Structure theorems beyond Z-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.