This is the title of my plenary talk given at the Association for Symbolic logic winter meeting (joint with the JMM) on January 12, 2018. Here are the slides.
The classification theory of elementary classes was started by Michael Morley in the early sixties, when he proved that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof of this result, now called Morley's categoricity theorem, led to the development of forking, a joint generalization of linear independence in vector spaces and algebraic independence of fields, which is now a central pillar of modern model theory.
In recent years, it has become apparent that the theory of forking can also be developed in several non-elementary contexts. Prime among those is the axiomatic framework of abstract elementary classes (AECs), encompassing the class of models of any L∞, ω-theory and closely connected to the more general framework accessible categories. A test question to judge progress in this direction is the forty year old eventual categoricity conjecture of Shelah, which says that a version of Morley's categoricity theorem should hold of any AEC. I will survey recent developments, including the connections with category theory and large cardinals as well as my resolution of the eventual categoricity conjecture for classes of models axiomatized by a universal L∞, ω-theory.
John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vols. 18 and 20, College Publications, 2009.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes. Part I, Annals of Pure and Applied Logic 168 (2017), no. 9, 1609–1642. Publisher version pdf arXiv.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes. Part II, Selecta Mathematica 23 (2017), no. 2, 1469–1506. Publisher version pdf arXiv.
Sebastien Vasey, Tameness from two successive good frames, Submitted. Preprint: pdf arXiv, 25 pages.