Structure theorems beyond Z-systems

Z 系统之外的结构定理

• 批准号: 2247331
• 负责人: Wenbo Sun
• 金额: $ 16.05万
• 依托单位: Virginia Polytechnic Institute and State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-07-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2247331&HistoricalAwards=false
• 关键词: Structure; theorems; beyond; systems

Ergodic theory is a rapidly evolving area of mathematical research which investigates the long-term behavior of dynamical systems. It encompasses deep connections across many mathematical fields, including analysis, combinatorics, and number theory. Structure theorems in ergodic theory, an essential tool in understanding the average behavior of dynamical systems over space and time, have been especially valuable in advancing the field over the past two decades. While most research on structure theorems has focused on systems with a single transformation, our understanding of systems with multiple transformations remains limited. In this project, the PI aims to establish new structure theorems for systems with multiple transformations, offering a fresh perspective on open problems in ergodic theory and combinatorics. The PI will introduce new courses and seminars, and provide guidance to undergraduate and graduate students, as well as postdoctoral fellows. Additionally, the PI will engage in activities aimed at promoting mathematics to the broader community as well as reaching out to underrepresented groups.The project can be divided into two main parts. The first part aims to deepen our understanding of concatenation theory, a new tool introduced by Tao and Ziegler in recent years to study the intersections of different factors of a dynamical system. A new framework for concatenation theory will be pursued which would apply in a wider range of settings, leading to new structure theorems. The second part of the project will use the structure theorems developed in the first part to investigate two specific open questions in ergodic theory and combinatorics. The first question pertains to the joint ergodicity conjecture, which concerns the convergence of multiple ergodic averages. Recent advances, including work of the PI, have established powerful tools for studying such questions. The second question focuses on the geometric Ramsey conjecture, a long-standing open question in combinatorics which studies geometric patterns that cannot be destroyed by partitioning Euclidean space into finitely many parts. To address this question, methods from higher order Fourier analysis, along with newly developed tools derived from structure theorems, will be employed.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.

遍历理论是数学研究的一个快速发展的领域，它研究动力系统的长期行为。它涵盖了许多数学领域的深刻联系，包括分析、组合学和数论。 遍历理论中的结构定理是理解动力系统在空间和时间上的平均行为的重要工具，在过去二十年中对于推动该领域的发展特别有价值。虽然大多数关于结构定理的研究都集中在具有单一变换的系统，但我们对具有多个变换的系统的理解仍然有限。在这个项目中，PI 旨在为具有多重变换的系统建立新的结构定理，为遍历理论和组合学中的开放问题提供新的视角。 PI 将推出新课程和研讨会，并为本科生、研究生以及博士后提供指导。此外，PI 将参与旨在向更广泛的社区推广数学以及接触代表性不足的群体的活动。该项目可分为两个主要部分。第一部分旨在加深我们对级联理论的理解，级联理论是陶和齐格勒近年来提出的一种新工具，用于研究动力系统中不同因素的交叉。 将寻求一种新的级联理论框架，该框架将适用于更广泛的环境，从而产生新的结构定理。 该项目的第二部分将使用第一部分中开发的结构定理来研究遍历理论和组合学中的两个具体的开放问题。第一个问题涉及联合遍历猜想，涉及多个遍历平均值的收敛。 最近的进展，包括 PI 的工作，已经建立了研究此类问题的强大工具。 第二个问题集中于几何拉姆齐猜想，这是组合学中一个长期存在的开放问题，它研究不能通过将欧几里得空间划分为有限多个部分来破坏的几何模式。为了解决这个问题，将采用高阶傅立叶分析方法以及从结构定理衍生的新开发工具。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查进行评估，被认为值得支持标准。