Banach Algebras in Abstract Harmonic Analysis

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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.