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: 2006
• 资助国家: 加拿大
• 起止时间: 2006-01-01 至 2007-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=339618
• 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.

用于研究复杂系统的研究最广泛和富有成果的技术之一是利用系统中的任何潜在的对称性，因为这些对称性影响系统以深刻的方式行事。对称人形成了一个称为一个组的数学对象，该组充当了正在研究的系统各个重要方面的转换。通过这样的行动，可以使用希尔伯特空间上的单一操作员的抽象理论（与感兴趣的方面相关的矢量空间）来形成和研究该组的代表。在这个项目中，我们利用这一一般理论来研究两个类别的问题，这两个问题都有可能有趣的应用。在一项工作中，我们使用翻译和扩张对称性来照亮小波分析的基础结构，这是一种用于存储，压缩，改进和分析信号和图像的新兴技术。作为一种特殊应用，我们将将结果应用于电子诊断工具，用于研究视网膜条件的诊断工具。小波分析将用于开发可能有助于眼科医生发现新兴条件的工具。在另一个流中，我们将研究与碳纳米管相关的对称组的表示理论。这些奇妙的大分子正在世界各地的许多实验室中合成，并且在纳米技术的新科学中，它们有望成为重要的基础。重要的是要知道某些物理特性（例如电导率）何时为了给定的管形形状。此类属性受到情况的对称性控制或至少受约束。与感兴趣的化学家合作，我们计划促进对这些物理特性如何出现的理解。