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

职业：模型理论分类理论、傅立叶分析和超图正则性

• 批准号: 2239737
• 负责人: Caroline Terry
• 金额: $ 47.27万
• 依托单位: Ohio State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-05-01 至 2028-04-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2239737&HistoricalAwards=false
• 关键词: CAREER; Model; theoretic; classification; theory

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.

模型理论是数学逻辑的一个分支，研究了不同类型的数学结构共享的共同特性。在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环境中发现了广泛的应用。相比之下，极端和算术组合学领域主要集中在限制问题上，但与模型理论具有共同的主题元素。近年来，模型理论与这些领域之间已经开始了广泛的互动，从而导致了令人惊讶的新结果。 该项目将通过从算术和极端组合学中建立对重要工具的模型理论理解来进一步探索这些联系。 从模型理论的角度了解这些工具有可能导致在这两个领域的新应用。该项目的教育组成部分着重于扩大利用俄亥俄州州基础设施和研究生的暑期学校的组织。在极端和添加剂的组合学中，事实证明，超图的规律性和高阶傅立叶分析是强大的工具。 该项目的目的是发展这些工具与稳定理论的概括之间的联系。 PI将证明定理将超图和算术规则性引理与稳定性的新概括相连。 对此进行补充，PI将开发这些高阶稳定概念的纯模型理论，以及稳定群体理论的高阶类似物。该奖项反映了NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的审查标准通过评估来通过评估来支持的。