# Very NIP Ordered Groups

## Alf Dolich

### The City University of New York

In recent years there has been renewed interest in theories without the independence property (NIP theories), a class of theories including all stable as well as all o-minimal theories. In this talk we concentrate on theories, T, which expand that of divisible ordered Abelian groups (a natural situation to consider if one is motivated by the study of o-minimal theories) and consider the problem determining the consequences of assuming that T is NIP on the structure of definable sets in models of T. One quickly realizes that given the great generality of the NIP assumption in order to address this type of question one wants to consider much stronger variants of not having the independence property. Thus we are led to the study of definable sets in models of theories T expanding that of divisible ordered Abelian groups satisfying various very strong forms of the NIP condition such as finite dp-rank, convex orderability, and VC minimality. In this talk I will survey results in this area and discuss many open problems.

Professor Dolich (Ph.D. 2002 University of Maryland, M.A. Columbia University, B.A. University of Pennsylvania) held a VIGRE Van Vleck Assistant Professorship at the University of Wisconsin, Madison, before coming to the New York area, where he now holds an Assistant Professor position at Kingsborough CC of CUNY. Professor Dolich conducts research in model theory, simple theories, and o-minimal theories with secondary interests in algebraic geometry and set theory.