Mathematical Sciences: Operator Algebras

9401252 Jorgensen教授将继续研究变形理论和量子群对偶中代数的表示和调和分析。重点将放在识别正能量表示上，放在希尔伯特空间中的算子的处理方法上。这些问题包括：变形变化下同构类的稳定性，以及稳定性区域内C*-代数的变化作为域的连续性。他还将研究n维有限测度欧几里得空间中Borel集合的谱对偶性；更一般地讲，关于算子代数的对偶理论，例如，关于周期椭圆算子。在调和分析中发现谱对的同构不变量是具有自对偶的仿射分形。Jorgensen计划对此进行研究，因为自对偶似乎也是几何迭代理论中出现的其他分形对象的重要工具。这些方法包括最近发现的（由Jorgensen与S. Pedersen共同研究的）Cuntz-C*-代数的对偶表示。这个项目涉及到多维小波发生器。小波方法是一个新的数学领域，近年来受到了自然科学家、工程师和信号处理专家的广泛关注。在新的数学学科中，它的独特之处在于在如此多的不同领域提供了如此多的直接和中期应用。例如，有了小波，人们现在可以期望在蜂窝电话、高清电视和高保真音乐再现等实际应用中，传输更清晰、频带更多。小波似乎模仿了像眼睛和耳朵这样的生物过程，因为它们集中在边界或边缘上。因此，它们可能是神经网络的自然界面，神经网络是计算机世界的大脑模仿者。虽然小波理论作为一门科学学科“诞生”于1988年，但它的根源在于地球物理学家Jean Morlet的工作，他设计了一种特殊的小波类工具来帮助寻找地下石油矿藏。所涉及的问题是一个熟悉的逆散射问题。这就是乔根森在算符理论和谐波分析方面的研究可以应用的地方。＊＊＊

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. ***