Dynamics with a combinatorial flavor

Overview: The investigator proposes a program of research, education, and outreach on problems in dynamics, broadly falling in two categories: ergodic, studying problems of convergence and recurrence, and topological, studying properties of sub- shifts and complexity. Throughout, the motivations are of a combinatorial flavor, inspired by questions in arithmetic combinatorics and computer science. The research goals are a deeper understanding of the connections among these fields and the outreach goals are to broaden the cohort of researchers working in these areas. Intellectual Merit: A basic problem in this area is 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 PI proposes building on past results symbolic and topological dynamics, further understanding 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. Broader Impacts: The outreach portion of the project involves conference organization, mentoring, advising, and public speaking. The PI is actively involved in education of researchers in fields with links to dynamics, organizing interdisciplinary meetings and lecturing for researchers in related fields, and in education of the general public, giving expository lectures aimed at non-mathematicians. The PI will continue working 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.