Recurrence for Measure Preserving Systems and Infinitary Ramsey Theory

University Of Memphis, Memphis TN

Investigators

Abstract

Proposal Number: DMS- 0200700 PI: Randall McCutcheon ABSTRACT This project focuses on multiple recurrence in ergodic theory, with applications to combinatorial number theory, specifically density Ramsey theory. The analysis of the returns of positive measure sets under measure preserving actions along combinatorially benign trajectories, such as the ranges of polynomials or IP-systems, has led to a variety of strong theorems not accessible by other methods. Theoretical dynamics originated with the investigation of the long term average behavior of physical systems evolving over time, and abstract ergodic theory grew out of such developments. Study of the applicability of this theory to seemingly unrelated, non-physical entities, such as those arising in combinatorics, strengthens our sense of the connectedness of and the unification of the landscape in the whole of mathematics.