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.

模型论是数理逻辑的一个分支，研究和分类数学结构的类别，例如向量空间、图和群的类别。经典模型理论侧重于研究可由有限句子集（一阶逻辑）定义的结构类别。尽管许多类是由有限句子集定义的，但也有许多类只能使用无限句子集（无限逻辑）来定义。该项目的背景是抽象基本类（简称AEC），这是一种可以研究由无限句子集定义的类的背景。 AEC的研究从七十年代末开始，近来该理论发展非常迅速。该项目的目标是继续 PI 的工作，寻找 AEC 在代数中的相互作用和应用。更准确地说，该项目的重点是寻找 AEC 与模块理论和行为（多边形、G 集）理论的相互作用和应用。该项目的第一部分重点是继续开发模块的 AEC。一个关键问题是确定纯嵌入模块的 AEC 的稳定性行为。该项目的第二部分重点是开发一个与 PI 能够完成的模块行为并行的理论。 将研究行为 AEC 的一个基本概念是独立关系（AEC 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点和局限性，以便他将来可以将其应用到其他环境中。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，认为值得支持。优点和更广泛的影响审查标准。