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.