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，可以保证一定数量的均匀性。 在20世纪初的弗兰克·拉姆西（Frank Ramsey）开创性的作品之后，该主题的变化称为“拉姆齐定理”。 Ramsey定理最近在计算机科学中用于设计某些类型的数据集的有效算法。 该项目涉及结构性拉姆齐理论，该理论使用模型理论的工具来描述所研究的对象。 模型理论是数学基础的一个领域，该区域研究通常在数学领域跨越大量数学对象的属性。 该研究项目将扩展模型理论对结构性拉姆齐理论的贡献。 将采用饱和模型理论，假芬仪模型理论，类别理论和拓扑动态的方法来在这些不同的数学领域中创建新的知识途径。 该项目还将为加州州立大学圣贝纳迪诺大学的学生提供更多的培训机会。本研究项目研究了如何在可能不同的一阶语言中的两个无限结构之间的某些地图可以转移有关这些结构的自动形态群体的信息，并最终，最终，这些无限生成的结构的Ramsey属性在这些无限结构中嵌入了这些结构。 特别是，该项目研究了两个无限结构之间的一对图，称为“半撤退”，该结构引起了一个结构的类型空间的缩回，以缩回另一个结构的类型空间。 该项目还研究了Ramsey定理从一类有限结构转移到这些有限结构的超副作用。 该项目将确定驯服分区的概念，并研究具有这些驯服分区的Ramsey定理的结构类别，但通常不是。 由于有限的大拉姆西学位的计算，由于最近的结果汇合而获得了许多测试用例。预计这项工作将在模型理论以及拓扑动力学中具有分类理论中的应用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响来通过评估来支持的。