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

模型论是数理逻辑的一个分支，它研究不同类型的数学结构所共有的共同性质。这一领域的一个关键思想是分界线的概念，这意味着数学结构之间的基本二分法。从历史上看，这些分界线中最重要的是稳定性，这一概念在无限结构的设置中得到了广泛的应用。相比之下，极值和算术组合学领域主要关注有限性问题，但与模型论有共同的主题元素。近年来，模型理论和这些领域之间开始了广泛的相互作用，导致了令人惊讶的新结果。 本项目将通过建立对算术和极值组合学重要工具的模型理论理解来进一步探索这些联系。 从模型理论的角度理解这些工具有可能导致这两个领域的新应用。该项目的教育部分侧重于利用俄亥俄州的基础设施和为研究生组织暑期学校来扩大参与。在极值和加法组合数学中，超图正则性和高阶傅立叶分析已被证明是强有力的工具。 这个项目的目标是发展这些工具和稳定性理论的概括之间的联系。 PI将证明超图中的驯服行为和算术正则引理与稳定性的新推广之间的联系。 作为补充，PI将开发这些高阶稳定性概念的纯模型理论，以及稳定群理论的高阶类似物。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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 on broadening participation efforts 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.