During the last decade, spectacular achievments have been completed in Algebraic and Arithmetic Geometry and in Representation Theory by using powerful new tools provided by Motivic Integration and Berkovich spaces. We propose to develop a general framework for geometry over non archimedean valued field that will provide common foundations for Motivic Integration and Berkovich spaces. This will allow to broaden the range of potential applications. A main originality of our approach is the use of advanced tools from modern Model Theory, like definability and stable domination, together with methods from Algebraic Geometry. The relevance of Model Theory to non archimedean geometry may be illustrated as follows: geometry over valued fields ultimately combine geometry over the residue field and geometry over the value group. Model theorically these geometries correspond respectively to stable and o-minimal theories. These are of a very different nature and Model Theory provides unifying concepts allowing to treat them on equal footing. This approach will in particular allow us to solve several fundamental open questions on the tame nature of the topology of Berkovich spaces and should open new perspectives towards outstanding conjectures like the Monodromy conjecture. Our goal is also to use model theoretic tools in order to give new applications of Motivic Integration to Algebraic Geometry and Singularity Theory.
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