Seminar

I will describe joint work with Dario Garcia and Charles Steinhorn on the Hrushovski-Wagner notion of pseudofinite dimension, for, e.g., definable sets in an ultraproduct of finite structures. The focus will be on rather general conditions which ensure that an ultraproduct is (super)simple, or stable, and on the relation between dimension-drop and forking, and on examples.

I will also discuss preliminary work with William Anscombe, Charles Steinhorn, and Daniel Wood on rather general notions of `asymptotic class' of finite structures'. These concern contexts where there is a strong uniformity on the cardinalities of definable sets in a class of finite structures, and take into account multi-sorted behaviour, allow infinite rank, and do not imply simplicity of the ultraproduct theory.