Mathematical Sciences: Operator Algebras, Quantization and Supersymmetry

本文提出用算子代数和非交换微分几何的方法来研究量子化空间和超空间，并在新的方向上加以继续和扩展。该计划的目标是：(1)了解具有复杂几何相空间的系统（涉及有限和无限多个自由度）的量化数学；(2)研究微观世界物理学的非交换几何原理，特别是对量子引力的可能影响；(3)探索分析与其他数学领域相关的新的数学现象。非交换几何是80年代产生的一个新的数学方向。它的主要目标是研究被称为非交换空间或流形的对象，这些对象不是普通的几何对象，但仍然具有几何风味。这类结构的例子自然出现，例如在亚原子现象的物理理论中。非交换几何方法成为研究现代分析中各种问题和理解微观世界物理基础的有力工具。提出了非交换流形的研究实例，重点是在量子物理中的应用。此外，还提出将量子理论的一些思想与数论和现代分析中的问题联系起来。* * *

9500463 Klimek It is proposed to continue and expand in new directions an ongoing program of studying quantized spaces and superspaces by means of the techniques of operator algebras and non-commutative differential geometry. The goals of the program are: (1) To understand the mathematics of quantization of systems (involving both finitely and infinitely many degrees of freedom) with complicated geometry of phase space; (2) To study the non-commutative geometric principles underlying the physics of the microworld, and in particular possible implications for quantum gravity; (3) To explore new mathematical phenomena relating analysis to other fields of mathematics. Non-commutative geometry is a new direction of mathematics created in the eighties. Its prime objective is to study objects, called non-commutative spaces or manifolds, which are not ordinary geometric objects but still have a geometric flavor. Examples of structures of this sort arise naturally e.g. in physical theories of subatomic phenomena. Methods of non-commutative geometry became a powerful tool in studying various problems in modern analysis and understanding the fundamentals of the physics of the microworld. It is proposed to study examples of non-commutative manifolds with emphasis put on applications to quantum physics. Also, it is proposed to relate some of the ideas of quantum theory to problems in number theory and modern analysis. ***