Géométrie et Théorie des Modèles
Année 2024 - 2025
Organisateurs : Zoé
Chatzidakis, Raf
Cluckers et George
Comte, Antoine
Ducros, Tamara
Servi.
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 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.
Vendredi 16 mai 2025 (salle
Maryam Mirzakhani, IHP). Orateurs ayant exposé :
11h00. Ludovic Rifford
(Université de Nice-Côte d'Azur). Sur la conjecture de Sard
de rang minimal en ge?ome?trie sous-riemannienne.
La Conjecture de Sard
classique prévoit que l’image de toutes les courbes
singulières partant d’un point fixé sur une variété équipée
d’une structure sous-riemannienne est de mesure nulle. Nous
discuterons dans cet exposé d’une conjecture plus faible
portant uniquement sur les courbes singulières de rang minimal.
Nous expliquerons comment ce problème est relié, dans le cas
analytique réel, aux propriétés de certains feuilletages
sous-analytiques et présenterons des résultats positifs dans le
cas de feuilletages dit « splittable ». Ceci est tiré d’un
travail en collaboration avec André Belotto et Adam Parusinski.
14h15. Elliott Kaplan (Max
Planck Institute, Bonn). Towards an o-minimal asymptotic
differential algebra.
Recently, Aschenbrenner, van
den Dries, and van der Hoeven showed that all maximal Hardy
fields have the same first-order theory as the field of
LE-transseries as differential fields. As a consequence, they
deduce a transfer theorem for algebraic ODEs. I will discuss
extensions of this transfer theorem to other classes of "tame"
ODEs, including restricted elementary and signomial ODEs. I will
also describe partial progress for extending this to ODEs
definable in a polynomially bounded o-minimal expansion of the
real field.
16h00. Antoine Chambert-Loir
(Université Paris-Cité). Continuité des intégrales-fibre
en géométrie analytique non archimédienne.
En
géométrie complexe, un théorème de Stoll (1966) exprime la
continuité de l'intégrale d'une (q,q)-forme à support compact
dans les fibres d'une application à fibres purement de dimension
q, à condition de prendre en compte la multiplicité de cette
application. J'expliquerai comment la théorie des formes
différentielles réelles sur les espaces analytiques non
archimédiens donne lieu à un théorème analogue où,
cependant, la multiplicité est prise en compte automatiquement
par la notion d'intégrale sur un espace analytique. Il s'agit
d'un travail avec Antoine Ducros, fondé sur notre article
prépublié en 2012, et sur la révision substantielle que nous
en avons faite depuis.
4 avril 2025, IHP, amphi
Darboux. Orateurs ayant exposé :
10h30 : Adele Padgett
(Vienna). O-minimal definability and some functional
transcendence properties of the Gamma function
O-minimality has been used to
prove functional transcendence results for important periodic
functions like exponentiation and the j function. The Gamma
function, which is not periodic but which satisfies simple
functional equations, is definable in an o-minimal structure when
restricted to certain unbounded regions in the complex plane. In
the first part of the talk, I will present work with P.
Speissegger on definable holomorphic continuations of functions
definable in two particular o-minimal structures, with an
application to definability of the complex Gamma function. Then I
will discuss some functional transcendence properties of the
Gamma function.
Après-midi
d'hommage à Zoé
14h30 : Ouverture
14h45 : Ehud Hrushovski
(Oxford). On the basic structures of difference equations.
This will be a personal talk
surveying some of my work with Zoé Chatzidakis over the last
three decades. Mathematically it forms a rather coherent chapter
in the basic model theory of difference equations, combining
ideas from geometric stability and simplicity, model-theoretic
algebra and algebraic geometry. The signposts include
axiomatization, two theories of dimension, higher amalgamation,
elimination of imaginaries, stable embeddedness; a structure
theorem for the basic geometries, taking the form of a
trichotomy; beyond finite dimensional difference varieties, a
stationarity theorem; and applications to algebraic dynamics. To
the extent that time permits, I will discuss a forthcoming joint
result refining the trichotomy a little: finite-dimensional
difference varieties admit dévissage to a combination of
one-dimensional ones, and ones coding the dynamics of
multiplication by a fixed group element in an algebraic
homogeneous space, and a soon to be published work of Zoé's on
the structure of locally modular groups.
16h30 : Martin Hils (Münster).
A tour through the model theory of pseudofinite fields and
other PAC fields. Definable measures and groups, and amalgamation
properties.
In this second of the two
afternoon talks in hommage to Zoé, I will revisit some of her
major research contributions to the model theory of PAC fields
and more specifically pseudofinite fields, mentioning also
connections to some recent work of myself. While, undoubtedly,
the model theory of difference fields (and the long-term
mathematical collaboration with Udi Hrushovski) takes central
stage in Zoé's research career, the model theoretic study of
pseudofinite and more general PAC fields has been an important
topic throughout Zoé's entire mathematical life, from the
construction of a well-behaved measure on definable sets in
pseudofinite fields (obtained in 1992 in a highly influential
joint work with Lou van den Dries and Angus Macintyre), the study
of their amalgamation properties and the relation to simple
theories (joint with Anand Pillay in the late 90's) to her recent
work, joint with Nick Ramsey, on measures in e-free PAC fields
and the definable amenability of definable groups therein.
17 janvier 2025, IHP, salle
Olga Ladyjenskaïa. Orateurs ayant exposé :
11h00 : Faustin Adiceam
(Université de Paris-Est Créteil), Homogeneous forms
inequalities.
The talk is concerned with
counting the number of solutions to a system of inequalities of
the form |F(x)| < A and ||x|| < B, where F is a real
homogeneous form in n variables and A, B are parameters,
requiring that the vector x should lie in a lattice. The
presented results deal with the case where the lattice is chosen
either randomly or deterministically. This is joint work with
Oscar Marmon (Lund University).
14h15 : Ulla Karhumäki
(Helsinki), Pseudofinite primitive permutation groups of
finite SU-rank.
A (definably) primitive
permutation group (G,X) is a group G together with a transitive
faithful and definable action on X such that there are no proper
nontrivial (definable) G-invariant equivalence relations on X.
Definably primitive permutation groups of finite Morley rank are
well-studied: in particular, it is shown by Macpherson and Pillay
that such a group with infinite point stabilisers is actually
primitive and by Borovik and Cherlin that, given such a group
(G,X), the Morley rank of G can be bounded in terms of the Morley
rank of X. We show similar results for a pseudofinite definably
primitive permutation group (G,X) of finite SU-rank: we first
show that (G,X) is primitive if and only if the point stabilisers
are infinite. This then allows us to apply a classification
result by Liebeck, Macpherson and Tent on (G,X) so that we may
bound the SU-rank of G in terms of the SU-rank of X. This is
joint work in with Nick Ramsey.
16h00 : Olivier Benoist
(ENS), Sums of squares of real-analytic functions.
Artin solved Hilbert's 17th
problem by showing that any real polynomial in n variables that
is nonnegative is a sum of squares of rational functions. Pfister
improved quantitatively Artin's theorem by showing that 2^n
squares suffice. In this talk, we will present new quantitative
results à la Pfister in the real-analytic setting (where
polynomials are replaced with real-analytic functions).
13 décembre 2024 (IHP, Salle
Pierre Grisvard, Bâtiment Borel). Orateurs ayant exposé :
11h : Martin Hils (U.
Münster), Lang-Weil type point estimates in finite difference
fields
In the talk, I will present a
recent result, establishing Lang-Weil type bounds for the number
of rational points of difference varieties over finite difference
fields, in terms of the transformal dimension of the variety and
assuming the existence of a smooth rational point. It follows
that in (certain) non-principle ultraproducts of finite
difference fields the coarse dimension of a quantifier-free type
equals its transformal transcendence degree.
The proof uses
a strong form of the classical Lang-Weil estimates and, as key
ingredient to obtain equidimensional Frobenius specializations,
the recent work of Dor and Hrushovski on the non-standard
Frobenius acting on an algebraically closed non-trivially valued
field, in particular the pure stable embeddedness of the residue
difference field in this context.
This is joint work with
Ehud Hrushovski, Jinhe Ye and Tingxiang Zou.
14h15 : Mickaël
Matusinski (Bordeaux), About power series expansions of
Pfaffian functions in one variable
We consider Pfaffian functions
(after Khovanski) defined in a neighbourhood of 0. On the one
hand, we investigate how the coefficients of their MacLaurin
expansion are determined by the coefficients of the equations. On
the other hand, given the MacLaurin expansion of a Pfaffian chain
of a given order and degree, we reconstruct explicitly the space
of equations it satisfies. Altogether, these leads to an explicit
parametric description of the space of monovariate Pfaffian
functions stratified by their order and degree. This is a work in
progress with Siegfried Van Hille.
16h : Adam Parusinski
(Nice), Perturbation of Polynomials and Linear Operators.
The Perturbation of
Polynomials and Linear Operators is a classical subject which
started with Rellich's work in the 1930s. The parameter
dependence of the polynomials (resp. operators) ranges from real
analytic over C^∞ to differentiable of finite order with often
drastically different regularity results for the roots (resp.
eigenvalues and eigenvectors). In this talk I will present
several recent results such as an optimal estimate of Sobolev
regularity of roots monic complex polynomials of one variable
with coefficients depending smoothly on one real parameter,
multiparameter versions of this result, and the problem of
continuity of “coefficients to roots map” with respect to the
C^d and the Sobolev norms. Recently, these results were
reinterpreted by Antonini, Cavalletti, and Lerario in terms of
Wasserstein metric in order to study optimal transport between
algebraic hypersurfaces in the complex projective space.
In
some cases better regularity of the roots can be obtained under
additional assumptions of non-oscilation or finiteness of ordrer
of contact between the roots, that is an interesting property if
one works with coefficients definable in o-minimal structures.
(based mainly on the joint work with Armin Rainer).
25 octobre 2024 (IHP, Salle
Pierre Grisvard, 3e étage, Bâtiment Borel). Programme :
11h - Akash Hossain
(Paris-Saclay, akash.hossain_at_universite-paris-saclay.fr), A
low-level description of types in DOAG, with applications to
independence.
Motivated by connections with
questions from model theory of valued fields, we investigate
problems of geometric nature in the model theory of divisible
ordered Abelian groups (DOAG). We are particularly interested in
finding algebraic characterizations of a model-theoretic
independence relation, called non-forking independence. There was
in previous literature an unsuccessful attempt to find such
characterizations in DOAG, using standard techniques from
o-minimal theory. We carried out a lower-level study of the
geometric properties of ordered Abelian groups, and we found
“invariants” which give us more control on types than what
o-minimality allows, in particular we did compute forking in
DOAG.
In this talk, we will present the geometric aspects
of our work, describe those invariants, and explain the
connections to forking.
14h15 - Rémi Jaoui
(Lyon 1, jaoui_at_math.univ-lyon1.fr), Integration in finite
terms and exponentially algebraic functions (séminaire
itinérant EFI).
Liouville introduced the class
of elementary functions to study analogues of the notion of
resolubility by radicals for algebraic equations for
transcendental and differential equations. Can the primitive of
an algebraic function be expressed as an elementary function? Is
the restricted (real-analytic) cosine function definable in the
structure (R,+,x, exp)? Does a planar vector field admits an
elementary integral?
In my talk, I will describe how the
(omega-stable) theory of blurred exponential fields axiomatized
by Kirby around 2007 provide a new framework for the development
of model-theoretic techniques to unify and study the various
notions of integrability by elementary functions. This is joint
work with Jonathan Kirby.
16h - Vincent Bagayoko
(IMJ-PRG, bagayoko_at_imj-prg.fr), Some valuation theory of
functional equations over regular growth rates
Groups under composition of
regular growth rates, together with an ordering or an
exponentiation in the sense of Miasnikov-Remeslennikov, naturally
appear in o-minimal geometry and asymptotic differential algebra.
Yet little is known about their first-order properties. There is
no compositional analog of the now well-studied first-order
theory of H-fields, and no good theory of extensions of such
expansions of groups.
Given a word w(y) over a group G with
a single variable y, the existence of a solution to w(y)=1 in an
extension of G is in general a difficult problem. It fails even
for certain specific types of equations if one wants to preserves
certain first-order properties of G, such as orderability. I
expect that this question is more traceable within an elementary
class of ordered groups that contains certain groups of o-minimal
germs. I will explain how to use of valuations on groups, ordered
groups and exponential groups as tools to study equations over
such groups, and show how one can recover more general results
about unary equations over torsion-free groups.
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