CAREER: Compactness in Incompact Worlds

The field of model theory studies classes of structures: groups, fields, graphs. This is a broad field to study, so important distinctions are made based on how the class of structures is described (or axiomatized). If the class is describable in first-order logic, it is called an elementary class. First-order logic has many powerful properties, especially the property of compactness. Compactness allows model theorists to build structures with exotic properties and has driven much of the model theory of elementary classes, most notably classification theory. However, many classes of structures are not describable in first-order logic (these are called nonelementary classes). Lacking compactness, the development of nonelementary model theory and classification theory has proceeded much slower than its elementary counterpart. Recent work in nonelementary classes has shown that various fragments of compactness can still hold in some nonelementary classes and are still powerful enough to prove various results of elementary classification theory. The PI will develop more of these fragments in nonelementary classes. Additionally, the PI will run a program to build research infrastructure at their home institution (an R2 institution and HSI). This program will support undergraduates conducting research in logic, supported by a speaker series that will build connections between expert logicians and faculty and students. This research will develop fragments of compactness in a variety of ways. From category theory, methods from accessible categories and from topoi will be used to find compactness principles in certain nonelementary classes. Drawing on model theory and set theory, generalized indiscernibles and generalizations of the Erdos-Rado theorem will find compactness principles that hold in all nonelementary classes. Further drawing on set theory, the connections between large cardinals and compactness principles (along with other model-theoretic ideas) will be extended. The connections formed in categorical logic will be extended to higher category theory to open new areas for model theory. 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.