Mathematical Sciences: Topological Dynamics

遍历理论是现代分析学的一个活跃的中心领域。它起源于19世纪末经典统计力学的理论表述。现代的观点源于深刻的数学理论的递归和遍历，如庞加莱，伯克霍夫，冯·诺伊曼和其他巨人在本世纪初发展。随着这门学科的发展和复杂性的不断提高，遍历理论与数学的其他分支建立了密切的关系，如动力系统、概率论、泛函分析、数论、微分拓扑和微分几何，并广泛应用于数学物理、信息论、还有计算机设计。遍历理论研究的主要对象是底层集合或空间上的转换或流动，最终目标是确定长期行为。埃利斯教授是拓扑动力学领域的权威专家，该领域涉及对连续流的研究。他是抽象的、先验的观点的大师，用他特殊的洞察力来获得深刻的、基本的结果。他最近的工作展示了如何将遍历理论纳入拓扑动力学。这种遍历性和拓扑性的等价性实现了一种很大的统一，通过这种统一，拓扑动力学中的一般定理可以直接推广到测量理论中，简化了证明，并得到了相当大的推广。在他目前的项目中，埃利斯教授建议利用他的新观点来研究遍历理论和动力系统中许多具有重大影响的重要主题。这些包括Furstenberg的泊松边界理论，Moore的遍遍定理，以及流形上各种测地流的margulis - mostow - zimmerr - ratner刚性型定理。

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.