Dynamics, Geometry, and Operator Algebras

AbstractKerrThe计划中提出的建议涉及一系列问题，旨在加深和扩大最近发现的和建立良好的动力学，metricgeometry和功能分析之间的联系。C*-代数理论将为各个组成部分提供基本的参考点和交汇点，其主要目标之一是推动C*-动力学的抽象理论的发展，该理论本着拓扑动力学的精神，将研究系统的长期行为（如熵和混合）视为其主要对象。这个项目将通过建立与Banach空间的局部理论的联系和新颖地利用C*-代数分类理论的扰动技术来进行，以期在非交换度量和微分几何，C*-代数结构和表示理论中应用，混沌概念的一种表现方式是通过许多复杂系统所表现出的不可预测性，它们会随着时间而进化这种不可预测性可以通过几种不同的方式在数学上精确化，这取决于人们想要强调的确切行为类型，例如混合或对初始条件的敏感性。确定各种现象之间的关系所涵盖的和有关的长期混沌，包括最显着的熵，是一个重大的正在进行的项目在理论的动力系统，并有实际利益，例如在努力衡量准确性的计算机模拟。虽然存在大量的结果，用于模拟经典物理的系统类型，该理论的发展要少得多的量子系统的数学描述的算子代数。该建议的广泛目标是打开新的视角在量子动力学的长期行为，并在同一时间，通过功能分析工具阐明经典动力学现象。

AbstractKerrThe program set forth in the proposal involves a nexus of problems aimed at deepening and broadening both recently discoveredand well established connections between dynamics, metricgeometry, and functional analysis. The theory of C*-algebras will provide the basic reference point and meeting ground for the various components.One of the main objectives is to push forward thedevelopment of an abstract theory of C*-dynamics which, in the spirit of topological dynamics, sees as its primary object of study the long-term behavior of systems as captured by properties like entropy and mixing. This project will be pursued by building on links to thelocal theory of Banach spaces and making novel use of the perturbation techniques of C*-algebra classification theory, with a view towards applications in noncommutative metric and differential geometry, C*-algebra structure and representation theory, and operator spacegeometry.One way in which the notion of chaos manifests itself is through the unpredictability exhibited by many complex systems as they evolve with time. This unpredictability can be made mathematically precise in several distinct ways depending on the exact kind of behavior one wantsto emphasize, such as mixing or sensitivity to initial conditions. Determiningthe relationships between the various phenomena encompassed by and related to the term chaos, including most notably entropy, is a major ongoing project in the theory of dynamical systems, and has practical interest for example in the effort to gauge the accuracy of computer simulations. While there exists a vast array of results for the type ofsystems which are used to model classical physics, the theory is much less developed for quantum systems as described mathematically by operator algebras.The broad aim of the proposal is to open up new perspectives on long-termbehavior in quantum dynamics and at the same time to shed new light on classical dynamical phenomena by means of functional-analytic tools.