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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708288
• 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.

抽象谐波分析的领域是数学的两个重要分支的融合：经典傅立叶分析和操作者代数理论。 傅立叶分析以法国数学家约瑟夫·傅里耶（Joseph Fourier）的名字命名。除了成为19世纪最重要的数学家之一外，傅立叶还是拿破仑军队的工程师，他的工作不仅对数学和物理学的理论问题产生了深远的影响，而且还是许多未来应用的基石。他最著名的作品是“热的分析理论”（1822），他表明，可以通过将问题分解成其基本组件部分，与声波可以通过识别其核心谐波来重现，可以通过将问题分解为基本组件的方式来分析其热量的方式。数学家拉格朗日（Lagrange）早于傅立叶（Fourier）大约三十年，他使用了类似的方法来分析振动字符串的行为。 （因此，“谐波分析”这个名字。）实际上，他的理论可以用来研究许多涉及周期性或波浪样行为的现象。如今，这种分析的变体用于不同的应用程序中，从CD和DVD上的图像和音频到执法部门使用的手指打印读取器，以及通过CT扫描进行图像重建。 对我们三维世界的较高甚至无限的维度类似物的“运算符”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，通常与肉眼根本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。量子力学是一种美丽的理论，可以为我们提供对宇宙最复杂部分的运作的难以置信的洞察力。 此外，它当然并非没有其实际应用。晶体管的发现和现代激光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，可以改变我们的世界运作方式。例如，我们现在处于量子计算机开发的风口浪尖上。这些机器具有非凡的计算潜力，使任何目前存在的经典计算机相形见war。如果他们完全实现，量子计算机有可能解决目前远远超出我们当前功能的问题。 我研究的对象是由局部紧凑型组引起的Banach代数。对此类物体的研究是傅立叶分析的自然抽象。我研究这些物体及其构成的群体的方法中的关键工具主要来自操作者代数理论。 尽管我自己的利益通常不在于在我的研究范围内立即应用于现实世界中的问题，但仍有这样的结果。特别是我的博士学位之一。已经拥有电气工程博士学位的学生，我们将研究某些本地紧凑型组的表示理论如何影响小波理论，这是傅立叶经典方法分析周期性现象的现代变体。 我的研究目的是进一步了解抽象谐波分析的核心对象。在这种情况下，培训高素质的人员有很大的机会。