Topology and Measure in Dynamics and Operator Algebras

现代分析的结构基础中的三个基本成分及其与理论物理的联系是测度、拓扑和群的概念。其中第一个涉及体积和大小的概念，第二个涉及邻近和收敛，第三个涉及对称和空间或时间位移的概念。当它们结合在一起时，就形成了动力系统的主题，在其经典起源中，动力系统模拟了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的转换的广泛现象。该项目将追求新的措施和拓扑结构之间的关系，最近开始出现在这个动态的框架，并紧密联系到相关领域的算子代数。我们的目标是使用这种观点来发展新的方法来理解一个给定的动力系统的特定对称性如何可能条件不同类型的渐近行为，从确定性到混沌。虽然拓扑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，在过去的几年里，这种共生关系不仅通过令人惊讶的深远类比的阐述，而且还通过发现冯诺依曼代数技术在C* 上的新应用而重新活跃起来。代数与结构深刻的后果。该项目的中心目标是在动态方面重新想象这些类比和技术。这一方面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-代数的努力的一部分。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。

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.