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。它涵盖了许多数学领域的深层联系，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基本工具，在过去的二十年里在这一领域的发展中特别有价值。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旨在为具有多个变换的系统建立新的结构定理，为遍历理论和组合学中的公开问题提供一个新的视角。PI将推出新的课程和研讨会，并为本科生和研究生以及博士后研究员提供指导。此外，国际数学协会将参与旨在向更广泛的社区推广数学以及接触代表不足的群体的活动。该项目可分为两个主要部分。第一部分旨在加深我们对串联理论的理解，串联理论是陶和齐格勒近年来引入的一种新工具，用于研究动力系统中不同因素的交集。将寻求一种新的串联理论框架，该框架将适用于更广泛的设置，从而导致新的结构定理。项目的第二部分将使用第一部分中发展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。第一个问题与联合遍历猜想有关，它涉及多个遍历平均的收敛。最近的进展，包括国际和平研究所的工作，已经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Ramsey猜想上，这是组合学中一个长期悬而未决的问题，它研究的是不能通过将欧几里德空间划分为有限多个部分而被摧毁的几何模式。为了解决这个问题，将采用高阶傅立叶分析的方法，以及新开发的源自结构定理的工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.