Dynamical Systems, C*-Algebra Theory, and K-Theory

动力系统的研究是对演化系统长期行为的研究。现代动力系统理论起源于19世纪末有关太阳系稳定性和进化的基本问题，并迅速发展到物理学、生物学、气象学、经济学和其他领域。本课题是利用C*-代数理论对某些拓扑动力系统进行数学分析。C*代数是无限维线性代数。该项目是一个尝试使用一些可计算的数据来确定某些C*-代数的结构产生的动力系统。该项目还研究了密切相关的C*-代数理论，该理论将用于计算动力系统产生的数据。该项目的成功将揭示动力系统理论与C*-代数理论之间深刻的内在联系，并为这些理论的进一步应用铺平道路。该项目也可以被描述为对简单可服从C*-代数分类在拓扑动力系统研究中的理论应用的研究。该项目的中心目标是使用k理论相关数据来分析最小动力系统的结构，并开发新的一般方法来计算可分可调C*-代数的k理论相关群。设G是作用于紧度量空间X上的群，X上连续函数的代数与群作用产生一个交叉积C*-代数。该项目将研究的一个具体问题是确定两个这样的作用何时是渐近共轭的。密切相关的问题包括C*-代数上的自同构的研究。提出了利用k理论和kk理论以及跟踪信息来描述群体行为的方法。在简单可调C*-代数的分类理论中发展的方法将进一步丰富。此外，将在动力系统和C*-代数理论之间建立新的桥梁。该项目还将研究某些简单C*-代数的不可约表示。作者的长期目标是通过本课题和相关研究以及其他研究者的相关研究，使C*-代数理论及其与动力系统研究的相互作用在遍历理论、非交换同伦理论、抽象谐波分析以及物理学和生物学等其他相关研究领域得到更深入的理论理解和更广泛的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The study of dynamical systems is the study of the long-term behavior of evolving systems. The modern theory of dynamical systems originated at the end of 19th century with fundamental questions concerning the stability and evolution of the solar system, which rapidly led to developments of applications to physics, biology, meteorology, economics and other areas. This project is a mathematical analysis of certain topological dynamical systems via C*-algebra theory. C*-algebras are infinite dimensional linear algebras. The project is an attempt to use a few computable data to determine the structure of certain C*-algebras arising from the dynamical systems. The project also studies closely related C*-algebra theory which will be used in the computation of data generated by dynamical systems. The success of the project would reveal the deep internal relationship between the theory of dynamical systems and theory of C*-algebras and pave the way for further applications of these theories.The project may also be described as a study of theoretical applications of the classification of simple amenable C*-algebras to the study of topological dynamical systems. The central goal of the project is to use K-theory related data to analyze the structure of minimal dynamical systems and to develop new general methods to compute K-theory related groups for separable amenable C*-algebras. Let G be a group acting on a compact metric space X. The algebra of continuous functions on X together with the group action generate a crossed product C*-algebra. One specific problem that the project will study is to determine when two such actions are asymptotically conjugate. Closely related problems include the study of automorphisms on C*-algebras. It is proposed to use K-theory and KK-theory as well as tracial information to characterize the group actions. Methods developed in the theory of classification of simple amenable C*-algebras will be further enriched. Moreover new bridges between dynamical systems and C*-algebra theory will be built. The project will also study irreducible representations of certain simple C*-algebras. It is the proposer's long term goal to provide, from this project and related research as well as via related studies by other researchers, some deeper theoretical understandings and wider applications of C*-algebra theory and its interplay with the study of dynamical systems to various other related research areas, such as ergodic theory, non-commutative homotopy theory, abstract harmonic analysis, as well as physics and biology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.