Mathematical Sciences: Ergodic Theory and Related Topics

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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. ***