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BIRS Workshop Lecture Videos

Interpolative Fusions Kruckman, Alex


Fix languages L_1 and L_2 with intersection L_\cap and union L_\cup. An L_\cup structure M is interpolative if whenever X_1 is an L_1-definable set and X_2 is an L_2-definable set, X_1 and X_2 intersect in M unless they are separated by L_\cap-definable sets. When T_1 is an L_1 theory and T_2 is an L_2 theory, we say that a theory T_\cup^* is the interpolative fusion of T_1 and T_2 if it axiomatizes the class of interpolative models of the union theory T_\cup. If T_1 and T_2 are model-complete, this is exactly the model companion of T_\cup. Interpolative fusions provide a unified framework for studying many examples of "generic constructions" in model theory. Some, like structures with generic predicates, or algebraically closed fields with several independent valuations, are explicitly interpolative fusions, while others, like structures with generic automorphisms (e.g. ACFA), or fields with generic operators (e.g. DCF), are bi-interpretable with interpolative fusions. In joint work with Erik Walsberg and Minh Tran, we study two basic questions: (1) When does the interpolative fusion exist, and how can we axiomatize it (2) How can we understand properties of the interpolative fusion T_\cup^* in terms of properties of the theories T_1, T_2, and T_\cap In this talk, I will focus on the latter question. Under mild stability-theoretic assumptions on the base theory T_\cap, we show preservation of a weak form of quantifier-elimination. And using this, we show that the interpolative fusion of NSOP_1 theories is NSOP_1. I will also discuss sufficient conditions for the preservation of other properties of interest (e.g. stability, NIP, simplicity, and NTP_2).

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