The Interplay of Ergodic Theory, Additive Combinatorics, and Harmonic Analysis

调查人员提出了一个研究计划的问题在于交叉的egrodic理论，添加剂组合数学，和谐波分析。这一领域的一个基本问题是了解系统的长期行为，这些系统的动态太复杂或太混乱而无法局部理解。具体来说，PI建议建立在过去的结果在多重递归和收敛，进一步理解连接到幂零群和动力系统，可以定义在其齐次空间（零系统）。这样的系统在理解多个遍历平均的极限行为方面发挥着关键作用，PI建议进行研究以扩展我们对它们作用的认识。所提出的研究是在几个相互关联的数学分支的接口。 虽然大多数提出的问题都在遍历理论范围内，但这一领域与组合学、数论和调和分析中的问题有着密切的关系。PI将继续探索这些深层联系，发展遍历理论在这些其他领域的应用，并利用这些其他领域的最新进展来解决遍历理论中的问题。 PI积极参与会议组织、指导和建议。她计划继续参与教育领域的研究人员与链接到遍历理论，组织跨学科会议和讲座遍历理论的研究人员在数论，组合学和谐波分析的相关领域。PI将继续与西北大学的研究生和本科生课程合作，指导本科生和研究生进行研究，并在西北大学的数学，统计和应用数学系内为各级女性开设辅导组。

The investigator proposes a research program in problems that lie at the intersection of egrodic theory, additive combinatorics, and harmonic analysis. A basic problem in this area it to understand the long term behavior of systems whose dynamics is too complicated or too chaotic to be understood locally. Specifically, the PI proposes to build on past results in multiple recurrence and convergence, further understanding the connections to nilpotent groups and the dynamical systems that can be defined on their homogeneous spaces (nilsystems). Such systems play a key role in understanding the limiting behavior of multiple ergodic averages and the PI proposes research to extend our knowledge of their role. The research proposed is at the interface of several interrelated branches of mathematics. While most of the problems proposed are within ergodic theory, this area has strong relations to problems in combinatorics, number theory, and harmonic analysis. The PI will continue to explore these deep links, developing applications of ergodic theory to these other areas and making use of recent advances in these other areas to address problems within ergodic theory. The PI is actively involved in conference organization, mentoring, and advising. She plans to to continue being involved in education of researchers in fields with links to ergodic theory, organizing interdisciplinary meetings and lecturing on ergodic theory for researchers in the related fields of number theory, combinatorics, and harmonic analysis. The PI will continue work with the graduate and undergraduate programs at Northwestern University, directing undergraduates and graduates in research, and running a mentoring group for women at all levels within the Mathematics, Statistics, and Applied Mathematics departments at Northwestern University.