Mathematical Sciences: Homological Methods in the Deformation Theory of Operator Algebras

9623314 PI：Nagy堪萨斯州立大学Nagy将研究C*代数形变量子化中的刚性现象。为了解决这些问题，Nagy计划使用和开发适合于某些C*-代数分类的同调技术。这项研究的动机之一是最近世界范围内对前景看好的新量子群理论的兴趣，该理论将数学的几个分支联系在一起，并与数学物理密切相关。这个项目的另一个动机和灵感是与非交换几何有关的C*-代数的K-理论的发展。这个项目的数学领域以希尔伯特空间上的算子代数理论为基础。算符代数的研究是由约翰·冯·诺伊曼于1930年在S开创的，目的是为量子力学提供一个数学框架。由于其广泛的应用，算子代数是目前数学中最活跃的领域之一。例如，由沃恩·琼斯在20世纪80年代的S中发现的算符代数与纽结理论的显著相互作用，导致了对DNA分子研究的有趣应用。由于算符代数在非对易几何、量子概率论和共形量子场论中的基础作用，它在物理学中变得越来越重要。

DMS-9623314 PI: Nagy Kansas State University Nagy will investigate rigidity phenomena in C*-algebraic deformation quantization. In order to address such problems, Nagy plans to use and develop homological techniques adequate for the classification of certain classes of C*-algebras. One motivation of this research is the recent worldwide interest in the promising new theory of quantum groups which links together several branches of mathematics and is closely related to mathematical physics. Another motivation and inspiration for this project is the development in the K-theory of C*-algebras in connection with non-commutative geometry. The area of mathematics for this project has its basis in the theory of algebras of operators on Hilbert spaces. The study of operator algebras was initiated in the 1930's by John von Neumann in order to provide a mathematical framework for quantum mechanics. Due to a broad range of applications, operator algebras are presently one of the most dynamic fields in mathematics. For example, the remarkable interactions of operator algebras with knot theory, discovered by Vaughan Jones in the 1980's, lead to interesting applications to the study of the DNA molecules. Due to their fundamental role in non-commutative geometry, in quantum probability theory, and in conformal quantum field theory, operator algebras have become increasingly important in physics.