Operator Algebras and Representations

Jorgensen光谱对最初是由Palle E.T. Jorgensen和Steen Pedersen在他们关于Fuglede猜想的工作中研究的，但后来发现它们也与小波理论、分形迭代、Julia集和准晶体有联系。所需的调和分析将是提案的项目1，它是基于对昆兹代数的某些表示的分析，以及冯·诺伊曼代数的自同态。这种表征理论在具体几何平铺问题的研究中有进一步的应用价值。但是所需要的表示是类型III（一般地），而早期工作中的一些方法是类型i。尽管如此，它们已经被证明可以适应无限情况。Jorgensen还计划与他的同事Florin Radulescu合作研究Berezin量化中出现的自同态的动力系统。对于非时变系统，指标理论可以帮助我们理解由Berezin量化的算子代数方法得到的非时变情况。该方案是在基础数学及其技术应用的接口。后者是指谐波分析和小波理论中应用于当前数据压缩算法的数学工具。这些应用衍生产品包括“指纹电子存储”和高分辨率电视；但这一提议还停留在理论层面。数学中的两个基本操作（平移和缩放）会进入控制算法的过滤器，递归过程中的重复是指从一步到下一步保持相同的最小数据。因此才有了“滤波器组”这个术语。令人惊讶的是，并且由提议者在最近的研究论文中注意到，这些过滤器结构可以被理解为构成提议研究基础的代数类型的表示。

Abstract Jorgensen Spectpal pairs were studied first by Palle E.T. Jorgensen and Steen Pedersen in connection with their work on the Fuglede conjecture, but they turned out later to also have connections to wavelet theory, fractal iterations, Julia sets, and quasi-crystals. The harmonic analysis which is needed will be project 1 of the proposal, and it is based on analysis of certain representations of the Cuntz algebras, and on endomorphisms of von Neumann algebras. This representation theory is further expected to be useful in connection with the study of concrete geometric tiling questions. But the representations needed are type III (generically), while some of the methods from earlier work are type I. Nonetheless they have already proved amenable for adaptation to the infinite case. Jorgensen also plans to collaborate with his colleague Florin Radulescu on dynamical systems of endomorphisms which arise in Berezin quantization. The index theory which is available for the time-independent systems is likely to help us understand the time-dependent case which is the result of the operator algebraic approach to Berezin quantization. The proposal is in the interface of basic mathematics with its technology applications. The latter refer to mathematical tools from harmonic analysis and wavelet theory which are applied in current data-compression algorithms. These application spin-offs include "fingerprint electronic storage" and high-resolution television; but the proposal is on the theoretical side of the equation. There are two basic operations from mathematics (translation and scaling) which go into the filters that govern algorithms, and the repetition in the recursive process refers to the preservation of the same minimal data from one step to the next. Hence the term "filter bank". Surprisingly, and noted by the proposer in recent research papers, these filter-constructions may be understood as representations of the kinds of algebras which form the basis for the proposed researc h.