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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=611683
• 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.

抽象调和分析领域融合了两个重要的数学分支：经典傅立叶分析和算子代数理论。 傅立叶分析因法国数学家约瑟夫·傅立叶而得名。除了是19世纪最重要的数学家之一，傅立叶还是拿破仑军队中的一名工程师，他的工作不仅对数学和物理的理论问题产生了深远的影响，而且也是许多未来应用的基石。他最著名的著作是《热的分析理论》(1822年)，他在书中指出，通过将问题分解为基本组成部分，可以分析通过固体进行热传导的方式，就像通过识别其核心谐波可以再现声波一样。数学家拉格朗日比傅立叶早了大约30年，他使用了类似的方法来分析振动弦的行为。(因此有了“调和分析”这个名字。)事实上，他的理论可以用来研究许多涉及周期性或类波浪行为的现象。如今，这种分析的变体被用于各种应用，从CD和DVD上的图像和音频编码，到执法部门使用的指纹识别器，以及通过CT扫描进行图像重建。 对我们三维世界的更高维，甚至是无限维类比上的“算符”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上。量子力学是一个美丽的理论，它可以为我们提供令人难以置信的洞察我们宇宙中最复杂部分的工作原理。此外，它当然也不是没有实际应用的。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，可能会改变我们世界的运转方式。例如，我们现在正处于量子计算机发展的尖端。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相形见绌。如果它们完全实现，量子计算机有潜力解决目前远远超出我们当前能力的问题。 我研究的对象是产生于局部紧群的Banach代数。对这类物体的研究是傅立叶分析的自然抽象。我研究这些对象的方法中的关键工具，以及它们所源自的群，主要来自于算子代数理论。 虽然我自己的兴趣通常不在于直接应用于现实世界问题的可能性，但在我的研究范围内，有可能出现这样的结果。特别是，我的一名博士生已经拥有电子工程博士学位，我们将研究某些局部紧群的表示理论如何影响小波理论，小波理论是傅里叶分析周期现象的经典方法的现代变体。 加深对抽象调和分析核心对象的理解是我研究的目的。在这方面，培训高素质人员的机会很大。