Combinatorial Ergodic Theory

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). This project involves a program of research in the intersections of ergodic theory and additive combinatorics. Most of the problems are within ergodic theory and are related to the analysis of the long-term behavior of systems whose dynamics are too complicated or too chaotic to be understood locally. The research component of the project builds on the principal investigator's results of the past few years in multiple recurrence and convergence and, in particular, on the new unerstanding these results have provided on the role of certain systems (nilsystems) in the analysis. This area of ergodic theory has strong relations to problems in additive combinatorics and harmonic analysis. The project will continue to explore these deep links, developing the applications of ergodic theory to such areas.The outreach portion of the project involves conference organization, mentoring, advising, and development of new courses. The principal investigator has been actively involved in education of researchers in fields with links to ergodic theory and will continue to do so, primarily by organizing interdisciplinary meetings. Ergodic theory is an area of mathematics in which probability and dynamical systems meet. As such, it is relevant to the mathematical modeling of many phenomena that arise in the physical sciences, engineering, and economics.