Mathematical Sciences: Operator Algebras

小行星9401252 乔根森教授将继续他的陈述和变形理论和量子群对偶代数所产生的谐波分析的研究。重点将是确定正能量表示，在希尔伯特空间中的运营商的处置方法。这些问题包括：同构类的稳定性下的变化的变形和连续性作为领域的C*-代数的变化范围内的稳定域。他还将致力于光谱对偶的博雷尔集在n维欧氏空间的有限措施;和，更一般地说，对偶理论的算子代数，例如，周期椭圆算子 谱对的同构不变量在调和分析中是具有自对偶性的仿射分形。 乔根森计划研究这一点，因为自对偶性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（Jorgensen与S. Pedersen）对偶。这个项目涉及多维小波发生器。小波方法是近年来引起自然科学家、工程师和信号处理专家广泛关注的一个新的数学领域。在新的数学学科中，它是独一无二的，在许多不同的领域提供了如此多的即时和中期应用。有了小波，人们现在可以期待，例如，更清晰的传输和更多的频带在这样的脚踏实地的应用，如蜂窝电话，高清晰度电视和高保真音乐再现。波纹似乎模仿生物过程，如眼睛和耳朵，因为它们专注于边界或边缘。因此，它们可能是神经网络的天然接口，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是在1988年“诞生”的，但它起源于地质学家Jean Morlet的工作，他设计了一种特殊的小波类型的工具来帮助寻找地下石油矿床。所涉及的问题是一个熟悉的逆散射问题。这一点上，乔根森的研究在运营商理论和谐波分析可以应用。 ***

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. ***