Géométrie et Théorie
des Modèles
Année 2022  2023
Organisateurs :
Zoé
Chatzidakis, Raf Cluckers.
Pour recevoir le programme par email, 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 quelquesuns 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 modeltheoretic
approach to tilting via ultraproducts, which allows to
transfer many firstorder properties between a perfectoid
field and its tilt (and conversely). In particular, our
method yields a simple proof of the FontaineWintenberger
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
AxKochen/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
(BlazquezSanz, 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
modeltheoretic 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.
16h15 : Hector Pasten (PUC
Santiago), On nonDiophantine 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 wellunderstood 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 GarciaFritz and
Pheidas on showing that several sets and relations over
rings of polynomials and rational functions that are not
Diophantine.
Programme des séances
passées : 200607,
200708,
200809,
200910,
201011,
201112,
201213,
201314,
201415,
201516,
201617,
201718,
201819,
201920,
202021,
202122.
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