CAREER: Harmonic Analysis, Ergodic Theory and Convex Geometry

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Ergodic theory originated in the study of the statistical behavior of dynamical systems that evolve in time. It is now a vital and growing area of research in mathematical analysis with connections to a broad range of subjects, including geometry, number theory, and combinatorics. The main purpose of this project will be to develop new tools in harmonic analysis and combinatorics to investigate questions central to ergodic theory and convex geometry. In ergodic theory, the PI will consider a variant of the widely studied Furstenberg-Bergelson-Leibman conjecture, for dynamical systems with the underlying structure of nilpotent groups. In harmonic analysis, maximal operators over high-dimensional convex bodies will be investigated in connection with the isotropic constant conjecture, a major open problem in convex geometry. The educational component of this CAREER project will contribute to the training of students and postdoctoral fellows while promoting mathematics to the broader community and encouraging the participation of individuals from underrepresented groups. The PI will continue to supervise undergraduate and graduate students and run his widely subscribed Ergodic Theory and Analysis online seminar series. The PI will also organize five online, one-week workshops, which will combine research training with professional development for undergraduate and graduate students interested in pursuing further education and academic careers in mathematics. This interdisciplinary project aims to develop new methods in harmonic analysis, number theory, and probability to understand central problems in ergodic theory and convex geometry. The primary focus in ergodic theory will be to understand norm and pointwise convergence phenomena for linear polynomial ergodic averages, toward the goal of proving a linear variant of the Furstenberg-Bergelson-Leibman conjecture in the context of all nilpotent groups. The project will also investigate the maximal functions corresponding to the Hardy-Littlewood averaging operators associated with convex symmetric bodies. The longstanding question of whether dimension-free estimates may be obtained for these maximal functions is related to the isotropic constant conjecture in high-dimensional convex geometry, which in turn has inspired deep and unexpected connections to many challenging questions in convex geometry, Banach space theory, and beyond. Describing the optimal constant in the Hardy-Littlewood maximal inequality in this setting would establish a new link between the dimension-free conjecture and the isotropic constant conjecture, and a new point of view on the latter problem, which has not yet been explored using tools from harmonic analysis. In addition, the project will develop tools in Fourier analysis and additive number theory toward a study of Weyl-type inequalities in the nilpotent setting and their applications to a nilpotent Waring problem as well as to a dimension-free variant of the classical Waring problem for squares.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.