C*-algebras Associated to Minimal and Hyperbolic Dynamical Systems

数学家使用不变量来研究和分类高度抽象的对象。一个有用的不变量的关键属性是它是可计算的，并且它可以区分许多不同的对象。动力系统是一类重要的数学对象，其不变量的构造是非常困难的。这个项目使用抽象算子代数（特别是C*-代数）研究动力系统的不变量。特定的不变量是相关算子代数的K-理论，而这又是相关动力系统的不变量。该项目将涉及早期职业研究人员，研究生和本科生的重要贡献，他们将通过参与研究而受益于培训。主要研究人员将研究由极小和双曲动力系统构造的C*-代数的Elliott不变量的范围，该范围由K理论和迹信息组成。这项研究是研究人员以前工作的继续。然而，除了研究作为抽象分次群的K-理论之外，研究者还将研究K-理论的序结构和相关C*-代数的迹。此外，这项研究超越了一致双曲动力系统的研究（例如，Smale空间）来研究一般的可扩展系统。具体项目包括研究奇维球面上的极小同胚、极小交叉积的K-理论的序结构、Smale空间C*-代数的精细结构以及与动力系统相关的C*-代数的HK-猜想。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估而被认为值得支持。

Mathematicians use invariants to study and classify highly abstract objects. The key properties of a useful invariant are that it is computable and that it distinguishes many different objects. For dynamical systems, an important class of mathematical objects, the construction of invariants is very difficult. This project studies invariants of dynamical systems using abstract operator algebras (in particular C*-algebras). The specific invariant is the K-theory of the relevant operator algebra, which in turn is an invariant of the relevant dynamical system. The project will involve significant contributions from early-career researchers, graduate students, and undergraduate students, who will benefit from training through research involvement.The principal investigator will study the range of the Elliott invariant, which consists of K-theory and tracial information, for C*-algebras constructed from minimal and hyperbolic dynamical systems. This research represents a continuation of the investigator’s previous work. However, in addition to the study of K-theory as an abstract graded group, the investigator will also study the order structure on K-theory and the traces of the relevant C*-algebras. Furthermore, this research goes beyond the study of uniformly hyperbolic dynamical systems (e.g., Smale spaces) to study general expansive systems. Specific projects include the study of minimal homeomorphisms on odd dimensional spheres, the order structure on K-theory for minimal crossed products, the finer structure of Smale space C*-algebras, and the HK-conjecture for C*-algebras associated to dynamical systems.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.