Mathematical Sciences: Ergodic Theory of Unipotent Translations on Homogeneous Spaces

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的变换。例如，如果变换是由微分方程产生的，与时间有关，或者更普遍地根据比时间更广泛的过程发展，人们就会说到底层几何上的流动。这是现代动力系统理论的背景，其中一个基本的关键问题是均匀空间上流动的分类。拉特纳教授是研究某些几何流动及其动力学的世界领军人物之一。她在环圈流的“刚性”（根据最小数据对流进行分类）方面的工作是遍历理论和动力系统最近取得的最高成就之一。它包括迄今为止在均匀流形上的流动分类方面最重要的工作。她的作品产生了例子和反例子，并为这个主题带来了一个重要的几何观点。在目前的提案中，她计划研究更一般的，所谓的“单性流”。她希望对有限体积齐次空间上由单幂变换保存的所有Borel概率测度进行分类；研究这类变换的刚性性质；并建立了变负曲率紧致曲面的单位切束上遍历流的指数混合。

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 of an underlying set or space. If for example, the transformation results from a differential equation, is time dependent, or more generally evolves according to a more extensive process than time, one speaks of a flow on the underlying geometry. This is the context of the modern theory of dynamical systems, in which a fundamentalcritical question is the classification of flows on homogenous spaces. Professor Ratner is one of the world leaders in the study of certain geometric flows and their dynamics. Her work on "rigidity" -- the classification of flows in terms of minimal data -- of horocycle flows is one of the top recent achievements in ergodic theory and dynamical systems. It comprises the most significant work to date in the classification of flows on homogeneous manifolds. Her work has produced examples and counterexamples, and it has brought an important geometrical point of view to the subject. In the current proposal she plans to study more general, so-called unipotent, flows. She hopes to classify all Borel probability measures preserved by unipotent tranformations on finite volume homogeneous spaces; to investigate the rigidity properties of such transformations; and to establish exponential mixing for ergodic flows on unit tangent bundles of compact surfaces of variable negative curvature.