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

动力系统、C* 代数理论和 K 理论

基本信息

批准号：
1954600
负责人：
Huaxin Lin
金额：
$ 18万
依托单位：
University of Oregon Eugene
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-07-01 至 2023-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954600&HistoricalAwards=false
关键词：
Dynamical Systems Algebra Theory

项目摘要

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.

动力学系统的研究是研究不断发展的系统的长期行为。动力系统的现代理论起源于19世纪末，其基本问题涉及太阳系的稳定性和演变，该问题迅速导致了对物理，生物学，气象学，经济学，经济学和其他领域的应用的发展。该项目是通过C* - 代数理论对某些拓扑动力学系统的数学分析。 C*代数是无限的尺寸线性代数。该项目试图使用一些可计算数据来确定由动态系统引起的某些C* - 代数的结构。该项目还研究了密切相关的C*代数理论，该理论将用于动态系统生成的数据。该项目的成功将揭示动力系统理论与C* - 代数理论与这些理论的进一步应用的方式之间的深厚内部关系。该项目还可以描述为对简单合理的C*-Algebras分类对拓扑动态系统进行分类的理论应用。该项目的核心目标是使用相关数据来分析最小动力学系统的结构，并开发新的通用方法来计算可分离的可分开的C*-Algebras的K理论相关组。令G为作用于紧凑的度量空间X的组。X上连续函数的代数与组动作一起生成交叉产物C*-Algebra。该项目将研究的一个具体问题是确定何时渐近地结合了两个这样的动作。密切相关的问题包括研究C* - 代数上的自动形态。建议使用K理论和KK理论以及曲折信息来表征小组动作。在简单的c* - 代数分类理论中开发的方法将进一步丰富。此外，将建立动态系统与C* - 代数理论之间的新桥梁。该项目还将研究某些简单的C* - 代数的不可约描述。提议者的长期目标是通过该项目和相关的研究以及其他研究人员的相关研究提供一些更深入的理论理解和更广泛的c* - 代数理论的应用，以及与其他相关研究领域的动态系统的相互作用，以及与其他相关研究领域的相互作用，例如脱节理论，诸如非官方理论，非传授同型统计学分析，摘要和绘制的统计学分析，以及摘要的物理学和物理学，以及物理学和物理学。使用基金会的知识分子优点和更广泛的审查标准，被认为值得通过评估来支持。