Banach Algebras in Abstract Harmonic Analysis

抽象调和分析领域是数学的两个重要分支：经典傅立叶分析和算子代数理论的融合。傅里叶分析得名于法国数学家约瑟夫·傅里叶。傅里叶不仅是19世纪最重要的数学家之一，他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。他最著名的作品是《热的分析理论》（1822年），他在书中指出，热通过固体传导的方式可以通过将问题分解为基本组成部分来分析，就像声波可以通过识别其核心谐波来重现一样。数学家拉格朗日比傅里叶早了大约30年，他用类似的方法分析了振动弦的行为。（因此得名“谐波分析”。）事实上，他的理论可以用来研究许多涉及周期或波状行为的现象。今天，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读取器，以及通过CT扫描进行图像重建。对更高维度，甚至无限维度的三维世界的“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，它可以让我们对宇宙中最复杂部分的运作方式有不可思议的了解。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，可以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都相形见绌。如果它们完全成熟，量子计算机有可能解决目前远远超出我们目前能力的问题。我研究的对象是由局部紧群产生的巴拿赫代数。对这些对象的研究是傅里叶分析的自然抽象。我研究这些对象的方法的关键工具，以及它们所源于的群体，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现实世界问题的潜力，但在我的研究范围内，有可能产生这样的结果。特别是，我的一个博士生，他已经拥有电气工程的博士学位，我们将研究某些局部紧群的表示理论如何影响小波理论，小波理论是傅里叶经典方法的现代变体，用于分析周期现象。进一步理解抽象谐波分析的核心对象是我的研究目标。在这种情况下，有大量的机会训练高度合格的人员。

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.