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Tuesday, May 15, 2018

**Abstract:** 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 an $L_\cap$-definable set. Examples of interpolative structures abound in model theory, but only recently did Minh Tran and Erik Walsberg begin studying the class in the abstract. When $T_1$ is an $L_1$ theory and $T_2$ is an $L_2$ theory, we are interested in the class of interpolative fusions: interpolative structures which are models of the union theory $T_\cup$. Putting aside the nontrivial question of whether this class is elementary (I will assume that it is, axiomatized by a theory $T^{*}$), I will explain how stability-theoretic assumptions on the base $L_\cap$-theory $T_\cap$ lead to preservation results of the form "If $T_1$ and $T_2$ both satisfy property $P$, then $T^{*}$ satisfies property $P$". This is joint work with Minh and Erik.