Applied Abstract Elementary Classes

应用抽象初级班

• 批准号: 2348881
• 负责人: Marcos Mazari Armida
• 金额: $ 13万
• 依托单位: Baylor University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2027-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2348881&HistoricalAwards=false
• 关键词: Applied; Abstract; Elementary; Classes

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.

模型论是数学逻辑的一个分支，研究数学结构的分类，如向量空间、图和群。经典模型论专注于研究可以由有限句集（一阶逻辑）定义的结构类。虽然许多类是由有限句的集合定义的，但也有许多类只能使用无限句的集合（无限逻辑）来定义。这个项目的背景是抽象基本类（简称AEC），这是一个可以学习由无限句子集定义的类的背景。自70年代末以来，人们开始研究AEC，近年来，该理论发展非常迅速。这个项目的目标是继续PI的工作，寻找相互作用和应用的AEC代数。更确切地说，该项目的重点是寻找AEC的模块理论和行为（多边形，G集）理论的相互作用和应用。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的稳定性行为。该项目的第二部分侧重于开发一个平行的理论行为PI已经能够完成的模块。 一个基本的概念，将研究的行为AEC是独立关系（非分叉AEC）。PI希望这些研究将帮助他更好地理解独立关系的优势和局限性，以便他将来能够将其应用于其他环境。该奖项反映了NSF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。