Non-elementary model theory

This is the title of a talk given at the CMU graduate student seminar on February 21, 2017. A video of the talk is on Youtube (if you don't like Youtube, here is a direct link).


Classically, model theorists have studied classes of models satisfying a fixed set of axioms, where the axioms can be described using only quantification over elements (not sets), and the disjunctions and conjunctions can only be finite. For some obscure reason, such classes are now called elementary classes.

In this talk, I will explain why studying elementary classes may not be enough, and will discuss frameworks that allow one to study classes axiomatized by axioms with infinite conjunctions and disjunctions. This includes the so-called abstract elementary classes (AECs), developed by Shelah in the mid seventies, which are a kind of concrete category closed under directed colimits. I will present some of the results of my thesis on AECs: the eventual categoricity conjecture for universal classes and generalizations of linear independence from linear algebra to certain nice kind of AECs.