Topics in Noncommutative Geometry, Quantization and Dynamics

主要研究者Klimek建议继续并向新的方向扩展正在进行的利用算子代数和非对易微分几何的技巧研究量子化空间和此类空间上的动力系统的计划。该计划的最终目标是：*了解具有复杂动力学的系统(涉及有限和无限多个自由度)的量子化数学；*研究微观世界物理学背后的非对易几何原理，特别是对量子混沌的可能影响；*探索将算子理论与其他数学领域联系起来的新的数学现象。非对易几何是八十年代开创的一个新的数学方向。它的主要目的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，但仍然具有几何色彩。这种结构的例子是自然而然出现的，例如在亚原子现象的物理理论中。非对易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学基本原理的有力工具。建议研究非对易流形和非对易流形上的动力系统的例子，重点是在量子物理和算子论中的应用。

AbstractKlimek The principal investigator proposes to continue and expand in new directions an ongoing program of studying quantized spaces and dynamical systems on such spaces by means of the techniques of operator algebras and non-commutative differential geometry. The ultimate goals of the program are:* To understand the mathematics of quantization of systems (involving both finitely and infinitely many degrees of freedom) with complicated dynamics;* To study the non-commutative geometric principles underlying the physics of the microworld, and in particular possible implications for quantum chaos;* To explore new mathematical phenomena relating operator theory 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, callednon-commutative spaces or manifolds, which are not ordinary geometric objects but still have a geometric flavor. Examples of structuresof 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 tostudy examples of non-commutative manifolds and dynamical systems on them with emphasis put on applications to quantum physics and operator theory.