Analytic consequences of unitary representation theory

单一表示论的分析结果

• 批准号: 3176-2006
• 负责人: Taylor, Keith
• 金额: $ 1.17万
• 依托单位: Dalhousie University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2008
• 资助国家: 加拿大
• 起止时间: 2008-01-01 至 2009-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=388718
• 关键词: Analytic; consequences; unitary; representation; theory

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.

研究复杂系统最广泛使用和最富有成果的技术之一是利用系统中任何潜在的对称性，因为这些对称性深刻地影响着系统的行为方式。对称性形成一个称为群的数学对象，该群充当所研究系统的各个重要方面的变换。通过这样的动作，可以使用希尔伯特空间（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有趣的应用。在一个工作流中，我们使用平移和膨胀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。作为一项特殊应用，我们将把我们的结果应用于视网膜电图 (ERG)，这是研究视网膜状况的诊断工具。小波分析将用于开发可能帮助眼科医生检测新出现疾病的工具。在另一个方向中，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特性（例如电导率）何时适用于给定形状的管子。这些属性受情况的对称性控制，或至少受到限制。我们计划与感兴趣的化学家合作，加深对这些物理特性如何出现的理解。