Recurrence for Measure Preserving Systems and Infinitary Ramsey Theory

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Proposal Number: DMS- 0200700PI: Randall McCutcheon ABSTRACTThis project focuses on multiple recurrence in ergodic theory, with applications to combinatorial number theory, specificallydensity Ramsey theory. The analysis of the returns of positivemeasure sets under measure preserving actions along combinatoriallybenign trajectories, such as the ranges of polynomials or IP-systems,has led to a variety of strong theorems not accessible by othermethods. Theoretical dynamics originated with the investigation of the long term average behavior of physical systems evolving over time, andabstract ergodic theory grew out of such developments. Study of theapplicability of this theory to seemingly unrelated, non-physical entities, such as those arising in combinatorics, strengthens oursense of the connectedness of and the unification of the landscapein the whole of mathematics.