CAREER: Model theoretic classification theory, Fourier analysis, and hypergraph regularity

Model theory is a branch of mathematical logic which studies common properties shared by different types of mathematical structures. A crucial idea in this area is the notion of a dividing line, meaning a fundamental dichotomy among mathematical structures. Historically, the most important of these dividing lines is stability, a notion which has found extensive applications in the setting of infinite structures. In contrast, the fields of extremal and arithmetic combinatorics focus mainly on finitary problems, but have thematic elements in common with model theory. In recent years, extensive interactions have begun between model theory and these fields, leading to surprising new results. This project will further explore these connections by establishing a model theoretic understanding of important tools from arithmetic and extremal combinatorics. Understanding these tools from a model theoretic perspective has the potential to lead to novel applications in both fields. The educational component of this project focuses efforts to bring more students into the field utilizing Ohio State infrastructure and the organization of a summer school for graduate students. In extremal and additive combinatorics, hypergraph regularity and higher order Fourier analysis have proved to be powerful tools. The goal of this project is to develop connections between these tools and generalizations of stability theory. The PI will prove theorems connecting tame behavior in hypergraph and arithmetic regularity lemmas to new generalizations of stability. Complementing this, the PI will develop the pure model theory of these higher order notions of stability, as well as higher order analogues of stable group theory. 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.