Applied Abstract Elementary Classes

模型理论是数理逻辑的一个分支，它研究和分类数学结构的类，如向量空间、图和群的类。经典模型理论侧重于研究可由有限句集(一阶逻辑)定义的结构类别。虽然许多类是由有限句集定义的，但也有许多类只能使用无限句集(无限逻辑)来定义。这个项目的背景是抽象基础类(简称AEC)，这是一个可以学习由无限句子集合定义的类的背景。自上世纪七十年代末以来，AECS理论开始研究，并在近几年得到了迅速发展。这个项目的目标是继续PI在寻找AEC的相互作用和在代数中的应用方面的工作。更准确地说，该项目专注于寻找AEC与模理论和ACTS(多边形、G-集合)理论之间的相互作用和应用。该项目的第一部分侧重于继续开发模块的AEC。一个关键问题是确定具有纯嵌入的模的AEC的稳定性。该项目的第二部分集中于开发ACTS的并行理论，以达到PI已经能够完成的模块的功能。将研究AECs的一个基本概念是独立关系(AECs的非分叉)。PI希望这些研究将帮助他更好地了解独立关系的优势和局限性，以便他可以在未来将它们应用于其他环境。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。

Model theory is a branch of mathematical logic that studies and classifies classes of mathematical structures, such as the class of vector spaces, graphs, and groups. Classical model theory focuses on studying classes of structures that can be defined by sets of finite sentences (first-order logic). Although many classes are defined by sets of finite sentences, there are many that can only be defined using sets of infinite sentences (infinitary logic). The setting of this project is that of abstract elementary classes (AECs for short) which is a setting where one can study classes defined by sets of infinite sentences. AECs have been studied since the late seventies, and recently, the theory has developed very rapidly. The objective of this project is to continue the PI's work on finding interactions and applications of AECs to algebra. More precisely, the project focuses on finding interactions and applications of AECs to module theory and acts (polygons, G-sets) theory. The first part of the project focuses on continuing the development of AECs of modules. A key problem is to determine the stability behavior of AECs of modules with pure embeddings. The second part of the project focuses on developing a parallel theory for acts to what the PI has been able to accomplish for modules. A fundamental notion that will be studied on AECs of acts is independence relations (non-forking for AECs). The PI expects that these studies will help him better understand the strengths and limitations of independence relations, so he can apply them in other settings in the future.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.