Mathematical Sciences: Operator Algebras

点击翻译按钮获取中文摘要

9401252 Jorgensen Professor Jorgensen will continue his research on representations and harmonic analysis of algebras arising in deformation theory and quantum group duality. The emphasis will be on identifying positive energy representations, putting at disposal methods from operators in Hilbert space. The problems include: stability of isomorphism classes under variation of the deformations and also continuity as fields of C*-algebras for variations within the stability domain. He will also work on spectral duality for Borel sets in n-dimensional Euclidean space of finite measure; and, more generally, on duality theory for operator algebras, e.g., for periodic elliptic operators. The isomorphism invariants for spectral pairs turn out to be affine fractals with self-dualiy in their harmonic analysis. Jorgensen plans to study this, as self-duality appears also to be an important tool for other fractal objects arising in geometric iteration theory. The methods include a recently discovered (by Jorgensen in joint work with S. Pedersen) duality for pairs of representations of the Cuntz-C*-algebras. This project involves multi-dimensional wavelet generators. The method of wavelets is a new mathematical field which, in the past few years, has attracted much attention from natural scientists, engineers, and experts in signal processing. Among new mathematical disciplines, it is unique in providing so many immediate and medium-term applications in so many diverse fields. With wavelets, one can now expect, for example, clearer transmission and more bands in such down-to-earth applications as cellular phones, high-definition television and high-fidelity music reproduction. Wavelets appear to imitate biological processes like the eye and ear in that they concentrate on borderlines or edges. Thus they are potentially the natural interface for neural nets, the brain imitators of the computer world. Though wavelet theory as a scientific discipline was "born" in 1988, it has its roots in the work of geophysicist Jean Morlet, who devised an ad hoc wavelet-type tool to help find underground oil deposits. The problem involved was a familiar problem of inverse scattering. This is the point at which Jorgensen's research in operator theory and harmonic analysis can be applied. ***