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

研究人员提出了一项研究计划，研究的问题位于极点理论、加法组合学和调和分析的交叉点上。这个领域的一个基本问题是理解系统的长期行为，这些系统的动力学太复杂或太混乱，以至于无法在当地理解。具体地说，PI建议建立在多重递归和收敛的过去结果的基础上，进一步了解与幂零群和可以在其齐次空间(零系)上定义的动力系统的联系。这样的系统在理解多个遍历平均值的限制行为方面起着关键作用，PI建议进行研究，以扩展我们对它们作用的了解。提出的研究是在几个相互关联的数学分支的交界处进行的。虽然提出的大多数问题都在遍历理论范围内，但这个领域与组合学、数论和调和分析中的问题有很强的联系。国际和平研究所将继续探索这些深入的联系，将遍历理论应用于这些其他领域，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。协会积极参与会议的组织、指导和咨询工作。她计划继续参与与遍历理论相关领域的研究人员的教育，组织跨学科会议，并为数论、组合学和调和分析等相关领域的研究人员讲授遍历理论。PI将继续与西北大学的研究生和本科生项目合作，指导本科生和毕业生的研究，并在西北大学数学系、统计学系和应用数学系为所有级别的女性运营一个辅导小组。

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.