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Abstract: Morley's categoricity theorem says that a countable first-order theory with a single model in some uncountable cardinal has a single model in all uncountable cardinals. The proof has had numerous consequences, including the development of stability theory, and in particular the invention by Shelah of forking, a far reaching generalization of algebraic independence in fields. Shelah has conjectured that a generalization of Morley's result should hold in any abstract elementary class (AEC). Roughly, an AEC is a category that behaves like the category of models of a first-order theory with elementary embedding. The framework encompasses for example classes of models of logics with infinite conjunctions and disjunctions.
In this talk, I will survey progress on Shelah's eventual categoricity conjecture, including a proof from large cardinals (joint with Shelah) and a full characterization of the categoricity spectrum in AECs with amalgamation. These proofs have already suggested several new directions, including a theory of forking in accessible categories connecting the field with homotopy theory (joint with Lieberman and RosickÃ½).