Multidimensional asymptotic classes

A 1-dimensional asymptotic class (Macpherson-Steinhorn) is a class of finite structures which satisfies the theorem of Chatzidakis-van den Dries-Macintyre about finite fields: definable sets are assigned a measure and dimension which gives the cardinality of the set asymptotically, and there are only finitely many dimensions and measures in any definable family. There are many examples of these classes, and they all have reasonably tame theories. Non-principal ultraproducts of these classes are supersimple of finite rank.

Recently this definition has been generalised to `Multidimensional Asymptotic Class' (joint work with Macpherson-Steinhorn-Wood). This is a much more flexible framework, suitable for multi-sorted structures. Examples are not necessarily simple. I will give conditions which imply simplicity/supersimplicity of non-principal ultraproducts.

An interesting example is the family of vector spaces over finite fields with a non-degenerate bilinear form (either alternating or symmetric). If there's time, I will explain some joint work with Kestner in which we look in detail at this class.