DEFINABLE SETS IN DP-MINIMAL ORDERED ABELIAN GROUPS (Model theoretic aspects of the notion of independence and dimension)

This article surveys some recent results on ordered abelian groups (possibly with additional definable structure) from the subclass of NIP theories which are dp-minimal. To put these results in context, the first part of the article reviews and compares various other generalizations of a-minimality (such as local a-minimality and a-stability) and their consequences. It is useful to make the further assumption that there is a cardinal bound on the number of convex subgroups definable in elementary extensions of the structure. Under this hypothesis, some classic theorems on o-minimal structures, such as the monotonicity theorem for unary definable functions, can be suitably generalized

Model theory of special subvarieties and Schanuel-type conjectures

We use the language and tools available in model theory to redefine and clarify the rather involved notion of a {\em special subvariety} known from the theory of Shimura varieties (mixed and pure)