Dynamics, Groupoids, and C*-Algebras

纯数学的一个关键目标是对高度抽象的对象进行分类;在此过程中，不变量可以用来区分不同的类型。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结构，确定这样的不变量是一个挑战。这个项目使用算子代数，特别是C*-代数的抽象概念来研究动力系统的不变量。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，他们将通过参与研究从培训中受益。更详细地说，给定一个动力系统，人们可以（经常）使用广群构造来构造一个C*-代数。这个C*-代数的K-理论是原动力系统的不变量。一个重要的问题是，以确定在动力系统的水平上，这个不变量的区分能力和可计算性。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，该不变量由K-理论和迹信息组成。研究者和合作者将系统地构建最小动力系统与规定的K-理论，并将计算与双曲动力系统相关的C*-代数的具体例子的K-理论称为Smale空间。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

One key goal of pure mathematics is to classify highly abstract objects; in doing so, invariants can serve to distinguish different types. The key properties of a useful invariant are that it is computable and that it distinguishes many different objects. It is challenging to determine such invariants for dynamical systems, an important class of mathematical structure. This project studies invariants of dynamical systems using the abstract concepts of operator algebras, in particular C*-algebras. The project will involve significant contributions from early-career researchers, graduate students, and undergraduate students, who will benefit from training through research involvement. In more detail, given a dynamical system one can (often) construct a C*-algebra using a groupoid construction. The K-theory of this C*- algebra is an invariant of the original dynamical system. An important question is to determine the distinguishing power and computability of this invariant at the dynamical system level. The 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. The investigator and collaborators will systematically construct minimal dynamical systems with prescribed K-theory and will compute the K-theory of specific examples of C*-algebras associated to hyperbolic dynamical systems called Smale spaces.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.