Dissertation Enhancement: Noncommutative Dynamical Systems

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9802696 Dadarlat This award supports a twelve month NSF Dissertation Enhancement Award for Dr. Marius Dadarlat at the University of Tokyo, Tokyo, Japan. Dr. Dadarlat's student, Mr. Nathaniel P. Brown, will work with Dr. Yasuyuki Kawahigashi, Professor in the Department of Mathematics, on a research project entitled, "Noncummutative Dynamical Systems." A dynamical system is a mathematical model through which one attempts to understand the behavior of physical phenomena. The study of classical, or commutative, dynamical systems is closely related to many aspects of modern physics including both chaos and quantum theories. The proposed research will attempt to apply classical dynamical systems theory to the noncommutative setting and will investigate a series of geometric and algebraic questions concerning crossed product C*-algebras. The first component of the research will examine when certain classes of crossed product C*-algebras are embedded into AF algebras, allowing deduction concerning the specific geometric and algebraic properties of the crossed product C*-algebra. The second component of the research will build on some of the known classification results for crossed product C*-algebras; specifically, it will attempt to determine when certain crossed products are essentially the same. Finally, the research will attempt to develop an innovative notion of topological entropy for these noncommutative systems. Entropy has played an important role in the study of commutative systems, but entropy in the noncommutative setting is relatively new and not yet well understood. The research is designed to study this facet of mathematics with the goal of achieving concrete calculations for future examination. ***