Mathematical Sciences: Topics in Modern Analysis

Rieffel将继续他对C*-代数量子化的研究。学习的主题包括量子化和量子群，量子拓扑和微分几何的方面，以及李群上的调和分析。还将注意与量子力学和场论、信号处理和模式识别等领域的应用之间的关系。这个项目的一般数学领域以希尔伯特空间算子的代数理论为基础。运算符可以被认为是复数的有限或无限矩阵。特殊类型的运算符通常放在一个代数中，自然称为运算符代数。这些抽象对象具有令人惊讶的各种应用。例如，它们在纽结理论中扮演着关键的角色，而纽结理论目前正被用于研究DNA的结构，并且它们在非对易几何中具有基本的重要性，这在物理学中变得越来越重要。

Rieffel will continue his investigation of C*-algebraic quantization. Topics to be studied include quantization and quantum groups, aspects of quantum topology and differential geometry, and harmonic analysis on Lie groups. Attention will also be given to relationships with applications in such areas as quantum mechanics and field theory, and signal processing and pattern recognition. The general area of mathematics of this project has its basis in the theory of algebras of Hilbert space operators. Operators can be thought of as finite or infinite matrices of complex numbers. Special types of operators are often put together in an algebra, naturally called an operator algebra. These abstract objects have a surprising variety of applications. For example, they play a key role in knot theory, which in turn is currently being used to study the structure of DNA, and they are of fundamental importance in noncommutative geometry, which is becoming increasingly important in physics.