Ergodic Theory and Additive Combinatorics

研究人员提出了一个研究、教育和探索组合遍历理论中的问题的计划。该计划的背景是研究系统的长期行为，这些系统的动力学太复杂或太复杂而不能被局部理解。动力系统被用来模拟自然科学和社会科学中的问题，它们的研究导致对所模拟现象的演化的理解。所提出的研究建立在过去几年在多重递归和收敛方面所获得的结果的基础上，PI证明了某些多重遍历平均的存在，并计划研究几个相关的平均。幂零群和可定义在其齐次空间上的动力系统在理解动力系统的混合性质中起着关键作用。遍历理论的这一领域与加法组合学中的问题有很强的联系，PI将继续探索这些深层次的联系。

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.