November 8
Artem Chernikov,
University of Maryland
External definability
An object (e.g. a set, a relation, a group, etc.) is externally definable in a structure $M$ if it is given by the intersection with $M$ of an object definable (with parameters) in some elementary extension of $M$. If all types over $M$ are definable (for example, if the theory of $M$ is stable), then all externally definable sets are already definable. This fails beyond stability, e.g. in linear orders (take a cut of some irrational number over the rationals) or in the Rado graph (where all subsets of a model are externally definable). An important theorem of Shelah shows that at least the expansion of an NIP structure $M$ by all externally definable sets $M^{\text{ext}}$ remains NIP. While externally definable sets in NIP structures are well behaved, partially explained by the existence of 'honest definitions' introduced in joint work with Simon, many questions remain open. In this talk I will survey some topics in the study of externally definable sets and discuss some new results on externally definable groups in NIP structures.