cooperation with partner institutions and beyond, leading to development of new long-term cooperation patterns and international initiatives.

Sergei Silvestrov
Lund University

Senast uppdaterad: 07-01-26 11:18