Dynamical Systems, C*-Algebra Theory, and K-Theory

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The study of dynamical systems is the study of the long-term behavior of evolving systems. The modern theory of dynamical systems originated at the end of 19th century with fundamental questions concerning the stability and evolution of the solar system, which rapidly led to developments of applications to physics, biology, meteorology, economics and other areas. This project is a mathematical analysis of certain topological dynamical systems via C*-algebra theory. C*-algebras are infinite dimensional linear algebras. The project is an attempt to use a few computable data to determine the structure of certain C*-algebras arising from the dynamical systems. The project also studies closely related C*-algebra theory which will be used in the computation of data generated by dynamical systems. The success of the project would reveal the deep internal relationship between the theory of dynamical systems and theory of C*-algebras and pave the way for further applications of these theories.The project may also be described as a study of theoretical applications of the classification of simple amenable C*-algebras to the study of topological dynamical systems. The central goal of the project is to use K-theory related data to analyze the structure of minimal dynamical systems and to develop new general methods to compute K-theory related groups for separable amenable C*-algebras. Let G be a group acting on a compact metric space X. The algebra of continuous functions on X together with the group action generate a crossed product C*-algebra. One specific problem that the project will study is to determine when two such actions are asymptotically conjugate. Closely related problems include the study of automorphisms on C*-algebras. It is proposed to use K-theory and KK-theory as well as tracial information to characterize the group actions. Methods developed in the theory of classification of simple amenable C*-algebras will be further enriched. Moreover new bridges between dynamical systems and C*-algebra theory will be built. The project will also study irreducible representations of certain simple C*-algebras. It is the proposer's long term goal to provide, from this project and related research as well as via related studies by other researchers, some deeper theoretical understandings and wider applications of C*-algebra theory and its interplay with the study of dynamical systems to various other related research areas, such as ergodic theory, non-commutative homotopy theory, abstract harmonic analysis, as well as physics and biology.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.