Dynamics, Groupoids, and C*-Algebras

University Of Colorado At Boulder, Boulder CO

Investigators

Abstract

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.