Dynamics with a combinatorial flavor

对象的变化受确定性的、与时间无关的更新规则控制的集合称为动态系统。许多动力系统过于复杂，无法局部理解，但我们可以预测这些系统长期行为的全局特性。该项目的总体目标是利用某些动力系统的性质来回答由组合学和计算机科学问题引起的问题，并对这些不同领域之间的联系有更深入的了解。该项目的研究部分将辅以指导、咨询、会议组织和为一般听众提供数学讲座。具体来说，研究者计划建立在过去的多次递归和收敛结果的基础上，进一步理解幂零群和可以在其齐次空间（幂零系统）上定义的动力系统的联系。这样的系统在理解多重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，以扩展我们对其作用的认识。PI建议在过去关于符号和拓扑动力学的结果的基础上，进一步理解位移系统的复杂性、周期性和自同构群之间的关系。虽然提出的大多数问题都在动力学范围内，但研究与组合学，数论和计算机科学中的问题有很强的关系。PI将继续探索这些深层联系，开发这些其他领域的应用程序，并利用这些其他领域的最新进展来解决动力学中的问题。

A collection of objects whose changes are governed by a deterministic, time-independent update rule is called a dynamical system. Many dynamical systems are too complicated to be understood locally, but nonetheless we can predict global properties of the long term behavior of these systems. The overall goal of this project is to use the properties of certain dynamical systems to answer questions motivated by problems in combinatorics and computer science, and to obtain a deeper understanding of the connections among such diverse fields. The research component of the project will be supplemented by mentoring, advising, conference organization and giving lectures on mathematics for a general audience. Specifically, the investigator plans building 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 PI proposes building on past results concerning symbolic and topological dynamics to further understand relations between complexity, periodicity, and automorphism groups of shift systems. While most of the problems proposed are within dynamics, the research has strong relations to problems in combinatorics, number theory, and computer science. The PI will continue 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.