Ergodic Theory and Additive Combinatorics

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ABSTRACTThe investigator proposes a program of research, education, and outreachon problems in combinatorial ergodic theory.The setting is the study of the long termbehavior of systems whose dynamics is too complicated or toochaotic to be understood locally. Dynamical systems areused to model problems in natural and social sciences and their studyleads to an understanding of the evolution of the modeled phenomena.The proposed research builds on results obtained over the past few years in multiple recurrence and convergence, The PI has proven the existence of certain multipleergodic averages and plans to study several related averages. Nilpotent groups and the dynamical systems that can be defined on their homogeneous spaces play a key role in understanding the mixing properties of dynamical systems. This area of ergodic theory has strong relations to problems in additive combinatorics and the PI will continue to explore these deep links.