Topology and Measure in Dynamics and Operator Algebras

现代分析的结构基础及其与理论物理的联系中的三个基本成分是度量、拓扑学和群的概念。其中第一个涉及体积和大小的概念，第二个涉及接近和收敛，第三个涉及对称性和空间或时间移位的概念。当它们结合在一起时，就形成了动力系统的主题，在它的经典起源中，动力系统建立了物理系统的时间演化模型，但现在适用于涉及从一种状态到另一种状态的转换的广泛现象。该项目将寻求度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联系。我们的目标是利用这个观点来发展新的方法来理解给定动力系统的特定对称性如何条件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长期以来在算子代数理论中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阐述令人惊讶的深远的类比而重新焕发生机，而且还发现了冯·诺依曼代数技术在C*-代数上的新类型的应用，具有结构上的深刻后果。该项目的中心目标是以动态的方式重新想象这些类比和技术。这将一方面在拓扑动力学中形成一条新的研究路线，将有限逼近性质与平均维度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，作为了解一般类型的交叉积及其K-理论分类的努力的一部分。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.