Topology and Measure in Dynamics and Operator Algebras

动力学和算子代数中的拓扑和测度

• 批准号: 1800633
• 负责人: Grigoris Paouris
• 金额: $ 18万
• 依托单位: Texas A&M University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-07-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800633&HistoricalAwards=false
• 关键词: Topology; Measure; Dynamics; Operator; Algebras

Three of the basic ingredients in the structural foundations of modern analysis and its connections with theoretical physics are the concepts of measure, topology, and group. The first of these deals with the notions of volume and size, the second with proximity and convergence, and the third with symmetry and the idea of displacement in space or time. When combined together they form the subject of dynamical systems, which in its classical origins models the time evolution of physical systems but is nowadays applicable to a wide range of phenomena involving transformations from one state to another. The project will pursue novel relationships between measure and topology that have recently begun to emerge within this dynamical framework and are tightly linked to the related field of operator algebras. The goal is to use this perspective to develop new ways of understanding how the particular symmetries of a given dynamical system may condition different types of asymptotic behavior, from the deterministic to the chaotic.While the topological and measure-theoretic perspectives have long between fruitfully intertwined in the theory of operator algebras, in the last several years this symbiosis has been reinvigorated through not only the elaboration of surprisingly far-reaching analogies but also the discovery of new kinds of applications of von Neumann algebra techniques to C*-algebras with structurally profound consequences. The central aim of the project is to reimagine these analogies and techniques in dynamical terms. This will on the one hand forge a novel line of investigation in topological dynamics that intertwines finite approximation properties with asymptotic phenomena like mean dimension, and on the other establish broader connections between group actions and C*-algebras as part of the effort to understand general types of crossed products and their K-theoretic classifiability.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.

现代分析的结构基础及其与理论物理的联系的三个基本组成部分是测度、拓扑和群的概念。第一个是关于体积和大小的概念，第二个是关于接近和收敛，第三个是关于对称和空间或时间上的位移的概念。当它们结合在一起时，就形成了动力系统的主题，动力系统在其经典起源中模拟物理系统的时间演化，但现在适用于涉及从一种状态到另一种状态转换的广泛现象。该项目将追求测量和拓扑之间的新关系，这些关系最近开始在这个动态框架内出现，并与算子代数的相关领域紧密相关。目标是利用这一观点来开发新的方法来理解给定动力系统的特定对称性如何约束不同类型的渐近行为，从确定性到混沌。虽然拓扑学和测度论的观点在算子代数理论中长期以来卓有成效地交织在一起，但在过去的几年里，这种共生关系不仅通过对令人惊讶的深远类比的阐述，而且通过对冯·诺伊曼代数技术在C*代数中的新应用的发现，重新焕发了活力，并带来了结构上深远的影响。该项目的中心目标是在动态术语中重新想象这些类比和技术。一方面，这将在拓扑动力学中形成一条新的研究路线，将有限近似性质与平均维数等渐近现象交织在一起，另一方面，在群作用和C*代数之间建立更广泛的联系，作为理解交叉积的一般类型及其k理论可分类性的努力的一部分。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。