RUI: Model Theory and Structural Ramsey Theory

在一个足够大的数据集中，一定程度的一致性是有保证的。 在世纪早期弗兰克·拉姆齐的开创性工作之后，这个主题的变体被称为“拉姆齐定理”。 拉姆齐定理最近在计算机科学中被用来设计某些类型数据集的有效算法。 这个项目涉及结构拉姆齐理论，它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，它研究跨越不同数学领域的大量数学对象的性质。 本研究将模型理论的贡献扩展到结构Ramsey理论。 饱和模型理论，伪有限模型理论，范畴理论和拓扑动力学的方法将被应用于创建跨越这些不同数学领域的新知识途径。 这个项目也将为加州圣贝纳迪诺的州立大学的学生提供更多的培训机会。这个研究项目研究了两个可能不同的一阶语言的无限结构之间的某些映射如何传递关于这些结构的自同构群的信息，并最终研究可嵌入这些无限结构中的有限生成结构的Ramsey性质。 特别是，这个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。 本项目还研究了一类有限结构的Ramsey定理到这类有限结构的超积的迁移。 这个项目将确定驯服分区的概念，并研究具有这些驯服分区的拉姆齐定理的结构类，但可能不是一般的。 许多测试案例是可作为一个汇合的结果，最近的结果在计算有限大拉姆齐度。这项工作预计将在模型理论的分类理论，以及在拓扑dynamics.This奖项反映了NSF的法定使命的应用，并已被认为是值得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的支持。

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.