Multiple Pointwise Ergodic Theorems

多点遍历定理

基本信息

批准号：
2154712
负责人：
Mariusz Mirek
金额：
$ 31.71万
依托单位：
Rutgers University New Brunswick
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-05-01 至 2025-04-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2154712&HistoricalAwards=false
关键词：
Multiple Pointwise Ergodic Theorems

项目摘要

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.

从统计的角度来看，奇异理论从统计的角度研究了动力学系统的长期行为。它是数学的庞大且快速发展的领域，它对添加剂组合学，数量理论和傅立叶分析等领域以及其他领域以及其他领域产生了深远的影响。该项目的主要重点是理解多种多项式阵尾平均值的规范现象和尖锐的收敛现象，这些平均值自然出现在分析和千古理论的界面上。还将研究成添加剂组合学的广泛问题。在整个研究计划期间，该项目将有助于对本科生和研究生以及博士后研究员的培训。 PI还将向更广泛的社区促进数学，并鼓励代表性不足的群体的参与。一个主要的开放问题之一是Furstenberg-Bergelson-Leibman猜想，这断言，这是多种多种主体的平均几乎无处不在。 PI将开发傅立叶分析和添加剂组合学的工具，从而可以研究对应于具有不同程度的多项式的多个多项式厄贡平均值的侧面收敛。此外，PI将研究拓扑字段和多维整数网格中的多项式Szemeredi的型定理。这些调查的核心是来自高级傅立叶分析的所谓逆定理。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响评估标准通过评估来支持的。