Jacob Feldman - US grants

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, and the ultimate goal is to determine long-term behavior. Professor Feldman is a leader in ergodic theory whose research has been highly influential. His recent work has involved rigidity of flows on negatively curved manifolds, positive entropy for certain flows on Euclidean space, non- measure-preserving transformations and random walks, and amenable group actions and consequences for von Neumann algebras of operators. The current proposal continues this ongoing research. In particular, Professor Feldman will study problems related to horocycle and geodesic flows, Kakutani equivalence, recurrence for nonstationary processes, and ordered pairs of ergodic equivalence relations.

9500803 Feldman This project involves many facets of work on probability theory and ergodic theory. The research includes work on the following projects: A classification of filtrations of stochastic processes; Zero-one asymptotic properties of averages of functions transformed under ergodic measure preserving transformations; Cofiltrations and ergodic group actions; Reparametrization for Euclidean group actions and for foliations; Ergodic properties of multiplication on the circle. Collectively, these projects cover a broad portion of the current ergodic theory research landscape. Ergodic theory in general concerns understanding the average long-term behavior of systems that are too complicated or chaotic to be followed in real time. Under the heading of "dynamics" can be placed the modern theory of how repeated applications of transformations effect measurable quantities. In this way, groups of transformations and abstract measure theory form the mathematical cornerstones of ergodic theory. This proposal involves the leading edge of the mathematics of ergodic theory. ***