RUI: Model Theory and Structural Ramsey Theory

RUI：模型理论和结构拉姆齐理论

• 批准号: 2246995
• 负责人: Lynn Scow
• 金额: $ 29.21万
• 依托单位: University Enterprises Corporation at CSUSB
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2246995&HistoricalAwards=false
• 关键词: RUI; Model; Theory; Structural; Ramsey

In a large enough data set, a certain amount of uniformity is guaranteed. Variations on this theme are called "Ramsey theorems" after the groundbreaking work of Frank Ramsey in the early 20th century. Ramsey theorems have been used most recently in computer science to design efficient algorithms on certain types of data sets. This project concerns structural Ramsey theory, which uses tools from model theory to describe the objects under study. Model theory is an area of foundations of mathematics that studies properties holding generally on large sets of mathematical objects, across different areas of mathematics. This research project will extend the contribution of model theory to structural Ramsey theory. Methods from saturated model theory, pseudofinite model theory, category theory, and topological dynamics will be applied to create new knowledge pathways across these different areas of mathematics. This project will also provide increased training opportunities for students at California State University, San Bernardino.This research project studies how certain maps between two infinite structures in possibly different first-order languages can transfer information about the automorphism groups of these structures and, ultimately, the Ramsey properties of the finitely-generated structures embeddable in these infinite structures. In particular, this project studies a pair of maps between two infinite structures called a "semi-retraction" that induces a retraction of the type space of one structure onto the type space of the other. This project also studies which Ramsey theorems transfer from a class of finite structures to ultraproducts of these finite structures. This project will identify notions of tame partitions and investigate the classes of structures which have a Ramsey theorem for these tame partitions, but perhaps not in general. Many test cases are available as a result of a confluence of recent results in the calculation of finite big Ramsey degrees. This work is anticipated to have applications in classification theory in model theory, as well as in topological dynamics.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.

在足够大的数据集中，保证了一定程度的一致性。 根据弗兰克·拉姆齐 (Frank Ramsey) 在 20 世纪初的开创性工作，这一主题的变体被称为“拉姆齐定理”。 拉姆齐定理最近在计算机科学中被用来设计针对某些类型的数据集的有效算法。 该项目涉及结构拉姆齐理论，该理论使用模型理论的工具来描述所研究的对象。 模型论是数学基础的一个领域，研究跨数学领域的大量数学对象通常具有的属性。 该研究项目将把模型理论的贡献扩展到结构拉姆齐理论。 饱和模型理论、伪有限模型理论、范畴论和拓扑动力学的方法将被应用于在这些不同的数学领域创建新的知识路径。 该项目还将为加州州立大学圣贝纳迪诺分校的学生提供更多的培训机会。该研究项目研究可能不同的一阶语言中的两个无限结构之间的某些映射如何传递有关这些结构的自同构群的信息，并最终，可嵌入这些无限结构中的有限生成结构的拉姆齐性质。 特别是，该项目研究了两个无限结构之间的一对映射，称为“半收缩”，它导致一个结构的类型空间收缩到另一个结构的类型空间上。 该项目还研究拉姆齐定理从一类有限结构转移到这些有限结构的超乘积。 该项目将识别驯服分区的概念，并研究具有这些驯服分区的拉姆齐定理的结构类别，但可能不是一般情况。 由于有限大拉姆齐度计算的最新结果的汇合，许多测试用例都是可用的。这项工作预计将在模型理论的分类理论以及拓扑动力学中得到应用。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。