Shelah's Categoricity Conjecture for Abstract Elementary Classes in Dependent Theories

This monograph investigates Saharon Shelah's Categoricity Conjecture within the framework of Abstract Elementary Classes (AECs) that exhibit structural dependence properties analogous to NIP (Not the Independence Property) in first-order model theory. We analyze the stability spectrum and the behavior of Galois types in tame AECs, establishing a transfer theorem for categoricity. Specifically, we prove that for a tame, dependent AEC K with the amalgamation property, categoricity in a successor cardinal + (where is sufficiently large and related to the Hanf number) implies categoricity in all higher cardinals. The argument synthesizes Grossberg and VanDieren's upward transfer techniques with a generalized notion of dependence for Galois types, thereby reducing the upper bound on the categoricity threshold. We further explore the interaction between dependent forking relations and the existence of prime models, providing a partial solution to the conjecture for classes stable in the sense of dependence.