Existence over an arid predicate

I will talk more on the topic introduced in my Week 2 talk, though this talk will be self-contained. As mentioned then, with Itay Kaplan and Pierre Simon we prove the existence property for stably embedded predicates in simple (or just rosy) countable theories, going via the notion of an "arid" predicate. I will present this result and its proof. Time allowing, I may explain how this recovers the result (proven independently by Chatzidakis and Afshordel) that any difference field with pseudofinite fixed field extends to an existentially closed difference field without extending the fixed field, and/or discuss other examples and non-examples.