Applied Abstract Elementary Classes

模型论是数学逻辑的一个分支，它研究和分类数学结构的类别，如向量空间、图和群的类别。经典模型理论侧重于研究可以由有限句集定义的结构类（一阶逻辑）。尽管许多类是由有限句集定义的，但有许多类只能使用无限句集（无限逻辑）来定义。这个项目的设置是抽象初级类（简称aec），在这个设置中，人们可以学习由无限句集定义的类。自20世纪70年代末以来，人们对aec进行了研究，近年来，该理论发展非常迅速。该项目的目标是继续PI在寻找aec在代数中的相互作用和应用方面的工作。更准确地说，该项目侧重于寻找aec在模块理论和行为（多边形、g集）理论中的相互作用和应用。该项目的第一部分重点是继续开发模块的aec。一个关键问题是确定纯嵌套模块的aec的稳定性行为。该项目的第二部分侧重于开发一个与PI已经能够为模块完成的功能平行的行为理论。对行为的aec进行研究的一个基本概念是独立关系（aec的非分叉）。PI希望这些研究可以帮助他更好地理解独立关系的优势和局限性，以便他可以在未来的其他环境中应用它们。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.