Géométrie et Théorie des Modèles

Année 2024 - 2025

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.

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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.

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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).

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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.

Vidéo

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.

Vidéo

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).

Vidéo

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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.

Vidéo

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.

Vidéo

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.

Vidéo

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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, 2022-23, 2022-23, 2023-24.
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