My main scientific research areas are abstract harmonic analysis, wavelet analysis and spectral problems arising in chemistry. This research is funded by NSERC with my grant currently at $18,000 per year.
Abstract Harmonic Analysis: This theory is a common generalization of the theory of the Fourier Transform and the Representation Theory of Finite Groups. The theory is generally concerned with analysis on locally compact groups which usually arise as the symmetries of some physical situation. Hence there is a rich interplay between abstract harmonic analysis and areas of physics and chemistry. My abstract work has mainly concentrated on the structure of mathematical objects (irreducble representations, dual spaces, operator algebras) that are constructed to help us analyze the groups of interest. I have had a long standing research collaboration with Prof. E. Kaniuth, University of Paderborn, Germany in this area. This collaboration has been funded by two NATO Collaborative Research Grants, the German Research Foundation and my NSERC grant.
Wavelet Analysis: Around 1985, a new approach to analyzing, storing, denoising, and compressing signals and images emerged. Since then, both the theory and applications of wavelet analysis have developed rapidly with a tremendous impact on almost every area of science, engineering and medicine that involves the manipulation of signals or images. As an example, the FBI collection of fingerprints are now compressed, stored and recovered, as necessary, with a wavelet based algorithm. It turned out that my knowledge in abstract harmonic analysis had an immediate application to the fundamental theory of wavelets.
Spectral Problems Arising in Chemistry: Large matrices with a special structure can be associated with families of hydrocarbon molecules and similar matrices can be associated with certain chemical reaction systems. Kenichi Fukui initiated a research project to study problems connected to the spectra of these matrices. His student, Shigeru Arimoto, joined our Mathematical Chemistry Research Unit and we have been working together since then. As we developed novel tools for the problems from quantum chemistry, we realized these same tools could be applied to study the spectra of large Töplitz matrices, which are of considerable interest in several areas of pure and applied mathematics. This is a satisfying illustration of the advantages of interdisciplinary research. The Mathematical Chemistry Research Unit was founded by myself and Prof. Paul Mezey after I became involved in some chemistry related problems.