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New Methods in Non Archimedean Geometry

Final Report Summary - NMNAG (New Methods in Non Archimedean Geometry)

The aim of the project was to import advanced tools from modern Model Theory, like definability and stable domination, to provide a new general framework for the geometry over non-archimedean valued fields in order to make progress on fundamental issues in non-archimedean geometry and to develop new applications. Using this approach, we succeeded in solving several fundamental open questions on the tame nature of the topology of Berkovich spaces. We also opened new perspectives in the study of the monodromy and in non-archimedean diophantine geometry.

Our main achievement is the study of the topology of non-archimedean spaces via advanced tools from Model Theory. In this joint work with E. Hrushovski we solved a long-standing open problem by proving that Berkovich analytifications of quasi-projective varieties strongly retract to polyhedra and are locally contractible. This was done by using an advanced notion of Model Theory, "stably dominated types". Our main idea was to assign to an algebraic variety over a valued field a new space, its stable completion, which is a pro-definable topological space and should be viewed as a model-theoretic version of the Berkovich analytification.