Séminaires : Géométrie et Théorie des Modèles

Equipe(s) : lm,
Responsables :Zoé Chatzidakis, Raf Cluckers, Georges Comte
Email des responsables : zoe.chatzidakis@imj-prg.fr
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http://gtm.imj-prg.fr/

 

Pour recevoir le programme par e-mail, écrivez à : zoe.chatzidakis@imj-prg.fr
Pour les personnes ne connaissant pas du tout de théorie des modèles, des notes introduisant les notions de base (formules, ensembles définissables, théorème de compacité, etc.) sont disponibles ici : https://webusers.imj-prg.fr/~zoe.chatzidakis/papiers/MTluminy.dvi/MTluminy.dvi. Ces personnes peuvent aussi consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998.Retour ligne automatique
Les notes de quelques-uns des exposés sont disponibles.

Orateur(s) Yatir Halevi et Franziska Jahnke - Ben Gurion et Münster,
Titre On dp-finite fields
Date12/02/2021
Horaire09:00 à 12:00
Diffusion
RésumeShelah's conjecture predicts that any infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation. Recently, Johnson proved that Shelah's conjecture holds for fields of finite dp-rank, also known as dp-finite fields. The aim of these two talks is to give an introduction to dp-rank in some algebraic structures and an overview of Johnson's work.
In the first talk, we define dp-rank (which is a notion of rank in NIP theories) and give examples of dp-finite structures. In particular, we discuss the dp-rank of ordered abelian groups and use them to construct multitude of examples of dp-finite fields. We also prove that every dp-finite field is perfect and sketch a proof that any valued field of dp-rank 1 is henselian.
In the second talk, we give an overview of Johnson's proof that every infinite dp-finite field is either algebraically closed, real closed or admits a non-trivial henselian valuation. Crucially, this relies on the notion of a W-topology, a natural generalization of topologies arising from valuations, and the construction of a definable W-topology on a sufficiently saturated unstable dp-finite field.
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