Mathematical Sciences: Analysis of the Quasi-Regular Representation of Lie Groups and Index Theory on Non-CompactManifolds

现代科学最基本的发展之一是量子力学理论。对物理学和基础数学的理解的尝试直接导致了现代分析中两个重要领域的前沿：表示理论和算子代数。表征理论是利用固有的物理对称性的一种手段，而算子代数的概念是为了提供量化的正确框架而发明的。最近，我们对量化背后的数学的理解有了很大的扩展。这源于物理概念和数学结构之间的深刻互动。特别是，表示理论和算子代数最近被结合在一个强大的理论中，这个理论以著名的Atiyah-Singer指数定理为中心。福克斯教授是一位年轻的研究人员，他掌握了为这一领域做出贡献所必需的广泛的数学知识。他是表示理论、算子代数和指标理论方面的专家。他提出了一个非紧流形的指标定理，推广了cones对紧流形的Atiyah-Singer指标定理的推广。该指标定理将用于研究作用于有限体积局部对称空间的半简单李群的拟正则表示的离散谱。除了它对表示理论和算子代数的内在兴趣之外，解决这个问题将导致更好地理解量化的各个方面。研究者之前的工作，在一定程度上是合作的，为开展这项研究奠定了坚实的基础。

One of the most fundamental developments in modern science has been the theory of quantum mechanics. The attempt to understand both the physics and the underlying mathematics has led directly to current frontiers in two important areas of Modern Analysis, representation theory and operator algebras. Representation theory is a means of exploiting the inherent physical symmetries, while the concept of operator algebras was invented to provide the correct framework for quantization. Recently, there has been a great expansion in our understanding of the mathematics underlying quantization. This has come from deep interactions between physical ideas and mathematical constructs. In particular, representation theory and operator algebras have been brought together recently in a powerful theory that centers around the celebrated Atiyah-Singer index theorem. Professor Fox is a young researcher who has mastered the broad range of mathematics necessary to contribute to this field. He is an expert in representation theory, operator algebras, and index theory. He proposes to develop an index theorem for noncompact manifolds that generalizes Connes' extension of the Atiyah-Singer index theorem for compact manifolds. This index theorem would be used to study the discrete spectrum of the quasi-regular representation of a semi-simple Lie group acting on a locally symmetric space of finite volume. In addition to its intrinsic interest in representation theory and operator algebras, a solution to this problem would lead to a better understanding of the various facets of quantization. The investigator's prior work, in part collaborative, has laid a solid foundation for undertaking this study.