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

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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.