RUI: Model Theory and Structural Ramsey Theory

在足够大的数据集中，一定程度的一致性是有保证的。这一主题的变种被称为“拉姆齐定理”，源于弗兰克·拉姆齐在20世纪初的开创性工作。拉姆齐定理最近在计算机科学中被用来在某些类型的数据集上设计有效的算法。这个项目涉及结构拉姆齐理论，它使用模型理论中的工具来描述被研究的对象。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集合数学对象的一般性质。本研究将拓展模型理论对结构拉姆齐理论的贡献。来自饱和模型理论、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的知识路径。这个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。这个研究项目研究了两个可能不同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，并最终传递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”，它诱导一个结构的类型空间到另一个结构的类型空间的收缩。这个项目还研究了Ramsey定理从一类有限结构到这些有限结构的超积的转换。这个项目将识别TAME划分的概念，并研究对于这些TAME划分具有Ramsey定理的结构类，但可能不是一般的。由于有限大Ramsey度计算的最新结果的汇合，许多测试用例都是可用的。这项工作预计将在分类理论、模型理论以及拓扑动力学中得到应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.