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NMNAG: Non-Archimedean geometry - François Loeser

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Direction de la communication


Marie Pinhas-Diena, in charge of scientific communications l tel: +33 (0)1 44 27 22 89 l email: marie.pinhas@upmc.fr

NMNAG: New Methods in Non-Archimedean Geometry - François Loeser

The project’s objective is to develop new methods and techniques from model theory, a branch of mathematical logic in the context of non-Archimedean geometry, giving a different perspective for new applications in algebraic geometry and singularity theory.


Some major findings and their applications

Working with E. Hrushovski, François Loeser introduced the notion of steady completion of an algebraic variety over a valued field, a model-theoretic version of the Berkovich spaces. They demonstrated the existence of strong contractions in this context, which has enabled them to demonstrate that the analytification in the sense of Berkovich of an algebraic variety over an ultrametric body shrinks sharply on a polyhedron and is locally contractible. The establishment of such properties of topological moderation was a longstanding open problem.


In collaboration with R. Cluckers and G. Comte, François Loeser succeeded in establishing the existence of the Yomdin and Gromov parameterizations in a non-Archimedean context. In an Archimedean context, the use of these parameterizations enabled Pila and Wilkie and Pila to achieve spectacular applications in diophantine geometry. Cluckers, Comte and Loeser also obtained a p-adic version of the results of Pila and Wikie which is expected to show new results of functional independence in Ax-Lindemann for certain varieties admitting p-adic uniformations.


Using the model theory, Hrushovski and Loeser got a new approach, based on non-Archimedean geometry, to a Denef and Loeser fixed-point theorem. This enabled them, in particular, to develop a new construction of the motivic Milnor fiber. The new construct has enabled Lê Quy Thuong (in a thesis under the direction of F. Loeser) to demonstrate a conjecture by Kontsevich and Soibelman that plays a crucial role in the theory of invariants of Donaldson-Thomas motivic.


For more information:

The Mathematics Instute of Jussieu (IMJ, CNRS/UPMC/Université Paris Diderot/FSMP)Nouvelle fenêtre (In French)


Read the presentation of François Loeser by the Mathematical Sciences Foundation: François Loeser ou l’art de l’analogieNouvelle fenêtre (In French)


© Cyril Frésillon CNRS Images - Photothèque