CAREER: Harmonic Analysis, Ergodic Theory and Convex Geometry

职业：调和分析、遍历理论和凸几何

• 批准号: 2236493
• 负责人: Mariusz Mirek
• 金额: $ 44.04万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2028-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2236493&HistoricalAwards=false
• 关键词: CAREER; Harmonic; Analysis; Ergodic; Theory

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.

千古理论起源于对时间发展的动力学系统的统计行为的研究。 现在，它是数学分析研究的重要且不断增长的研究领域，与广泛的受试者的联系，包括几何，数字理论和组合学。 该项目的主要目的是开发谐波分析和组合学中的新工具，以调查沿厄尔贡理论和凸几何的核心问题。 在崇高的理论中，PI将考虑具有尼尔肽基团的基础结构的动力学系统的广泛研究的Furstenberg-Bergelson-Leibman猜想的变体。 在谐波分析中，将研究与各向同性恒定猜想有关的高维凸体的最大算子，这是凸几何的主要开放问题。 该职业项目的教育部分将有助于对学生和博士后研究员的培训，同时向更广泛的社区推广数学，并鼓励来自代表性不足的团体的个人参与。 PI将继续监督本科生和研究生，并运行他广泛订阅的崇高理论和分析在线研讨会系列。 PI还将组织五个在线，为期一周的研讨会，该研讨会将研究培训与有兴趣从事进一步的教育和数学学术职业的本科和研究生的专业发展相结合。 这个跨学科项目旨在开发谐波分析，数理论和可能性的概率中的新方法，以理解沿着厄贡理论和凸几何形状中的中心问题。 崇高理论的主要重点是理解线性多项式ergodic平均值的规范和方向收敛现象，以证明在所有nilpotent组的情况下证明Furstenberg-Bergelson-Leibman猜想的线性变体。 该项目还将研究与与凸对称体相关的硬质小木平均操作员相对应的最大功能。 对于这些最大功能是否可以获得无维度估计的长期问题与高维凸几何形状中的各向同性恒定猜想有关，这反过来又激发了与凸数学，Banach空间理论以及超越的许多挑战性问题的深刻和意外的联系。 在这种情况下，描述在这种情况下耐铁木中最大不平等的最佳常数将在无维度的猜想和各向同性常数猜想之间建立新的联系，以及关于后一种问题的新观点，尚未使用谐波分析中的工具来探索。此外，该项目将开发傅立叶分析和添加数理论中的工具，以研究静脉设置中的韦伊尔型不平等，及其在尼尔疗法的警告问题上的应用，以及对经典警告问题的无维度变体的适用于正方形的无效变异。 标准。