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。有用的不变量的关键属性是它是可计算的并且它区分许多不同的对象。对于动力系统这一重要的数学对象来说，构造不变量是非常困难的。该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。具体的不变量是相关算子代数的K理论，它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大贡献，他们将从参与研究的培训中受益。首席研究员将研究由最小和双曲动力系统构建的 C* 代数的艾略特不变量的范围，其中包括 K 理论和迹线信息。这项研究是研究者之前工作的延续。然而，除了将K-理论作为一个抽象的分级群进行研究外，研究者还将研究K-理论的序结构和相关C*-代数的踪迹。此外，这项研究超越了均匀双曲动力系统（例如 Smale 空间）的研究，还研究了一般的膨胀系统。具体项目包括奇数维球上最小同胚的研究、最小交叉乘积的 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，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.