Multiple Pointwise Ergodic Theorems

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Ergodic theory studies long-term behavior in dynamical systems from a statistical point of view. It is a large and rapidly developing area of mathematics, which has a profound impact on the development of additive combinatorics, number theory, and Fourier analysis, among other fields. The primary focus of this project is on understanding phenomena of norm and pointwise convergence for multiple polynomial ergodic averages that naturally arise at the interface of analysis and ergodic theory. A broad class of questions in additive combinatorics will also be investigated. Throughout the duration of the research program, the project will contribute to the training of undergraduate and graduate students and of postdoctoral fellows. The PI will also promote mathematics to the broader community and encourage the participation of underrepresented groups.One of the major open problems in pointwise ergodic theory is the Furstenberg-Bergelson-Leibman conjecture, which asserts that the multiple polynomial ergodic averages converge pointwise almost everywhere. The PI will develop tools in Fourier analysis and additive combinatorics allowing to study pointwise convergence for multiple polynomial ergodic averages corresponding to polynomials with distinct degrees. In addition, the PI will investigate polynomial Szemeredi's type theorems in topological fields and multidimensional integer grids. At the heart of these investigations are the so-called inverse theorems from higher order Fourier analysis.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.