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.
Vidéo.
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.
Notes de l'exposé
et vidéo.
Vendredi 9 décembre 2022, à l'ENS, salle
W. Orateurs :
14h15 : François
Loeser (IMJPRG), Un théorème de finitude pour
les fonctions tropicales sur les squelettes
Les squelettes sont des sousensembles linéaires par morceaux d'espaces analytiques nonarchimédiens 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, semianalytic everywhereincluding infinityand 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 nonalgebraic, 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, semianalytic) 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.
Vendredi 24 mars
2023, à l'IHP, salle 314. Programme :
11h : Blaise Boissonneau (KU
Leuven), Defining valuations using
this one weird trick
In this talk, we present classical methods to define valuations and use them to derive conditions on the residue fields and value groups guaranteeing definability, and discuss how close these conditions are to being optimal.
Vidéo
14h15 : Arno Fehm (TU
Dresden), Axiomatizing the
existential theory of F_p((t))
From a model theoretic point of view, local fields of positive
characteristic, i.e. fields of Laurent series over finite
fields, are much less well understood than their
characteristic zero counterparts  the fields of real,
complex and padic numbers. I will discuss different
approaches to axiomatize and decide at least their
existential theory in various languages and under various
forms of resolution of singularities. From a geometric
point of view, deciding the existential theory essentially
means to determine algorithmically which algebraic
varieties have rational points over these fields. Joint
work with Sylvy Anscombe and Philip Dittmann.
16h : Sebastian
Eterovic (Leeds), Generic solutions to
systems of equations involving functions from arithmetic
geometry
In arithmetic geometry one encounters many important transcendental
functions exhibiting interesting algebraic properties. Perhaps
the most famous example of this is the complex exponential
function, which is wellknown to satisfy the definition of a
group homomorphism. When studying these algebraic properties,
a very natural question that arises is something known as the
“existential closedness problem”: when does an
algebraic variety intersect the graph of the function in a
Zariski dense set?
In this talk I will introduce the existential closedness problem, we
will review what is known about it, and I will present results about a
strengthening of the problem where we seek to find a point in the
intersection of the algebraic variety and the graph of the function
which is generic in the algebraic variety.
Vidéo
Vendredi 21 avril 2023, à l'IHP,
amphithéâtre Hermite.
Programme :
11h : Margaret Bilu (IMB, Bordeaux), A motivic circle method
The HardyLittlewood circle method is a wellknown
technique of analytic number theory that has successfully
solved several major number theory problems. In particular, it
has been instrumental in the study of rational points on
hypersurfaces of low degree. More recently, a version of the
method over function fields, combined with spreading out
techniques, has led to new results about the geometry of
moduli spaces of rational curves on hypersurfaces of low
degree. In this talk I will show how to implement a circle
method with an even more geometric flavour, where the
computations take place in a suitable Grothendieck ring of
varieties, and explain how this leads to a more precise
description of the geometry of the above moduli spaces. This
is joint work with Tim Browning.
Vidéo
14h15 : Philipp Hieronymi (Bonn), Fractals and Model
Theory This talk is motivated by the following fundamental question: What is the logical/modeltheoretic complexity generated by fractal objects?
Here I will focus on fractal objects defined in firstorder expansions of the ordered real additive group. The main problem I want to address here is: If such an expansion defines a fractal object, what can be said about its logical complexity in the sense of Shelahstyle combinatorial tameness notions such as NIP and NTP2? The main results I will mention are joint work with Erik Walsberg.
Vidéo
16h : Martin Hils (Münster), Spaces of definable
types and beautiful pairs in unstable theories
By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is “meaningful” precisely when the set of all definable types in T is strict prodefinable, which is the case if and only if T is nfcp.
We transfer the notion of beautiful pairs to unstable theories and study them in particular in henselian valued fields, establishing AxKochenErshov principles for various questions in this context. Using this, we show that the theory of beautiful pairs of models of ACVF is “meaningful” and infer the strict prodefinability of various spaces of definable types in ACVF, e.g., the model theoretic analogue of the Huber analytification of an algebraic variety.
This is joint work with Pablo Cubides Kovacsics and Jinhe Ye.
Vidéo
Vendredi 16 juin 2023. Jussieu, salle 101, couloir
1516. Programme :
11h : Neer
Bhardwaj (Weizman), Approximate PilaWilkie type counting for
complex analytic sets
We develop a variation of the PilaWilkie counting theorem,
where we count rational points that approximate bounded complex
analytic sets. A unique aspect of our result is that it does not
depend on the analytic set (or family) in question. We apply
this approximate counting to obtain an effective PilaWilkie
type statement for analytic sets cut out by computable
functions. This is joint work with Gal Binyamini.
14h15 : Sylvy Anscombe (IMJPRG), Interpretations of fragments of
theories of fields
In previous work with Fehm, and then Dittmann and Fehm, we found that the existential theory of an equicharacteristic henselian valued field is axiomatised using the existential theory of its residue field, conditionally, similar to an earlier theorem of Denef and Schoutens  giving a transfer of decidability for existential theories. In this talk I'll describe parts of ongoing work with Fehm (in the main different to those discussed recently at CIRM) in which we use an “abstract” framework for interpreting families of incomplete theories in others in order to find transfers of decidability in various settings. I will discuss consequences for theories of PAC fields and parts of the universalexistential theory of equicharacteristic henselian valued fields.
16h : Tom Scanlon (Berkeley), (Un)likely intersections and
definable complex quotient spaces
The ZilberPink conjectures predict that if S is a special variety, X ⊆ S is an irreducible subvariety of S which is not contained in a proper special subvariety, then the union of the unlikely intersections of X with special subvarieties of S is not Zariski dense in X, where here, an intersection between subvarieties X and Y of S is unlikely if dim X + dim Y < dim S. To make this precise, we need to specify what is meant by “special subvariety”. We will do so through the theory of definable complex quotient spaces, modeled on those introduced by Bakker, Klingler, and Tsimerman. Using this formalism we will prove a complement to the ZilberPink conjecture to the effect that under some natural geometric conditions likely intersections will be Zariski dense in X (joint work with Sebastian Eterović) and in the other direction that a function field version of the ZilberPink conjecture holds effectively (joint work with Jonathan Pila).
Programme des séances
passées : 200607,
200708,
200809,
200910,
201011,
201112,
201213,
201314,
201415,
201516,
201617,
201718,
201819,
201920,
202021,
202122.
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