Banach Algebras in Abstract Harmonic Analysis

抽象调和分析中的巴纳赫代数

• 批准号: RGPIN-2014-05514
• 负责人: Forrest, Brian
• 金额: $ 1.68万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2014
• 资助国家: 加拿大
• 起止时间: 2014-01-01 至 2015-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=567218
• 关键词: Banach; Algebras; Abstract; Harmonic; Analysis

The field of abstract harmonic analysis is a blend of two important branches of mathematics: classical Fourier analysis and the theory of operator algebras. Fourier analysis gets its name from the French mathematician Joseph Fourier. In addition to being one of the foremost mathematicians of the 19th century, Fourier was an engineer in Napoleon's army who's work had a profound impact on not only theoretical problems in mathematics and physics but was also the cornerstone of many future applications. His most famous work was "The Analytical Theory of Heat" (1822) where he showed that the manner in which heat was conducted through a solid body could be analyzed by breaking down the problem into its fundamental component parts in much the same way as a sound wave can be reproduced by identifying its core harmonics. The mathematician Lagrange, who predated Fourier by roughly thirty years, used similar methods to analyze the behaviour of a vibrating string. (Hence the name "harmonic analysis".) In fact, his theory could be used to study many phenomena that involved periodic or wave-like behaviour. Today variants of this sort of analysis are used in diverse applications ranging from the encoding of images and audio on CDs and DVDs through to finger print readers used by law enforcement, and image reconstruction via CT scans. The study of ''operators'' on higher, or even infinite dimensional analogs of our three dimensional world provides the mathematical foundation for quantum mechanics, a branch of physics that generally relates to behaviours that are essentially not observable by the naked eye. Instead they occur at the atomic or subatomic level. Quantum mechanics is a beautiful theory that can provide us with incredible insight into the workings of the most complex parts of our universe. In addition, it is certainly not without its practical applications. The discovery of transistors and the development of modern lasers both have their roots in quantum theory. Quantum mechanics also has the potential for further revolutionary applications that could change how our world works. For example, we are now on the cusp of the development of quantum computers. These are machines with extraordinary computational potential, dwarfing any classical computer that is currently in existence. Should they come to full fruition, quantum computer have the potential to solve problems that at this point are far beyond our current capabilities. The objects I study are Banach algebras arising from locally compact groups. The study of such objects is a natural abstraction of Fourier analysis. The key tools in my approach to studying these objects, and the groups that they stem from, comes mainly from the theory of operator algebras. While my own interests generally do not lie in the potential for immediate applications to real world problems, within the scope of my research there is the potential for such an outcome. In particular, with one of my Ph.D. students, who already holds a doctoral degree in Electrical Engineering, we will look at how the representation theory of certain locally compact groups impacts the theory of wavelets, a modern variant of Fourier's classical approach to analysis of periodic phenomena. It is the goal of my research to further our understanding of the core objects of abstract harmonic analysis. Within this context there is substantial opportunity for the training of highly qualified personnel.

抽象调和分析领域是数学的两个重要分支的混合：经典傅立叶分析和算子代数理论。傅立叶分析的名字来自法国数学家约瑟夫·傅立叶。除了是一个最重要的数学家的19世纪，傅立叶是一个工程师在拿破仑的军队谁的工作产生了深远的影响，不仅理论问题的数学和物理，但也是基石，许多未来的应用。他最著名的工作是“热分析理论”（1822年），他表明，在其中热是通过一个固体进行的方式可以分析分解成其基本组成部分的问题，在很大程度上相同的方式声波可以复制确定其核心谐波。数学家拉格朗日比傅立叶早了大约30年，他用类似的方法分析了振动弦的行为。（因此称为“谐波分析”）。事实上，他的理论可以用来研究许多涉及周期性或波动行为的现象。今天，这种分析的变体用于各种应用，从CD和DVD上的图像和音频编码到执法部门使用的指纹读取器，以及通过CT扫描进行图像重建。对我们三维世界的更高甚至无限维类似物上的“算子”的研究为量子力学提供了数学基础，量子力学是物理学的一个分支，通常与肉眼基本上无法观察到的行为有关。相反，它们发生在原子或亚原子水平。量子力学是一个美丽的理论，它可以为我们提供对宇宙中最复杂部分的运作的难以置信的洞察力。此外，它当然不是没有实际应用。晶体管的发现和现代激光器的发展都源于量子理论。量子力学还具有进一步革命性应用的潜力，这些应用可能会改变我们世界的运作方式。例如，我们现在正处于量子计算机发展的风口浪尖。这些机器具有非凡的计算潜力，使目前存在的任何经典计算机都相形见绌。如果它们能够完全实现，量子计算机有可能解决目前远远超出我们现有能力的问题。我研究的对象是Banach代数所产生的局部紧群。对这些物体的研究是傅立叶分析的自然抽象。在我研究这些对象的方法中，以及它们所源自的群的关键工具主要来自算子代数理论。虽然我自己的兴趣一般不在于直接应用于真实的世界问题的潜力，但在我的研究范围内，有可能产生这样的结果。尤其是我的一位博士学生，谁已经拥有电气工程博士学位，我们将看看如何表示理论的某些局部紧群影响小波理论，现代变种傅立叶的经典方法来分析周期性现象。这是我的研究的目标，以进一步我们的抽象谐波分析的核心对象的理解。在这方面，有大量机会培训高素质的人员。