| Résume | I will discuss model-theoretic developments stemming from a theorem of Chatzidakis, van den Dries and Macintyre, which states that given a formula φ(x,y) in the language of rings, there are finitely many pairs (μ ,d) (μ rational, d a natural number) such that for any finite field F_q and parameter a, the definable set φ(F_q,a) has size roughly μ q^d for one of these pairs (μ ,d). A model-theoretic framework suggested by this was developed by Elwes, myself, and Steinhorn, with notions of an `asymptotic class' of finite structures, and `measurable' infinite structure: an ultraproduct of an asymptotic class is measurable (and in particular has supersimple finite rank theory).
I will discuss recent work with Anscombe, Steinhorn and Wolf on `multi-dimensional asymptotic classes' of finite structures and infinite `generalised measurable' structures which greatly extends this framework, to include classes of multi-sorted structures, which may have infinite rank ultraproducts, or even have ultraproducts with non-simple theory (though these ultraproducts can never have the strict order property). The key feature is the fixed bound, for each formula φ(x,y), on the number of approximate sizes of sets φ(M,a) as M ranges through a class of finite structures and the parameter a varies through M. The focus will be on naturally-arising examples.
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