Structure theorems beyond Z-systems

遍历理论是一个快速发展的数学研究领域，它研究动力系统的长期行为。它包含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，在过去的二十年里，它在推进该领域方面特别有价值。虽然大多数关于结构定理的研究都集中在具有单个变换的系统上，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的目的是建立新的结构定理的系统与多重变换，提供了一个新的视角开放的问题，遍历理论和组合。PI将引入新的课程和研讨会，并为本科生和研究生以及博士后研究员提供指导。此外，PI将参与旨在向更广泛的社区推广数学的活动，并接触代表性不足的群体。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，这是陶和齐格勒近年来引入的一种新工具，用于研究动力系统不同因素的交叉点。 一个新的框架串联理论将追求这将适用于更广泛的设置，导致新的结构定理。 第二部分的项目将使用结构定理开发的第一部分，调查两个具体的开放问题，遍历理论和组合。第一个问题涉及联合遍历猜想，它涉及多个遍历平均的收敛性。 最近的进展，包括PI的工作，为研究这些问题建立了强大的工具。 第二个问题集中在几何拉姆齐猜想，一个长期存在的开放问题，在组合学研究几何模式，不能被破坏的分割欧几里德空间成许多部分。为了解决这个问题，从高阶傅立叶分析的方法，沿着与新开发的工具来自结构定理，将采用。这个奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.