Dynamics, Geometry, and Operator Algebras

Texas A&M Research Foundation, College Station TX

Investigators

Abstract

Abstract Kerr The program set forth in the proposal involves a nexus of problems aimed at deepening and broadening both recently discovered and well established connections between dynamics, metric geometry, and functional analysis. The theory of C*-algebras will provide the basic reference point and meeting ground for the various components. One of the main objectives is to push forward the development of an abstract theory of C*-dynamics which, in the spirit of topological dynamics, sees as its primary object of study the long-term behavior of systems as captured by properties like entropy and mixing. This project will be pursued by building on links to the local theory of Banach spaces and making novel use of the perturbation techniques of C*-algebra classification theory, with a view towards applications in noncommutative metric and differential geometry, C*-algebra structure and representation theory, and operator space geometry. One way in which the notion of chaos manifests itself is through the unpredictability exhibited by many complex systems as they evolve with time. This unpredictability can be made mathematically precise in several distinct ways depending on the exact kind of behavior one wants to emphasize, such as mixing or sensitivity to initial conditions. Determining the relationships between the various phenomena encompassed by and related to the term chaos, including most notably entropy, is a major ongoing project in the theory of dynamical systems, and has practical interest for example in the effort to gauge the accuracy of computer simulations. While there exists a vast array of results for the type of systems which are used to model classical physics, the theory is much less developed for quantum systems as described mathematically by operator algebras. The broad aim of the proposal is to open up new perspectives on long-term behavior in quantum dynamics and at the same time to shed new light on classical dynamical phenomena by means of functional-analytic tools.