Géométrie et Théorie des Modèles
Année 2025 - 2026
Organisateurs : Raf
Cluckers et George
Comte, Antoine
Ducros, Tamara
Servi.
Pour recevoir le programme par e-mail, écrivez
à : antoine.ducros@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. 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 29 mai 2026 (salle Yvette
Cauchois, bâtiment Perrin, IHP). Orateurs ayant exposé :
11h00. André Belotto (UPC). Resolution of Singular Foliations via Principalization.
I will discuss resolution of singularities for foliations and present our approach
via a weighted principalization theorem for ideals on smooth orbifolds equipped with a foliation.
As an application, I will describe the resolution of certain foliations in arbitrary dimensions, including
Darboux totally integrable foliations. This is joint work with D. Abramovich, M. Temkin, and J. Wlodarczyk.
14h15. Alexi Block-Gorman (Universiteit van Amsterdam). Une caractérisation des géométries des réels automatiques.
Depuis que Julius Büchi a mis en évidence un lien entre les automates et certaines structures issues de la logique mathématique, cette correspondance a fait l’objet d’études approfondies.
Il existe un lien naturel entre l’algèbre booléenne des ensembles définissables et celle des automates et des « langages réguliers » qu’ils reconnaissent.
En franchissant une étape supplémentaire et en associant à chaque mot en base entière la valeur numérique qu’il représente, on peut également établir des connexions entre des phénomènes géométriques et des propriétés d’automates.
Par exemple, on dira qu’une partie X des nombres réels est k‑régulière s’il existe un automate de Büchi qui accepte les développements en base k de tous les éléments de X, et rejette les développements de tous les éléments de son complément.
Dans cet exposé, on considère les sous‑structures k‑régulières de l’expansion du groupe additif réel par toutes les parties k‑régulières, et on les caractérise à partir de leur géométrie, au sens des « topologies modérées » de la théorie des modèles, qui généralisent l’o‑minimalité.
16h00. Tim Santens (Cambridge). The leading constant in Malle's conjecture over function fields.
A conjecture of Malle gives a prediction for the asymptotic growth of the number of number
fields with a given Galois group when ordered by their discriminants. Recently, significant
progress has been made on the function field analogue of this problem by leveraging topological and
algebro-geometric techniques—specifically the homological stability of Hurwitz spaces. In this talk,
I will explore the connection between Malle’s Conjecture and the topology of these Hurwitz space.
I will then present my recent work, which utilizes these homological stability results to prove a version of
Malle's conjecture over function fields with an explicit leading constant.
Vendredi 17 avril 2026 (salle Yvette
Cauchois, bâtiment Perrin, IHP). Orateurs ayant exposé :
11h00. Konstantinos Kartas (Münster). Τame Banach rings.
The almost purity theorem is a foundational result in perfectoid geometry which, as the name suggests, holds only in an “almost” sense.
We aim to identify a class of Banach rings for which the theorem holds in a genuine (non-almost) form.
To this end, we introduce the notion of a tame Banach ring, extending the notion of a tame field, and prove that these rings satisfy the desired property for Galois covers of p-power degree. We also outline some ideas towards the case of general étale covers. Joint work with Franziska Jahnke.
14h15 Adrien Deloro (SU). Paires quasi-Frobenius en théorie des modèles.
Soient G un groupe et H < G un sous-groupe propre. Par analogie avec une configuration célèbre en théorie des groupes finis, la paire est dite quasi-Frobeniu
si H est à la fois disjoint de ses conjugués distincts et d'indice fini dans son normalisateur. Cette définition n'a rien d'arbitraire : elle capture le comportement des tores maximaux dans PGL(2, C), et leurs formes réelles.
On peut conjecturer qu'une paire quasi-Frobenius en contexte modèle-théorique donne soit G résoluble, soit G 《de type PGL(2, C) ou forme réelle》: ce serait une caractérisation groupe-théorique d'un système de racines.
Je présenterai l'état des lieux avec des travaux d'Altınel, Corredor, Onshuus, Rideau-Kikuchi, Wiscons et Zamour.
16h00. Sam Mattheus (Vrije Universiteit Brussel). Pseudorandomness in extremal graph theory.
Abstract: We will show how pseudorandomness plays an increasingly important
role in several areas of extremal graph theory. Interestingly, the sources of pseudorandomness can be quite varied, as we will provide examples from algebraic geometry,
finite group theory, and the theory of buildings. Nevertheless, novel constructions remain highly in demand. To motivate this need, we will survey some recent results where pseudorandom objects match, enhance, or surpass classical probabilistic approaches.
Vendredi 27 mars 2026 (salle Yvette
Cauchois, bâtiment Perrin, IHP). Orateurs ayant exposé :
11h00. Pierre Schapira (SU). Subanalytic sheaves and exponential D-modules.
En voici le résumé.
14h00. Tomas Ibarlucia (UPC). Affine logic and the geometry of simplices of invariant measures.
A Choquet simplex is a compact convex set in which every point is the barycenter of a unique boundary measure. Infinite-dimensional simplices are quite diverse, e.g., any
Polish space can be obtained as the set of extreme points of a metrizable Choquet simplex. However, those that arise naturally in functional analysis and ergodic theory tend to be either Bauer
(i.e., the extreme points form a compact set) or the Poulsen simplex (the unique metrizable simplex whose extreme points are dense). A famous result of Glasner and Weiss captures a precise instance of this dichotomy:
for any countable group G, the simplex Pr_G(2^G) of invariant probability measures of the topological Bernoulli shift is either Bauer or the Poulsen simplex. Moreover, it is Bauer if and only if G has Property (T).
I will discuss a generalization of the Glasner--Weiss Theorem to a larger class of simplices arising from permutation groups.
This addresses questions of Austin motivated by the theory of exchangeable random variables. The proof is based on continuous model theory, and more precisely on recent developments in affine logic.
15h30. Sylvy Anscombe (UPC). Geometric formulations of existential Ax-Kochen-Ershov-statements.
In joint work with Dittmann and Fehm (2023) we showed that decidability of the existential theory of (Fq((t)),t) follows from a property we called (R4),
studied before by Kuhlmann in connection with problems of local uniformization. The property (R4) is: every large field k is existentially closed in every extension that admits a k-rational k-place.
he result from 2023 was an improvement on a result of Denef and Schoutens (2003), who showed that Resolution of Singularities implies the same decidability problem, and related to a previous
result from other work of Fehm and I (2016). Recently Dittmann has proved a general statement, also dependent on (R4), that extends the known results for existential theories of henselian valuation rings in a range of settings.
More recently, with Fehm (2026), we studied weakenings of (R4) to deal only with existential closedness (again of large k in extensions with k-rational k-places)
restricted to existential formulas with at most a certain number of quantifiers. In turn this yields the decidability of the corresponding fragments of the existential theory of k((t)), relative to the corresponding fragment of the existential theory of k.
In this talk I will explain this newer work (previous GTM talks having already addressed the other results), and explore further potential extensions of this fragmented approach which seem to be related to Kuhlmann's work on valuation regular function fields over defectless fields, from the theory of tame valued fields.
Vendredi 30 janvier 2025 (salle Yvette
Cauchois, bâtiment Perrin, IHP). Orateurs ayant exposé :
11h00. Matteo Ruggiero (UPC). On the
Dynamical Manin Mumford problem for polynomial endomorphisms of
the plane.
The Dynamical Manin-Mumford problem is
a dynamical question inspired by classical results from
arithmetic geometry. In the setting of regular polynomial
endomorphisms of C^2 of degree d>=2, it tasks to determine
whether an algebraic curve containing infinitely many preperiodic
points must be itself preperiodic. In a work in collaboration
with Romain Dujardin and Charles Favre, we prove this conclusion
to hold, provided that: (★) the dynamics at infinity has no
superattracting periodic points. The proof is an interesting
blend of techniques from arithmetic geometry and
complex/non-archimedean dynamics. Condition (★) is crucial for
our approach: it ensures that we can work near the Julia set at
infinity at some place, and that the set W where orbits converge
at super-exponential speed d at a fixed point at infinity is a
(invariant) curve. If time allows, I will also present our recent
results about the properties of W in the superattracting case.
14h15. Matteo Verzobio (Institute of Science
and Technology, Austria). Counting rational points on smooth
hypersurfaces.
Let X be a smooth projective
hypersurface defined over Q. We provide new bounds for the
cardinality of rational points of bounded height on X. If X is
smooth and has degree at least 6, we improve the dimension growth
conjecture bound. We achieve an analogue result for affine
hypersurfaces whose projective closure is smooth.
16h00. Simon André (SU)). Homogeneity of
Coxeter groups
A group G is said to be
homogeneous if, for any two tuples u, v of elements of G that
have the same type, there exists an automorphism f of G such that
f(u)=v. I will present some homogeneity results for Coxeter
groups, a class of finitely presented groups generated by
involutions. This talk is based on a joint work with Gianluca
Paolini.
Vendredi 12 décembre 2025 (salle Yvette
Cauchois, bâtiment Perrin, IHP). Orateurs ayant exposé :
11h00. Martin Bays (Oxford). Groups from non-expansion in higher dimension.
.
Call a complex polynomial f(x,y) _expanding_ if there is e>0 such that
for all sufficiently large finite sets A and B of complex numbers with
|B| >= |A|, we have |f(A,B)| > |A|^{1+e}. A result of Elekes and Rónyai
shows that the only non-expanding polynomials f(x,y) are those obtained
from addition or multiplication by composing with unary polynomials.
Thinking of B as parametrising a family of unary polynomials f_b(x)
= f(x,b), we can see this conclusion as placing B in an algebraic group
acting via f. Generalising in these terms, arbitrary nilpotent algebraic
groups and their actions can arise. I will review some results
indicating that this should be the most general situation, including
work with Tingxiang Zou which confirms this in certain cases using
methods from model theory and from additive and incidence combinatorics.
14h15. Loïs Faisant (Ku Leuven). A Motivic Poisson Formula for Split Algebraic Tori
.
Over the past ten or twenty years, a number of works—by Bilu & Browning, Browning & Sawin, Browning & Vishe, Bourqui, Chambert-Loir & Loeser, Glas & Hase-Liu, Peyre,
among others—have demonstrated how number theory, particularly analytic number theory, can provide new insights into the study of moduli spaces of curves.
Notably, the recent development of motivic versions of number-theoretic tools (harmonic analysis on adèles, the circle method, lifting to universal torsors)
has paved the way for a motivic version of Manin’s program: the dictionary between number fields and function fields allows one to move from the fine study of the distribution of rational points
on Fano varieties to predictions about the virtual motive of the moduli space of morphisms from a given curve to a Fano variety over the complex numbers.
The purpose of this talk is to present the results of a collaboration with Margaret Bilu (CNRS/École Polytechnique), in which we develop a multiplicative version of the motivic Poisson formula.
This allows us to demonstrate a motivic stabilization phenomenon concerning the moduli space of morphisms from an algebraic curve (projective, smooth, of arbitrary genus) to a toric variety.
Over the past ten or twenty years, a number of works—by Bilu & Browning, Browning & Sawin, Browning & Vishe, Bourqui, Chambert-Loir & Loeser, Glas & Hase-Liu, Peyre,
among others—have demonstrated how number theory, particularly analytic number theory, can provide new insights into the study of moduli spaces of curves.
Notably, the recent development of motivic versions of number-theoretic tools (harmonic analysis on adèles, the circle method, lifting to universal torsors)
has paved the way for a motivic version of Manin’s program: the dictionary between number fields and function fields allows one to move from the fine study of the distribution of rational points
on Fano varieties to predictions about the virtual motive of the moduli space of morphisms from a given curve to a Fano variety over the complex numbers.
The purpose of this talk is to present the results of a collaboration with Margaret Bilu (CNRS/École Polytechnique), in which we develop a multiplicative version of the motivic Poisson formula.
This allows us to demonstrate a motivic stabilization phenomenon concerning the moduli space of morphisms from an algebraic curve (projective, smooth, of arbitrary genus) to a toric variety.
16h00. Vadim Lebovici (Sorbonne Université). Formules cinématiques additives et fonctions constructibles.
En géométrie intégrale, la formule cinématique additive exprime le volume moyen de la somme de Minkowski de deux compacts convexes placés aléatoirement dans l'espace euclidien.
Que se passe-t-il si on ne les suppose plus convexes ? Dans un travail en collaboration avec Andreas Bernig, nous montrons une formule cinématique additive pour les sous-variétés compactes sous-analytiques (potentiellement à bord, et même à coins) de l'espace euclidien.
La clé est de généraliser la somme de Minkowski par la convolution des fonctions constructibles définie par Viro et Schapira dans les années 80. Cette convolution, fondée sur des calculs de caractéristique d'Euler, est définie à l'aide d'un formalisme des opérations sur les fonctions constructibles.
Vendredi 7 novembre 2025 (salle Yvette
Cauchois, bâtiment Perrin, IHP). Orateurs ayant exposé :
11h00. Gareth Jones (University of Manchester).
Some effective special point results .
I'll
discuss some joint work with Binyamini, Schmidt, and Thomas, in
which we prove an effective and uniform version of Manin-Mumford
for products of CM elliptic curves. I'll show how we deduce from
this an effective Andre-Oort result for fibre powers of the
Legendre family. I'll then discuss some work in progress with
Schmidt , on extending the above to multiplicative extensions,
and perhaps some other similar results.
14h15. Vlerë Mehmeti (Sorbonne Université,
ENS Paris). Local-global principles and the u-invariant.
I will be speaking of local-global principles
obtained by working on non-Archimedean Berkovich analytic curves,
which are defined over complete rank 1 valued fields. By local
considerations in the case of quadratic forms, one can then
obtain upper bounds on a related invariant. I will also speak of
some recent generalizations obtained through this approach
together with K.J. Becher and N. Daans, which make it possible to
get rid of the "rank 1" assumption on the valuation.
16h00. Philip Dittmann (University of
Manchester). Existential theories of henselian valued fields
in positive characteristic with parameters .
While
the model theory of henselian valued fields in residue
characteristic zero is completely understood, the situation in
positive characteristic is rather more subtle, even for local
fields like the formal Laurent series field F_p((t)). This
applies even when analysing only existential theories, which are
arguably of the strongest interest in arithmetic, cf. Hilbert's
Tenth Problem. Notable progress was made here in particular by
Anscombe–Fehm, who showed that the existential theory of the
ring F_p((t)) without parameters is decidable, and
Denef–Schoutens, who showed the same when allowing the
parameter t assuming Resolution of Singularities. The latter
result was later improved in joint work of mine with
Anscombe–Fehm. I will report on ongoing work in this direction,
focussing on existential theories of henselian valued fields like
K((t)) for some base field K of positive characteristic with
parameters from a trivially valued base field. As an application,
it is possible to find wide classes of function fields F such
that it is decidable which polynomial equations over F have
solutions in almost all completions of F, as well as stronger
results under a Resolution of Singularities assumption. Some of
this was first explored in joint work with Fehm.
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