Dynamics with a combinatorial flavor

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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.