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

动力系统的研究是对演化系统的长期行为的研究。现代动力系统理论起源于世纪末，它以太阳系的稳定性和演化为基本问题，迅速发展到物理学、生物学、气象学、经济学等领域。这个项目是通过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。该项目试图使用一些可计算的数据来确定动力系统中某些C *-代数的结构。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生成的数据。该项目的成功将揭示动力系统理论与C~*-代数理论之间的深层内在联系，并为这些理论的进一步应用铺平道路，也可以说是研究简单顺从C~*-代数分类在拓扑动力系统研究中的理论应用。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，并开发新的通用方法来计算可分离顺从C *-代数的K-理论相关群。设G是作用在紧度量空间X上的群. X上连续函数的代数与群作用一起生成交叉积C *-代数。该项目将研究的一个具体问题是确定两个这样的行动是渐近共轭。密切相关的问题包括C *-代数上的自同构的研究。本文提出用K-理论和KK-理论以及迹信息来刻画群作用。进一步丰富了单顺从C~*-代数分类理论的方法。此外，新的桥梁动力系统和C *-代数理论之间将建立。该项目还将研究某些简单C *-代数的不可约表示。长期目标是通过本项目和相关研究以及其他研究人员的相关研究，提供C *-代数理论及其与动力系统研究的相互作用的一些更深入的理论理解和更广泛的应用，以各种其他相关研究领域，如遍历理论，非交换同伦理论，抽象调和分析，该奖项反映了NSF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.