# Project A9: Geometric Model Theory of Valued Fields

## Principal investigator

## Participating scientists

## Summary

Since the work of Ax-Kochen and Ershov, the model theory of valued fields has seen several spectacular applications to number theory and algebraic geometry. But it is only with the more recent work of Haskell-Hrushovski-Macpherson on stable domination and the classification of imaginaries in algebraically closed valued fields that the powerful tools of stability theory and geometric model theory got available in the valued context.The present project concerns central topics in the geometric model theory of valued fields, and it is intended to start with the arrival of the PI who will move from Paris to Münster in summer 2016. The project is divided into three parts which are very much intertwined, as far as the questions asked and the research goals are concerned, and also because of the unity of the intended methods. One subproject deals with the model theory of abelian varieties over valued fields, a second one with the model theory of separably closed valued fields, and the last one is entitled valued difference fields, non-standard Frobenius and NTP

_{2}.

The latter two concern valued fields with extra structure - Hasse derivations and so-called "λ-functions" in the case of separably closed valued fields, an automorphism in the case of valued difference fields. In both contexts, we plan to investigate notions from pure model theory like stable domination and non-forking, as well as imaginaries and definable groups and fields, key features in view of potential applications to non-Archimedian geometry.

The first subproject deals with definable equivariant retractions of analytifications of abelian varieties over valued fields onto their skeleton. Moreover, it proposes the study of model- theoretic connected components for groups definable in algebraically closed valued fields.