Dynamics with a Combinatorial Bent

该奖项支持主要研究者在动态、教育和外展问题研究项目上的工作。研究部分是研究动力学过于复杂而无法局部理解的系统的长期行为。这些问题存在于动力学中，但与许多其他数学领域有着密切的联系。教育部分包括指导本科生学习高级课题，指导研究生学习博士生，在与动态相关的领域培养研究人员，以及组织跨学科会议。拓展活动包括组织会议，为西北大学的各级女性开办指导小组，以及努力增加申请数学研究生院的女性人数。首席研究员将研究遍历理论中的问题，研究收敛、递归和等分布问题，以及拓扑动力学中的问题，研究递归、移位系统和复杂性的性质。在符号和拓扑动力学中，这将建立在过去的结果之上，以进一步理解系统的复杂性、周期性、动力学性质和自然关联群的代数性质之间的关系。在遍历理论中，这将建立在先前对多重递归和收敛的研究基础上，研究不同的平均值，等分布结果，以及具有特定性质的系统的构建。提出的问题存在于动力学中，但与其他领域的问题有很强的联系，包括组合学、数论、单词组合学和计算机科学。我们的目标是探索这些深层联系，开发与这些其他领域相关的应用程序，并利用这些其他领域的最新进展来解决动力学中的问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports the principal investigator's work on a program of research on problems in dynamics, education, and outreach. The research portion is to study the long-term behavior of systems whose dynamics is too complicated to be understood locally. The problems lie within dynamics, but have strong connections to numerous other fields of mathematics. The education portion involves directing undergraduates in advanced topics and graduate students in doctoral students, educating researchers in fields with links to dynamics, and organizing interdisciplinary meetings. The outreach involves conference organization, running mentoring group for women all levels at Northwestern, and working to increase the number of women applying to graduate school in mathematics. The principal investigator will work on problems in ergodic theory, studying questions of convergence, recurrence, and equidistribution, and problems in topological dynamics, studying properties of recurrence, shift systems, and complexity. In symbolic and topological dynamics, this will build on past results to further understand relations between complexity, periodicity, dynamical properties of the system, and the algebraic properties of naturally associated groups. In ergodic theory, this will build on previous research in multiple recurrence and convergence, with the study of different averages, equidistribution results, and building of systems with particular properties. The problems proposed lie within dynamics, but there are strong connections to problems from other fields, including combinatorics, number theory, combinatorics of words, and computer science. The goal is to explore these deep links, developing applications to these other areas and making use of recent advances in these other areas to address problems within dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.