Mathematical Sciences: Topological Dynamics

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Ergodic theory is an active, central area of Modern Analysis. Its origins lie in the theoretical formulation for classical statistical mechanics at the end of the nineteenth century. The modern point of view derives from the profound mathematical theory of recurrence and ergodicity, as developed by Poincare, Birkhoff, von Neumann and other giants earlier in this century. As the subject has evolved and sophistication continually increased, ergodic theory has acquired close relationships with other branches of mathematics, such as dynamical systems, probability theory, functional analysis, number theory, differential topology, and differential geometry, and with applications as far ranging as mathematical physics, information theory, and computer design. The principal objects of study in ergodic theory are transformations, or flows, on an underlying set or space, and the ultimate goal is to determine long-term behavior. Professor Ellis is a leading expert in topological dynamics, which involves the study of continuous flows. He is a master of the abstract, transcendental viewpoint, using his special insights to obtain deep, fundamental results. His recent work shows how to subsume much of ergodic theory under topological dynamics. This equivalence of ergodic and topological properties effects a great unification by which general theorems from topological dynamics are carried over directly to measure theoretic situations, proofs are simplified, and considerable generalization is obtained. In his current project, Professor Ellis proposes to utilize his new viewpoint to investigate many significant topics of major impact in ergodic theory and dynamical systems. These include Furstenberg's theory of Poisson boundaries, Moore's ergodic theorem, and the Margulis-Mostow-Zimmer-Ratner rigidity type theorems for various geodesic flows on manifolds.