CAREER: Harmonic Analysis, Ergodic Theory and Convex Geometry

遍历理论起源于对随时间演化的动力系统的统计行为的研究。它现在是数学分析研究中一个重要的和不断发展的领域，与广泛的学科有联系，包括几何、数论和组合学。这个项目的主要目的是开发谐波分析和组合学的新工具，以研究遍历理论和凸几何的核心问题。在遍历理论中，PI将考虑广泛研究的Furstenberg-Bergelson-Leibman猜想的一个变体，用于具有幂零群底层结构的动力系统。在调和分析中，高维凸体上的极大算子将与各向同性常数猜想（凸几何中的一个主要开放问题）联系起来进行研究。这个CAREER项目的教育部分将有助于培养学生和博士后，同时将数学推广到更广泛的社区，并鼓励来自代表性不足群体的个人参与。PI将继续指导本科生和研究生，并举办他广泛订阅的遍历理论与分析在线系列研讨会。PI还将组织五次为期一周的在线研讨会，这些研讨会将把研究培训与专业发展结合起来，为有兴趣在数学领域继续深造和从事学术事业的本科生和研究生提供帮助。这个跨学科项目旨在发展谐波分析、数论和概率论的新方法，以理解遍历理论和凸几何中的核心问题。遍历理论的主要焦点将是理解线性多项式遍历平均的范数和点向收敛现象，以在所有幂零群的背景下证明Furstenberg-Bergelson-Leibman猜想的线性变体。该项目还将研究与凸对称体相关的Hardy-Littlewood平均算子对应的极大函数。长期存在的问题是，是否可以获得这些极大函数的无维估计与高维凸几何中的各向同性常数猜想有关，这反过来又激发了凸几何，巴拿赫空间理论等许多具有挑战性的问题的深刻和意想不到的联系。在这种情况下，描述Hardy-Littlewood极大不等式中的最优常数将在无量纲猜想和各向同性常数猜想之间建立新的联系，并为后者问题提供新的观点，后者尚未使用调和分析工具进行探索。此外，该项目将开发傅里叶分析和加性数论的工具，以研究幂零设置中的weyl型不等式，并将其应用于幂零Waring问题以及经典Waring问题的无维变体。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.