Analytic consequences of unitary representation theory

研究复杂系统最广泛使用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。对称性形成了一个称为群的数学对象，这个群充当了所研究系统的各个重要方面的变换。从这样的作用，可以形成群的表示，并使用希尔伯特空间（与感兴趣的方面相关的向量空间）上的酉算子的抽象理论进行研究。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。在一个工作流中，我们使用平移和伸缩对称性来阐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。作为一个特殊的应用，我们将把我们的结果应用于视网膜电图（ERG），这是一个诊断工具的研究条件的视网膜。小波分析将被用来开发工具，可能有助于眼科医生在他们的新出现的条件检测。在另一个流中，我们将研究与碳纳米管相关的对称群的表示理论。这些奇妙的大分子正在世界各地的许多实验室中合成，它们有望成为纳米技术新科学的重要组成部分。重要的是要知道某些物理性质，如导电性，将保持一个给定的管形状。这些属性是由情境的对称性所控制，或者至少是受情境的对称性所约束。与感兴趣的化学家合作，我们计划推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.