Géométrie et Théorie des Modèles

Année 2022 - 2023


Organisateurs : Zoé Chatzidakis, Raf Cluckers.
Pour recevoir le programme par e-mail, écrivez à : zchatzid_at_dma.ens.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. Elles 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.
Les notes de quelques-uns des exposés sont disponibles.

Vendredi 25 novembre 2022, à l'IHP, salle 01, et/ou Zoom. Programme :

11h : Franziska Jahnke (Münster), Taming perfectoid fields
Tilting perfectoid fields, developed by Scholze, allows to transfer results between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. We present a model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger Theorem which states that the absolute Galois group of a perfectoid field and its tilt are canonically isomorphic. A key ingredient in our approach is an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof).
This is joint work with Konstantinos Kartas.

14h15 : Rémi Jaoui (Lyon), Abundance of strongly minimal autonomous differential equations
In several classical families of differential equations such as the Painlevé families (Nagloo, Pillay) or finite dimensional families of Schwarzian differential equations (Blazquez-Sanz, Casale, Freitag, Nagloo), the following picture has been obtained regarding the transcendence properties of their solutions:

- (Strong minimality): outside of an exceptional set of parameters, the corresponding differential equations are strongly minimal,
- (Geometric triviality): algebraic independence of several solutions is controlled by pairwise algebraic independence outside of this exceptional set of parameters,
- (Multidimensionality): the differential equations defined by generic independent parameters are orthogonal.

Are the families of differential equations satisfying such transcendence properties scarce or abundant in the universe of algebraic differential equations?

I will describe an abundance result for families of autonomous differential equations satisfying the first two properties. The model-theoretic side of the proof uses a fine understanding of the structure of autonomous differential equations internal to the constants that we have recently obtained in a joint work with Rahim Moosa. The geometric side of the proof uses a series of papers of S.C. Coutinho and J.V. Pereira on the dynamical properties of a generic foliation.

Vidéo.

16h15 : Hector Pasten (PUC Santiago), On non-Diophantine sets in rings of functions (Zoom)
For a ring R, a subset of a cartesian power of R is said to be Diophantine if it is positive existentially definable over R with parameters from R. In general, Diophantine sets over rings are not well-understood even in very natural situations; for instance, we do not know if the ring of integers Z is Diophantine in the field of rational numbers. To show that a set is Diophantine requires to produce a particular existential formula that defines it. However, to show that a set is not Diophantine is a more subtle task; in lack of a good description of Diophantine sets it requires to find at least a property shared by all of them. I will give an outline of some recent joint work with Garcia-Fritz and Pheidas on showing that several sets and relations over rings of polynomials and rational functions that are not Diophantine.

Notes de l'exposé et vidéo.

Vendredi 9 décembre 2022, à l'ENS, salle W. Orateurs :

14h15 : François Loeser (IMJ-PRG), Un théorème de finitude pour les fonctions tropicales sur les squelettes
Les squelettes sont des sous-ensembles linéaires par morceaux d'espaces analytiques non-archimédiens apparaissant naturellement dans nombre de situations. Nous présenterons un résultat général de finitude, obtenu en collaboration avec A. Ducros, E. Hrushovski et J. Ye, concernant le groupe abélien ordonné des fonctions tropicales sur les squelettes des analytifiés de Berkovich de variétés algébriques. Notre approche utilise la version modèle théorique de l'analytification (la complétion stable) développée dans un travail antérieur avec E. Hrushovski.

Vidéo

16h : Alex Wilkie (U. of Oxford), Integer points on analytic sets
In 2004 I proved that that if C is a transcendental curve definable in the structure R_{an}, then the number of points on C with integer coordinates of modulus less than H, is bounded by k loglog H for some constsnt k depending only on C. (The situation is vastly different for rational points.) The proof used the fact that such sets C are, in fact, semi-analytic everywhere-including infinity-and so the crux of the matter was to bound the number of solutions to equations of the form

(*)    F(1/n) = 1/m
for n, m integers bounded in modulus by (large) H, and where F is a non-algebraic, analytic function defined on an open interval containing 0.
It turns out that there is probably no generalization of the 2004 result for arbitrary R_{an}-definable sets (which need not be globally, or even locally, semi-analytic) but inspired by observations of Gareth Jones and Gal Binyamini, the three of us began looking at equations of the form (*) in many variables and I shall be reporting on our results.

Vidéo.

Programme des séances passées : 2006-07, 2007-08, 2008-09, 2009-10, 2010-11, 2011-12, 2012-13, 2013-14, 2014-15, 2015-16, 2016-17, 2017-18, 2018-19, 2019-20, 2020-21, 2021-22.
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