Analytic consequences of unitary representation theory

研究复杂系统最广泛和最有成效的技术之一是利用系统中任何潜在的对称性，因为这些对称性以深刻的方式影响系统的行为。对称体形成一个称为群的数学对象，这个群充当所研究系统的各个重要方面的变换。从这样的作用出发，可以利用Hilbert空间(与感兴趣的方面相关的向量空间)上的酉算子抽象理论来形成和研究群的表示。在这个项目中，我们利用这个一般理论来研究两类问题，这两类问题都有潜在的有趣的应用。在一个工作流程中，我们使用平移和伸缩对称来说明小波分析的基本结构，小波分析是一种用于存储、压缩、改进和分析信号和图像的新兴技术。作为一种特殊的应用，我们将把我们的结果应用于视网膜电图(ERG)，这是一种研究视网膜状况的诊断工具。小波分析将被用来开发可能有助于眼科医生检测新出现的情况的工具。在另一组中，我们将研究与碳纳米管相关的对称性群的表示理论。这些令人惊叹的大分子正在世界各地的许多实验室合成，它们有望成为新的纳米技术科学的重要组成部分。重要的是要知道，对于给定形状的管子，何时某些物理性能，如导电性，将保持不变。这样的性质由情况的对称性控制，或至少受其约束。与感兴趣的化学家合作，我们计划促进对这些物理性质如何出现的理解。

One of the most widely used and fruitful techniques for the study of complicated systems is to take advantage of any underlying symmetries in the system, because these symmetries influence how a system behaves in profound ways. The symmeries form a mathematical object called a group and this group acts as transformations of various important aspects of the system under study. From such an action, a represntation of the group can be formed and studied using the abstract theory of unitary operators on a Hilbert space (a vector space related to the aspects of interest). In this project, we exploit this general theory to study two classes of problems, both of which have potentially interesting applications. In one stream of work, we are using the translation and dilation symmetries to illuminate the underlying structure of wavelet analysis, an emerging technique for storing, compressing, improving and analyzing signals and images. As a special application, we will be applying our results to electroretinagrams (ERGs) that are a diagnostic tool for the study of conditions of the retina. Wavelet analysis will be used to develop tools that could potentially assist ophthalmologists in their detection of emerging conditions. In the other stream, we will study the representation theory of the groups of symmetries associated with carbon nanotubes. These marvelous large molecules are being synthesized in a number of labs around the world and they hold promise to be important building blocks in the new science of nanotechnology. It is important to know when certain physical properties, such as conductivity, will hold for a given shape of tube. Such properties are controlled, or at least constrained, by the symmetries of the situation. Working with interested chemists, we plan on advancing understanding of how these physical properties emerge.