## Cardinalities of definable sets in finite structures

#### Dugald Macpherson (Leeds)

Frank Adams 1,

I will discuss recent work with Sylvy Anscombe, Charles Steinhorn and Daniel Wolf, related work with Gwyneth Harrison-Shermoen, and also the PhD thesis of Ricardo Bello Aguirre, building on the concept of asymptotic class’ of finite structures and ultimately on results of Chatzidakis, van den Dries and Macintyre on definable sets in finite fields. We consider classes $$C$$ of finite structures over a given language, such that for any formula $$\phi(x,y)$$ (with $$x$$ and $$y$$ tuples) the cardinalities of definable sets $$\phi(x,a)$$ are tightly constrained in structures $$M$$ in $$C$$ as $$a$$ varies through $$M$$, either asymptotically (multidimensional asymptotic classes’) or exactly (`multidimensional exact classes’).