Séminaires : Géométrie et Théorie des Modèles

Equipe(s) Responsable(s)SalleAdresse
Logique Mathématique
Zoé Chatzidakis, Raf Cluckers, Georges Comte



Pour recevoir le programme par e-mail, écrivez à : zoe.chatzidakis@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 : https://webusers.imj-prg.fr/~zoe.chatzidakis/papiers/MTluminy.dvi/MTluminy.dvi. Ces personnes 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.Retour ligne automatique
Les notes de quelques-uns des exposés sont disponibles.

Séances à suivre

Orateur(s)Titre Date DébutSalleAdresseDiffusion
+ Séances antérieures

Séances antérieures

Orateur(s)Titre Date DébutSalleAdresse
+ Emmanuel Peyre Espaces de modules asymptotiques 24/05/2024 11:00 15-16 salle 101 Campus Pierre et Marie Curie

La version géométrique du programme de Manin conduit à des prédictions très précises sur le comportement asymptotique des espaces de modules de morphismes d'une courbe dans une variété qui fait intervenir un produit eulérien motivique dû à M. Bilu.
Après la description d'un cas concret élémentaire, le but de cet exposé est d'expliquer la formule heuristique conjecturée ainsi que le principe d'équidistribution qui lui est associé.


+ Thomas Scanlon Manin maps. differential algebra, o-minimality, and intermediate Kodaira-Spencer rank 24/05/2024 14:15 15-16, salle 101 Campus Pierre et Marie Curie

In 1963, Manin used a construction related to the work of Fuchs, Gauss, and Picard on linear differential operators to prove a function field version of the Mordell conjecture about rational points on algebraic curves.  Over the years, Manin's construction has been reinterpreted in various ways , most notably using differential algebra.  Indeed, the analytic presentation is generally regarded as a heuristic, as it was even in Manin's proofs.   I will describe some work (joint with T. Dupuy and J. Freitag)  using o-minimality and differential algebra exploiting the analytic presentation to explain some known properties of the Manin maps (e.g. Manin's theorem of the kernel and the one basedness of Manin kernels of non-isotrivial simple abelian varieties), to produce examples of simple abelian varieties with Manin kernels of intermediate rank, and to show why such examples cannot come from low dimensional families of abelian varieties.

+ James Freitag Variations on the degree of nonminimality 24/05/2024 16:00 15-16 salle 101 Campus Pierre et Marie Curie

A few years ago, Rahim Moosa and I introduced the degree of nonminimality, which was designed for proving transcendence statements for solutions of differential equations. We'll spend the first half of the talk developing the notion and describing the applications. The degree of nonminimality is just one way to make quantitative the general model theoretic fact that dividing of any partial type can always be detected by some indiscernible sequence in the parameters of the formulas. In the second half of the talk, we will describe several variations on the notion which have recent applications beyond transcendence results.

+ Ehud Hrushovski Definable model equivalence relations and their invariants 15/03/2024 10:30 15-16, 413 Jussieu

An interpretation between theories can be presented as a composition of the construction of imaginary sorts, and the taking of reducts. In this work with Michael Benedikt, we consider more general ways of reducing structure, using definable equivalence relations on models with a given universe or, equivalently as it turns out, definable groupoids extending the groupoid of models and isomorphisms. We characterize the simplest ones from several points of view; continuous logic turns out surprisingly to play an intrinsic role. Examples seem to hint at a possibility of contact with categories that are usually inaccessible to definability considerations, notably from differential geometry. This is a preliminary investigation, and I hope to be able to give complete proofs of the main results.

+ Yohan Brunebarbe Algebraicity of Shafarevich morphisms 15/03/2024 14:30 16-26, 113 Jussieu

For a normal complex algebraic variety X equipped with a semisimple complex local system V, a Shafarevich morphism X → Y is a map which contracts precisely those algebraic subvarieties on which V has finite monodromy. The existence of such maps has interesting consequences on the geometry of universal covers of complex algebraic varieties. Shafarevich morphisms were constructed for projective X by Eyssidieux, and recently have been constructed analytically in the quasiprojective case independently by Deng--Yamanoi and myself using techniques from non-abelian Hodge theory. In joint work with B. Bakker and J. Tsimerman, we show that these maps are algebraic, and that in fact Y is quasiprojective. This is a generalization of the Griffiths conjecture on the quasiprojectivity of images of period maps, and the proof critically uses o-minimal geometry.

+ Lou van den Dries Analytic Hardy fields 15/03/2024 16:15 15-16, 101 Jussieu

Joint work of Matthias Aschenbrenner, Joris van der Hoeven, and me led to the following two theorems about maximal Hardy fields:
(1) they are all elementarily equivalent to the ordered differential field of transseries; 
(2) they are η_1 in the sense of Hausdorff.

This happened several years ago. As to (1), the proof goes through with “Hardy field” replaced by “analytic Hardy field” (with corresponding notion of “maximal”). This was not the case for (2), where we used gluing constructions and partitions of unity unavailable in the analytic context. Last year, Aschenbrenner and I did establish (2) also in the analytic case by reduction to the non-analytic setting, using Whitney's powerful approximation theorem. I will give an overview of this, recalling also the background about transseries and asymptotic differential algebra. There are further things to say about analytic Hardy fields that have no obvious analogue for arbitrary Hardy fields, such as analytic continuation to the complex plane. The second part of my talk will be about that. Some of this, in particular possible connections to o-minimality, will be partly speculative.

+ Alex Wilkie Analytic Continuation and Zilber's Quasiminimality Conjecture 26/01/2024 11:00 Amphitheatre Yvonne Choquet-Bruhat, Bâtiment Perrin IHP

This is the title of a paper that has recently been accepted for the volume of the journal “Model Theory” dedicated to Boris Zilber on the occasion of his 75th birthday. (The paper can be found on the GTM preprint server or on arXiv.) The conjecture asserts that every definable subset of the complex field expanded by the complex exponential function is either countable or cocountable. In the paper I propose a conjecture concerning the analytic continuation of o-minimally defined complex analytic functions which implies Zilber's conjecture (and much more) and in this talk I will give an outline of the main argument in the paper as well as some further remarks. (I was going to write “as well as some recent progress”, but that would be too strong!) 


+ Gabriel Conant Group compactifications in continuous logic, with applications to multiplicative combinatorics 26/01/2024 16:15 Salle Yvette Cauchois IHP

I will discuss recent work on the general theme of continuous logic as an environment well-suited for certain methods in multiplicative combinatorics (i.e., the extension of additive combinatorics to noncommutative groups). The starting point is Pillay's result that the connected component of a definable compactification of a pseudofinite group is abelian. In joint work with Hrushovski and Pillay, we give a short proof of this using only classical tools, including a result of A. Turing on finitely approximated Lie groups. Using a connection between Turing's theorem and a (relatively) more recent result of Kazhdan on approximate homomorphisms, one obtains a generalization of Pillay's theorem to ultraproducts of amenable torsion groups. In previous work on “tame arithmetic regularity”, the results of Pillay and of Kazhdan were instrumental for introducing classical Bohr neighborhoods into the setting of noncommutative groups. However, the execution of this approach was quite complicated due to certain drawbacks of classical first-order logic. In the paper with Hrushovski and Pillay, we build Kazhdan's result into continuous logic in order to remove these complications. As an illustration of the method, we use the stabilizer theorem to extend a fundamental result from additive combinatorics (called Bogolyubov's Lemma) to arbitrary amenable groups. More recently, in work with Pillay, we combine this continuous setting with local stability to prove a regularity lemma for “stable functions” on amenable groups. This result is an analytic analogue of the arithmetic regularity lemma for stable subsets of finite groups, proved first in the abelian case by Terry and Wolf, and then generalized by myself, Pillay, and Terry. As a consequence of stability of Hilbert spaces, the analytic stable arithmetic regularity lemma applies to convolutions of arbitrary functions on amenable groups. This allows one to deduce the previous generalization of Bogolyubov's Lemma as a quick corollary of analytic stable arithmetic regularity.

+ Antoine Ducros Stratification of the image of a map between analytic spaces 26/01/2024 14:15 Salle Yvette Cauchois (Batiment Perrin) IHP

Let f : Y → X be a morphism between compact Berkovich spaces over an arbitrary non-Archimedean field. In general, the structure of the image f(Y) appears to be rather mysterious, unless one makes strong assumption on f (like flatness, or properness). Nevertheless, I will explain how recent flattening results in non-Archimedean geometry allow to exhibit, under very weak assumptions on f (automatically fulfilled if Y is irreducible, for example) a finite stratification of f(Y) with reasonable pieces (each of them is a Zariski-closed subset of an analytic domain of X).


+ Jonathan Pila Ax-Schanuel and exceptional integrability 24/11/2023 11:00 salle Yvette Cauchois (Batiment Perrin) IHP

In joint work with Jacob Tsimerman we study when the primitive of a given algebraic function can be constructed using primitives from some given finite set of algebraic functions, their inverses, algebraic functions, and composition. When the given finite set is just {1/x} this is the classical problem of “elementary integrability” (of algebraic functions). I will discuss some results, including a decision procedure for this question, and further problems and conjectures.

+ Floris Vermeulen Dimension growth for affine varieties. 24/11/2023 16:30 Salle Yvette Cauchois (Batiment Perrin) IHP

Given a projective algebraic variety X over Q, the dimension growth conjecture predicts general upper bounds for the number of points of bounded height on X. It was originally conjectured by Serre, and independently in a uniform way by Heath-Brown. By work of Browning, Heath-Brown and Salberger, uniform dimension growth is now a theorem. 
I will give a general overview of dimension growth and explain some ideas of the proof. The main ingredient is the so-called determinant method, which goes back to Bombieri and Pila, and has been successfully applied to many counting problems. I will then turn to dimension growth for affine varieties, and report on recent work with Raf Cluckers, Pierre Dèbes, Yotam Hendel, and Kien Nguyen.

+ George Comte Inequalities for some metric motivic invariants 24/11/2023 14:15 Salle Yvette Cauchois (Bâtiment Perrin) IHP

In a joint work with Immanuel Halupczok we introduce, on one hand, a partial preorder on the set of motivic constructible functions, 
extending the one originally defined by Cluckers and Loeser, and, on the other hand, a notion of number of connected components, still in the definable 
nonarchimedean context. For the last one we use the existence special canonical stratifications. Those two notions meet, for instance, in a nonarchimedean 
version of a real inequality involving the metric entropy and integral-geometric invariants, called Vitushkin invariants. I will try to explain how. 

+ Neer Bhardwaj Approximate Pila-Wilkie type counting for complex analytic sets 16/06/2023 11:00 couloir 15-16, salle 101 Campus Pierre et Marie Curie

We develop a variation of the Pila-Wilkie 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 Pila-Wilkie type statement for analytic sets cut out by computable functions.

This is joint work with Gal Binyamini

+ Tom Scanlon (Un)likely intersections and definable complex quotient spaces 16/06/2023 16:00 Couloir 15-16, salle 101 Campus Pierre et Marie Curie

The Zilber-Pink 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 Zilber-Pink 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 Zilber-Pink conjecture holds effectively (joint work with Jonathan Pila).

+ Sylvy Anscombe Interpretations of fragments of theories of fields 16/06/2023 14:15 Couloir 15-16, salle 101 Campus Pierre et Marie Curie

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 universal-existential theory of equicharacteristic henselian valued fields.



+ Margaret Bilu A motivic circle method 21/04/2023 11:00 IHP, Amphithéâtre Hermite

The Hardy–Littlewood circle method is a well-known 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. 


+ Martin Hils Spaces of definable types and beautiful pairs in unstable theories 21/04/2023 16:00 IHP, Amphitheatre Hermite

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 pro-definable, 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 Ax-Kochen-Ershov 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 pro-definability 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.

+ Philipp Hieronymi Fractals and Model Theory 21/04/2023 14:15 IHP, Amphitheatre Hermite

This talk is motivated by the following fundamental question: What is the logical/model-theoretic complexity generated by fractal objects?
Here I will focus on fractal objects defined in first-order 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 Shelah-style combinatorial tameness notions such as NIP and NTP2? 
The main results I will mention are joint work with Erik Walsberg.

+ Blaise Boissonneau Defining valuations using this one weird trick 24/03/2023 11:00 IHP, salle 314 IHP

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.

+ Sebastian Eterovic Generic solutions to systems of equations involving functions from arithmetic geometry 24/03/2023 16:00 IHP, salle 314 IHP

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 well-known 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.

+ Arno Fehm Axiomatizing the existential theory of F_p((t)) 24/03/2023 14:15 IHP, salle 314 IHP

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 p-adic 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. 

+ Alex Wilkie Integer points on analytic sets 09/12/2022 16:00 Salle W, ENS

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, semi-analytic everywhere-including infinity-and 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 non-algebraic, 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, semi-analytic) 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.

+ François Loeser Un théorème de finitude pour les fonctions tropicales sur les squelettes 09/12/2022 14:15 Salle W, ENS

Les squelettes sont des sous-ensembles linéaires par morceaux d'espaces analytiques non-archimédiens 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.

+ Franziska Jahnke Taming perfectoid fields 25/11/2022 11:00 Salle 01 IHP

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 model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger 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 Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof). 
This is joint work with Konstantinos Kartas.

+ Rémi Jaoui Abundance of strongly minimal autonomous differential equations 25/11/2022 14:15 Salle 01 IHP

In several classical families of differential equations such as the Painlevé families (Nagloo, Pillay) or finite dimensional families of Schwarzian differential equations (Blazquez-Sanz, 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 model-theoretic 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.

+ Hector Pasten On non-Diophantine sets in rings of functions 25/11/2022 16:15 Zoom, et salle 01 IHP

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 well-understood 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 Garcia-Fritz and Pheidas on showing that several sets and relations over rings of polynomials and rational functions that are not Diophantine.

+ Thomas Scanlon Skew-invariant curves and algebraic independence 13/05/2022 14:15 Salle W, ENS, et Zoom ENS

A σ-variety over a difference field (K,σ) is a pair (X,φ) consisting of an algebraic variety X over K and φ:X → X^σ is a regular map from X to its transform Xσ under σ. A subvariety Y ⊆ X is skew-invariant if φ(Y) ⊆ Y^σ. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of σ-varieties of the form (𝔸^n,φ) where φ(x_1,...,x_n) = (P_1(x_1),...,P_n(x_n)). The most important case, from which the others may be deduced, is that of n = 2. In the present work we give a sharper description of the skew-invariant curves in the case where P_2 = P_1^τ for some other automorphism of K which commutes with σ. Specifically, if P in K[x] is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or ± Chebyshev (i.e. P is “nonexceptional”) then skew-invariant curves in (𝔸^2,(P,P^τ)) are horizontal, vertical, or skew-twists: described by equations of the form y = α^{σ^n} ∘ P^{σ^{n-1}} ∘ ⋅⋅⋅ ∘ P^σ ∘ P(x) or x = β^{σ{-1}}∘ P^{τ σ^{-n-2}}∘ P^{τ σ^{-n-3}}∘ ⋅⋅⋅ ∘ P^τ(y) where P = α ∘ β and P^τ = α^{σ^{n+1}}∘ β^{σ^n}} for some integer n.

+ Arthur Forey Complexity of l-adic sheaves 13/05/2022 11:00 Salle W (ENS) et Zoom ENS

To a complex of l-adic sheaves on a quasi-projective variety one associate an integer, its complexity. The main result on the complexity is that it is continuous with tensor product, pullback and pushforward, providing effective version of the constructibility theorems in l-adic cohomology. Another key feature is that the complexity bounds the dimensions of the cohomology groups of the complex. This can be used to prove equidistribution results for exponential sums over finite fields. This is due to Will Sawin, written up in collaboration with Javier Fresán and Emmanuel Kowalski.

+ Gal Binayamini Sharp o-minimality: towards an arithmetically tame geometry 13/05/2022 16:00 Salle W (ENS), et Zoom

Over the last 15 years a remarkable link between o-minimality and algebraic/arithmetic geometry has been unfolding following the discovery of Pila-Wilkie's counting theorem and its applications around unlikely intersections, functional transcendence etc. While the counting theorem is nearly optimal in general, Wilkie has conjectured a much sharper form in the structure R_exp. There is a folklore expectation that such sharper bounds should hold in structures "coming from geometry", but for lack of a general formalism explicit conjectures have been made only for specific structures.
I will describe a refinement of the standard o-minimality theory aimed at capturing the finer "arithmetic tameness" that we expect to see in structures coming from geometry. After presenting the general framework I will discuss my result with Vorobjov showing that the restricted Pfaffian structure is sharply o-minimal, and how this was used in our recent work with Novikov and Zack to prove Wilkie's conjecture for the restricted Pfaffian structure and for Wilkie's original case of R_exp. I will also discuss some conjectures on the construction of larger sharply o-minimal structures, and some partial results in this direction. Finally I will explain the crucial role played by these results in my recent work with Schmidt and Yafaev on Galois orbit lower bounds for CM points in general Shimura varieties, and subsequently in the recent resolution of general André-Oort conjecture by Pila-Shankar-Tsimerman-(Esnault-Groechenig).

+ Philipp Hieronymi Tameness beyond o-minimality (in expansions of the real ordered additive group) 18/03/2022 15:45 Zoom
In his influential paper “Tameness in expansions of the real field” from the early 2000s, Chris Miller wrote: “ What might it mean for a first-order expansion of the field of real numbers to be tame or well behaved? In recent years, much attention has been paid by model theorists and real-analytic geometers to the o-minimal setting: expansions of the real field in which every definable set has finitely many connected components. But there are expansions of the real field that define sets with infinitely many connected components, yet are tame in some well-defined sense [...]. The analysis of such structures often requires a mixture of model-theoretic, analytic-geometric and descriptive set-theoretic techniques. An underlying idea is that first-order definability, in combination with the field structure, can be used as a tool for determining how complicated is a given set of real numbers.” Much progress has been made since then, and in this talk I will discuss an updated account of this research program. I will consider this program in the larger generality of expansions of the real ordered additive group (rather than just in expansions of the real field as envisioned by Miller). In particular, I will mention in this context recent joint work with Erik Walsberg, in which we produce an interesting tetrachotomy for such expansions.
+ Tamara Servi Interdefinability and compatibility in certain o-minimal expansions of the real field 18/03/2022 14:00 Zoom
Let us say that a real function f is o-minimal if the expansion (R,f) of the real field by f is o-minimal. A function g is definable from f if g is definable in (R,f). Two o-minimal functions are compatible if there exists an o-minimal expansion M of the real field in which they are both definable. I will discuss the o-minimality, the interdefinability and the compatibility of two special functions, Euler's Gamma and Riemann's Zeta, restricted to the reals. If time allows it, I will present a general technique for establishing whether a function is definable or not in a given o-minimal expansion of the reals. Joint work with J.-P. Rolin and P. Speissegger.
+ James Freitag Not Pfaffian 18/02/2022 15:45 Zoom
This talk describes the connection between /strong minimality/ of the differential equation satisfied by an complex analytic function and the real and imaginary parts of the function being /Pfaffian/. The talk will not assume the audience knows these notions previously, and will attempt to motivate why each of them are important notions in various areas. The connection we give, combined with a theorem of Freitag and Scanlon (2017) provides the answer to a question of Binyamini and Novikov (2017). We also answer a question of Bianconi (2016). We give what seem to be the first examples of functions which are definable in o-minimal expansions of the reals and are differentially algebraic, but not Pfaffian.
+ Minh-Chieu Tran The Kemperman inverse problem 18/02/2022 14:00
Let G be a connected locally compact group with a left Haar measure μ, and let A,B ⊆ G be nonempty and compact. Assume further that G is unimodular, i.e., μ is also the right Haar measure; this holds, e.g., when G is compact, a nilpotent Lie group, or a semisimple Lie group. In 1964, Kemperman showed that
μ(AB) ≥ min {μ(A)+μ(B), μ(G)} .

The Kemperman inverse problem (proposed by Griesmer, Kemperman, and Tao) asks when the equality happens or nearly happens. I will discuss the recent solution of this problem, highlighting the connections to model theory. (Joint with Jinpeng An, Yifan Jing, and Ruixiang Zhang).
+ Konstantinos Kartas Decidability via the tilting correspondence 21/01/2022 15:30
We discuss new decidability and undecidability results for mixed characteristic henselian fields, whose proof goes via reduction to positive characteristic. The reduction uses extensively the theory of perfectoid fields and also the earlier Krasner-Kazhdan-Deligne principle. Our main results will be:
(1) A relative decidability theorem for perfectoid fields. Using this, we obtain decidability of certain tame fields of mixed characteristic.
(2) An undecidability result for the asymptotic theory of all finite extensions of ℚ_p (fixed p) with cross-section.
We will also discuss a tentative step towards understanding the underlying model theory of arithmetic phenomena in this area, by presenting a model-theoretic way of seeing the Fontaine-Wintenberger theorem.
+ Floris Vermeulen Hensel minimality and counting in valued fields 21/01/2022 14:00
Hensel minimality is a new axiomatic framework for doing tame geometry in non-Archimedean fields, aimed to mimic o-minimality. It is designed to be broadly applicable while having strong consequences. We will give a general overview of the theory of Hensel minimality. Afterwards, we discuss arithmetic applications to counting rational points on definable sets in valued fields. This is partially joint work with R. Cluckers, I. Halupczok and S. Rideau-Kikuchi, and partially with V. Cantoral-Farfan and K. Huu Nguyen.
+ Pierre Simon Monadically NIP ordered graphs and bounded twin-width 18/06/2021 15:00 Zoom
An open problem in theoretical computer science asks to characterize tameness for hereditary classes of finite structures. The notion of bounded twin-width was proposed and studied recently by Bonnet, Geniet, Kim, Thommasé and Watrignant. Classes of graphs of bounded twin-width have many desirable properties. In particular, they are monadically NIP (remain NIP after naming arbitrary unary predicates). In joint work with Szymon Torunczyk we show the converse for classes of ordered graphs. We then obtain a very clear dichotomy between tame (slow growth, monadically NIP, algorithmically simple ...) and wild hereditary classes of ordered graphs. Those results were also obtained by Bonnet, Giocanti, Ossona de Mendez and Thomassé. In this talk, I will focus on the model theoretic input.
+ André Belotto da Silva Real perspectives on monomialization 18/06/2021 16:30 Zoom
I will discuss recent work in collaboration with Edward Bierstone on transformation of a mapping to monomial form (with respect to local coordinates) by simple modifications of the source and target. Our techniques apply in a uniform way to the algebraic and analytic categories, as well as to classes of infinitely differentiable real functions that are quasianalytic or definable in an o-minimal structure. Our results in the real cases are best possible. The talk will focus on real phenomena and on an application to quantifier elimination of certain o-minimal polynomially bounded structure.
+ Will Johnson Curve-excluding fields 21/05/2021 09:00
Let T be the theory of fields K of characteristic 0 such that the equation x^4 + y^4 = 1 has only four solutions in K. We show that T has a model companion. More generally, if K_0 is a field of characteristic 0 and C is a curve (affine or projective) of genus ≥ 2 with C(K_0) = ∅, then there is a model companion CXF of the theory of fields K extending K_0 with C(K) = ∅.
We can use this theory to construct a field K with an interesting combination of properties. On the model-theoretic side, the theory of K is complete, decidable, model-complete, and algebraically bounded, and K is a “geometric structure” in the sense of Hrushovski and Pillay. Additionally, some classification-theoretic properties might hold in K. On the field-theoretic side, K is non-large---there is a smooth curve C such that C(K) is finite and non-empty. This is unusual; the vast majority of model-theoretically tractable fields are large or finite. On the other hand, K is “virtually large”---it has a finite extension which is large. In fact, every proper algebraic extension of K is pseudo algebraically closed (PAC). The absolute Galois group of K is an ω-free profinite group. This negatively answers a question of Junker and Koenigsmann (is every model-complete infinite field large?) and a question of Macintyre (does every model-complete field have a small Galois group?).
This is based on joint work with Erik Walsberg and Vincent Ye.
+ Silvain RIdeau Pseudo-T-closed fields, approximations and NTP2 21/05/2021 10:30 Zoom
Joint work with Samaria Montenegro
The striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a “local-global” principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in flavour, both in its presentation and in the results we aim for.
The first result I would like to present is an approximation result, generalising a result of Kollar on PAC fields, respectively Johnson on henselian fields. This result can be rephrased as the fact that existential closeness in certain topological enrichments come for free from existential closeness as a field. The second result is a (model theoretic) classification result for bounded pseudo-T-closed fields, in the guise of the computation of their burden. One of the striking consequence of these two results is that a bounded perfect PAC field with n independent valuations has burden n and, in particular, is NTP2.
+ Artem Chernikov Recognizing groups and fields in Erdős geometry and model theory 23/04/2021 16:30
Assume that Q is a relation on R^s of arity s definable in an o-minimal expansion of R. I will discuss how certain extremal asymptotic behaviors of the sizes of the intersections of Q with finite n × ... × n grids, for growing n, can only occur if Q is closely connected to a certain algebraic structure.
On the one hand, if the projection of Q onto any s-1 coordinates is finite-to-one but Q has maximal size intersections with some grids (of size >n^(s-1 - ε)), then Q restricted to some open set is, up to coordinatewise homeomorphisms, of the form x_1+...+x_s=0. This is a special case of the recent generalization of the Elekes-Szabó theorem to any arity and dimension in which general abelian Lie groups arise (joint work with Kobi Peterzil and Sergei Starchenko). On the other hand, if Q omits a finite complete s-partite hypergraph but can intersect finite grids in more that than n^(s-1 + ε) points, then the real field can be definably recovered from Q (joint work with Abdul Basit, Sergei Starchenko, Terence Tao and Chieu-Minh Tran).
I will explain how these results are connected to the model-theoretic trichotomy principle and discuss variants for higher dimensions, and for stable structures with distal expansions.
+ Gabriel Conant VC-dimension in model theory, discrete geometry, and combinatorics 23/04/2021 15:00
In statistical learning theory, the notion of VC-dimension was developed by Vapnik and Chervonenkis in the context of approximating probabilities of events by the relative frequency of random test points. This notion has been widely used in combinatorics and computer science, and is also directly connected to model theory through the study of NIP theories. This talk will start with an overview of VC-dimension, with examples motivated by discrete geometry and additive combinatorics. I will then present several model theoretic applications of VC-dimension. The selection of topics will focus on the use of finitely approximable Keisler measures to analyze the structure of algebraic and combinatorial objects with bounded VC-dimension.
+ Jason Bell Effective isotrivial Mordell-Lang in positive characteristic 26/03/2021 15:00
The Mordell-Lang conjecture (now a theorem, proved by Faltings, Vojta, McQuillan,...) asserts that if G is a semiabelian variety G defined over an algebraically closed field of characteristic zero, X is a subvariety of G, and Γ is a finite rank subgroup of G, then X ∩ Γ is a finite union of cosets of Γ. In positive characteristic, the naive translation of this theorem does not hold, however Hrushovski, using model theoretic techniques, showed that in some sense all counterexamples arise from semiabelian varieties defined over finite fields (the isotrivial case). This was later refined by Moosa and Scanlon, who showed in the isotrivial case that the intersection of a subvariety of a semiabelian variety G with a finitely generated subgroup Γ of G that is invariant under the Frobenius endomorphism F: G → G is a finite union of sets of the form S+A, where A is a subgroup of Γ and S is a sum of orbits under the map F. We show how how one can use finite-state automata to give a concrete description of these intersections Γ ∩ X in the isotrivial setting, by constructing a finite machine that identifies all points in the intersection. In particular, this allows us to give decision procedures for answering questions such as: is X ∩ Γ empty? finite? does it contain a coset of an infinite subgroup? In addition, we are able to read off coarse asymptotic estimates for the number of points of height ≤ H in the intersection from the machine. This is joint work with Dragos Ghioca and Rahim Moosa.
+ Rémi Jaoui Linearization procedures in the semi-minimal analysis of algebraic differential equations 26/03/2021 16:30
It is well-known that certain algebraic differential equations restrain in an essential way the algebraic relations that their solutions share. For example, the solutions of the first equation of Painlevé y'' = 6y^2 + t are “new” transcendental functions of order two which whenever distinct are algebraically independent (together with their derivatives).
I will first describe an account of such phenomena using the language of geometric stability theory in a differentially closed field. I will then explain how linearization procedures and geometric stability theory fit together to study such transcendence results in practice.
+ Yatir Halevi et Franziska Jahnke On dp-finite fields 12/02/2021 09:00
Shelah's conjecture predicts that any infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation. Recently, Johnson proved that Shelah's conjecture holds for fields of finite dp-rank, also known as dp-finite fields. The aim of these two talks is to give an introduction to dp-rank in some algebraic structures and an overview of Johnson's work.
In the first talk, we define dp-rank (which is a notion of rank in NIP theories) and give examples of dp-finite structures. In particular, we discuss the dp-rank of ordered abelian groups and use them to construct multitude of examples of dp-finite fields. We also prove that every dp-finite field is perfect and sketch a proof that any valued field of dp-rank 1 is henselian.
In the second talk, we give an overview of Johnson's proof that every infinite dp-finite field is either algebraically closed, real closed or admits a non-trivial henselian valuation. Crucially, this relies on the notion of a W-topology, a natural generalization of topologies arising from valuations, and the construction of a definable W-topology on a sufficiently saturated unstable dp-finite field.
+ Alessandro Berarducci An application of surreal numbers to the asymptotic analysis of certain exponential functions 15/01/2021 09:30
Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f+g, fg and f^g are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x). They did so by studying the possible limits at infinity of the quotient f(x)/g(x) of two functions in the fragment: if g is kept fixed and f varies, the possible limits form a discrete set of real numbers of order type omega. Using the surreal numbers, we extend the latter result to the whole class of Skolem functions and we discuss some additional progress towards the conjecture of Skolem. This is joint work with Marcello Mamino (http://arxiv.org/abs/1911.07576, to appear in the JSL).
+ Daniel Palacin Solving equations in finite groups and complete amalgamation 15/01/2021 11:00
Roth's theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length 3, that is, the equation x+z=2y has a solution in A. Finitary versions of Roth's theorem study subsets A of {0, ... , N}, and ask whether the same holds for sufficiently large N, for a fixed lower bound on the density. In a similar way, concerning finite groups, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation, or even a system of equations. For instance, for the equation xy=z, Gowers (2008) showed that any subset of a finite simple non-abelian group will contain many solutions to this equation, provided it has sufficiently large density. We will report on recent work with Amador Martin-Pizarro on how to find solutions to the above equations in the context of pseudo-finite groups, using techniques from model theory which resonate with (a group version of) the independence theorem in simple theories due to Pillay, Scanlon and Wagner. In this talk, we will not discuss the technical aspects of the proof, but present the main ideas to a general audience.
+ Annalisa Conversano Groups definable in o-minimal structures and algebraic groups 11/12/2020 09:00 Zoom
Groups definable in o-minimal structures have been studied by many authors in the last 30 years and include algebraic groups over algebraically closed fields of characteristic 0, semi-algebraic groups over real closed fields, important classes of real Lie groups such as abelian groups, compact groups and linear semisimple groups. In this talk I will present results on groups definable in o-minimal structures, demonstrating a strong analogy with topological decompositions of linear algebraic groups. Limitations of this analogy will be shown through several examples.
+ Pablo Cubides Kovacsics Cohomology of algebraic varieties over non-archimedean fields 11/12/2020 10:30 Zoom
I will report on a joint work with Mário Edmundo and Jinhe Ye in which we introduced a sheaf cohomology theory for algebraic varieties over non-archimedean fields based on Hrushovski-Loeser spaces. After informally framing our main results with respect to classical statements, I will discuss some details of our construction and the main difficulties arising in this new context. If time allows, I will further explain how our results allow us to recover results of V. Berkovich on the sheaf cohomology of the analytification of an algebraic variety over a rank 1 complete non-archimedean field.
+ Will Johnson The étale-open topology (suite) 27/11/2020 09:00 Zoom
Fix an abstract field K. For each K-variety V, we will define an “étale-open” topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties.
The étale-open topology is mostly of interest when K is large (also known as ample). On non-large fields, the étale-open topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some well-known facts about large fields, and classify the model-theoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with Chieu-Minh Tran, Erik Walsberg, and Jinhe Ye.
+ Will Johnson The étale-open topology 13/11/2020 09:00 Zoom
Fix an abstract field K. For each K-variety V, we will define an étale-open topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties. The étale-open topology is mostly of interest when K is large (also known as ample). On non-large fields, the étale-open topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some well-known facts about large fields, and classify the model-theoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with Chieu-Minh Tran, Erik Walsberg, and Jinhe Ye.
+ Jinhe (Vincent) Ye Belles paires of valued fields and analytification 13/11/2020 10:30
In their work, Hrushovski and Loeser proposed the space V̂ of generically stable types concentrating on V to study the homotopy type of the Berkovich analytification of V. An important feature of V̂ is that it is canonically identified as a projective limit of definable sets in ACVF, which grants them tools from model theory. In this talk, we will give a brief introduction to this object and present an alternative approach to internalize various spaces of definable types, motivated by Poizat's work on belles paires of stable theories. Several results of interest to model theorists will also be discussed. Particularly, we recover the space V̂ is strict pro-definable and we propose a model-theoretic counterpart Ṽ of Huber's analytification. Time permitting, we will discuss some comparison and lifting results between V̂ and Ṽ. This is a joint project with Pablo Cubides Kovacsics and Martin Hils.
+ Dmitry Novikov Complex Cellular Structures 16/10/2020 09:00 Zoom
Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of convenient semialgebraic images of standard cubes. The Gromov-Yomdin Lemma (later generalized by Pila and Wilkie) proves that the maps could be chosen of C^r-smooth norm at most one, and the number of such maps is uniformly bounded for finite-dimensional families. This number was not effectively bounded by Yomdin or Gromov, but it necessarily grows as r → ∞. It turns out there is a natural obstruction to a naive holomorphic complexification of this result related to the natural hyperbolic metric of complex holomorphic sets. We prove a lemma about holomorphic functions in annulii, a quantitative version of the great Picard theorem. This lemma allowed us to construct an effective holomorphic version of the cellular decomposition results in all dimensions, with explicit polynomial bounds on complexity for families of complex (sub)analytic and semialgebraic sets. As the first corollary we get an effective version of Yomdin-Gromov Lemma with polynomial bounds on the complexity, thus proving a long-standing Yomdin conjecture about tail entropy of analytic maps. Further connection to diophantine applications will be explained in Gal's talk.
+ Gal Binayamini Tame geometry and diophantine approximation 16/10/2020 10:30
Tame geometry is the study of structures where the definable sets admit finite complexity. Around 15 years ago Pila and Wilkie discovered a deep connection between tame geometry and diophantine approximation, in the form of asymptotic estimates on the number of rational points in a tame set (as a function of height). This later led to deep applications in diophantine geometry, functional transcendence and Hodge theory. I will describe some conjectures and a long-term project around a more effective form of tame geometry, suited for improving the quality of the diophantine approximation results and their applications. I will try to outline some of the pieces that are already available, and how they should conjecturally fit together. Finally I will survey some applications of the existing results around the Manin-Mumford conjecture, the Andre-Oort conjecture, Galois-orbit lower bounds in Shimura varieties, unlikely intersections in group schemes, and some other directions (time permitting).
+ Raf Cluckers Exponential sums modulo powers of primes, singularity theory, and local global principles 06/03/2020 14:15 ENS, Salle W

The theme of the talk is around the theory of Igusa's local zeta functions, his broader program on local global principles, and recent progress on these via singularity theory and the minimal model program with M. Mustata and K. H. Nguyen. I will also present some new open questions that push Igusa's program further, and partial evidence obtained with K. H. Nguyen.

+ Nick Ramsey Constructing pseudo-algebraically closed fields 06/03/2020 16:00 ENS, Salle W

A field K is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point. These fields were introduced by Ax in his characterization of pseudo-finite fields and have since become an important object of study in both model theory and field arithmetic. We will explain how the analysis of a PAC field often reduces to questions about the model theory of the absolute group and describe how these reductions combine with a graph-coding construction of Cherlin, van den Dries, and Macintyre together with to construct PAC fields with prescribed combinatorial properties.

+ Alex Wilkie Some remarks on complex analytic functions in a definable context 06/03/2020 11:00 ENS, Salle W

We fix an o-minimal expansion of the real field, M say. Definability notions are with respect to M. Let F = {f_x : x in X} be a definable family of (single valued) complex analytic functions, each one having domain some disk, D_x say, in ℂ, where the parameter space X is a definable subset of ℝ^m for some m. We present some finiteness theorems for such families F which are uniform in parameters and give some applications.
We also speculate on the notion of “definable” Riemann surface.


+ Bas Edixhoven Geometric quadratic Chabauty 31/01/2020 16:00 Salle W, ENS

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only `simple algebraic geometry' (line bundles over the jacobian and models over the integers).

+ Marie Françoise Roy Quantitative Fundamental Theorem of Algebra 31/01/2020 14:15 ENS, Salle W

Using subresultants, we modify a recent real-algebraic proof due to Eisermann of the Fundamental Theorem of Algebra ([FTA]) to obtain the following quantitative information: in order to prove the [FTA] for polynomials of degree d, the Intermediate Value Theorem ([IVT]) is requested to hold for real polynomials of degree at most d^2. We also explain that the classical algebraic proof due to Laplace requires [IVT] for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

+ Martin Hils Classification des imaginaires dans VFA 31/01/2020 11:00 ENS, Salle W

(travail en commun avec Silvain Rideau-Kikuchi)
Les imaginaires (c'est-à-dire les quotients définissables) dans la théorie ACVF des corps algébriquement clos non-trivialement valués sont classifiés par les sortes “géométriques”. Ceci est un résultat fondamental dû à Haskell, Hrushovski et Macpherson. En utilisant l'approche via la densité des types définissables/invariants, nous donnons une réduction des imaginaires dans des corps valués henséliens, sous des hypothèses assez générales, aux sortes géométriques et à des imaginaires de RV avec des sortes pour certains espaces vectoriels de dimension finie sur le corps résiduel.
Dans l'exposé, je vais principalement parler d'une application qui a été à l'origine de notre travail: Les imaginaires de la théorie VFA des corps algébriquement clos valués non-trivialement de caractéristique 0, munis d'un Frobenius non-standard, sont classifiés par les sortes géométriques. Entre autre, notre preuve passe par une étude fine des imaginaires dans une suite exacte courte (pure) ainsi que par un résultat clé du papier de Hrushovski sur les groupoïdes.

+ Victoria Cantoral Farfan The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture 08/11/2019 14:15 ENS, Salle W

The famous Mumford-Tate conjecture asserts that, for every prime number l, Hodge cycles are &Qopf;_l linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic Sato-Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized Sato-Tate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces.
Our main goal is to prove that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The relevance of this result lies mainly in the fact that the list of known cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture. This is a joint work with Johan Commelin.

+ Franziska Jahnke Characterizing NIP henselian fields 08/11/2019 11:00 ENS, Salle W

In this talk, we characterize NIP henselian valued fields modulo the theory of their residue field. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

+ Laurent Moret-Bailly Une construction d'extensions faiblement non ramifiées d'un anneau de valuation 08/11/2019 16:00 ENS, Salle W

Étant donné un anneau de valuation V de corps résiduel F et contenant un corps k, et une extension k' de k, on cherche à construire une extension V' de V contenant k', d'idéal maximal engendré par celui de V, et de corps résiduel composé de F et k'. On y parvient notamment si F ou k' est séparable sur k.

+ Lorenzo Fantini A valuative approach to the inner geometry of surfaces 11/10/2019 11:00 ENS, Salle W ENS

Lipschitz geometry is a branch of singularity theory that studies the metric data of a germ of a complex analytic space.
I will discuss a new approach to the study of such metric germs, and in particular of an invariant called Lipschitz inner rate, based on the combinatorics of a space of valuations, the so-called non-archimedean link of the singularity. I will describe completely the inner metric structure of a complex surface germ showing that its inner rates both determine and are determined by global geometric data: the topology of the germ, its hyperplane sections, and its generic polar curves.
This is a joint work with André Belotto and Anne Pichon.


+ Silvain RIdeau H-minimality 11/10/2019 16:00 ENS, Salle W ENS

My goal, in this talk, is to explain a new notion of minimality for (characteristic zero) Henselian fields, which generalizes C-minimality, P-minimality and V-minimality and puts no restriction on the residue field or valued group contrary to these previous notions. This new notion, h-minimality, can be defined, analogously to other minimality notions, by asking that 1-types, over algebraically closed sets, are entirely determined by their reduct to some sublanguage - in that case the pure language of valued fields. However, contrary to what happens with other minimality notions, particular care has to be taken with regards to the parameters. In fact, we define a family of notions: l-h-min for l a natural number or omega. My second goal in this talk will be to explain the various geometric properties that follow form h-minimality, among which the well-known Jacobian property, but also higher degree and higher dimensional versions of that property.

+ Antoine Ducros Quantifier elimination in algebraically closed valued fields in the analytic language: a geometric approach 11/10/2019 14:15 ENS, Salle W ENS

I will present a work on flattening by blow-ups in the context of Berkovich geometry (inspired by Raynaud and Gruson's paper on the same topic in the scheme-theoretic setting), and explain how it gives rise to the description of the image of an arbitrary analytic map between two compact Berkovich spaces, and why this description is (very likely) related to quantifier elimination in the Lipshitz-Cluckers variant of Lipshitz-Robinson's analytic language. (I plan to spend most of the talk discussing the results rather than their proofs.)

+ Dimitri Wyss Non-archimedean and motivic integrals on the Hitchin fibration 10/05/2019 16:00 ENS, Salle W
Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between `stringy' Hodge numbers for moduli spaces of SL_n/PGL_n Higgs bundles. With Michael Groechenig and Paul Ziegler we prove this conjecture using non-archimedean integrals on these moduli spaces, building on work of Denef-Loeser and Batyrev. Similar ideas also lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô.

In my talk I will outline the main arguments of the proofs and discuss the adjustments needed, in order to replace non-archimedean integrals by motivic ones. The latter is joint work with François Loeser.
+ Avraham Aizenbud Point-wise surjective presentations of stacks, or why I am not afraid of (infinity) stacks anymore 10/05/2019 14:15 ENS, Salle W
Any algebraic stack X can be represented by a groupoid object in the category of schemes: that is, a pair of schemes Ob, Mor and morphisms
source, target: Mor → Ob, inversion: Mor → Mor, composition: Mor ×_Ob Mor → Mor and identity: Ob → Mor that satisfy certain axioms. Yet this description of the stack X might be misleading.

Namely, given a field F which is not algebraically closed, we have a natural functor between the groupoid (Ob(F),Mor(F)) and the groupoid X(F). While this functor is fully faithful, it is often not essentially surjective.

In joint work with Nir Avni (in progress) we show that any algebraic groupoid has a presentation such that this functor will be essentially surjective for many fields (and under some assumptions on the stack, for any field). The results are also extended to Henselian rings.

Despite the title, the talk will be about usual stacks and not infinity-stacks, yet in some of the proofs it is more convenient to use the language of higher categories and I'll try to explain why.

No prior knowledge of infinity stacks will be assumed, but a superficial acquaintance with usual stacks will be helpful.
+ Katrin Tent Almost strongly minimal ample geometries 10/05/2019 11:00 ENS, Salle W
The notion of ampleness captures essential properties of projective spaces over fields. It is natural to ask whether any sufficiently ample strongly minimal set arises from an algebraically closed field. In this talk I will explain the question and present recent results on ample strongly minimal structures.

+ Jonathan Pila Independence of CM points in elliptic curves 22/03/2019 11:00 ENS, Salle W
I will speak about joint work with Jacob Tsimerman. Let E be an elliptic curve parameterized by a modular (or Shimura) curve. There are a number of results (..., Buium-Poonen, Kuhne) to the effect that the images of CM points are (under suitable hypotheses) linearly independent in E. We consider this issue in the setting of the Zilber-Pink conjecture and prove a result which improves previous results in some aspects
+ Per Salberger Counting rational points with the determinant method 22/03/2019 14:15 ENS, Salle W
The determinant method gives upper bounds for the number of rational points of bounded height on or near algebraic varieties defined over global fields. There is a real-analytic version of the method due to Bombieri and Pila and a p-adic version due to Heath-Brown. The aim of our talk is to describe a global refinement of the p-adic method and some applications like a uniform bound for non-singular cubic curves which improves upon earlier bounds of Ellenberg-Venkatesh and Heath-Brown.
+ Vler&euml; Mehmeti Patching over Berkovich Curves 22/03/2019 16:00 ENS, Salle W
Patching was first introduced as an approach to the Inverse Galois Problem. The technique was then extended to a more algebraic setting and used to prove a local-global principle by D. Harbater, J. Hartmann and D. Krashen. I will present an adaptation of the method of patching to the setting of Berkovich analytic curves. This will then be used to prove a local-global principle for function fields of curves that generalizes that of the above mentioned authors.
+ Pablo Cubides Kovacsics Definable subsets of a Berkovich curve 15/02/2019 16:00 ENS, Salle W
Let k be an algebraically closed complete rank 1 non-trivially valued field. Let X be an algebraic curve over k and let X^an be its analytification in the sense of Berkovich. We functorially associate to X^an a definable set X^S in a natural language. As a corollary, we obtain an alternative proof of a result of Hrushovski-Loeser about the iso-definability of curves. Our association being explicit allows us to provide a concrete description of the definable subsets of X^S: they correspond to radial sets. This is a joint work with Jérôme Poineau.
+ Chris Daw Unlikely intersections with E×CM curves in 𝒜_2 15/02/2019 11:00 ENS, Salle W
The Zilber-Pink conjecture predicts that an algebraic curve in A_2 has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety.
Under a large Galois orbits hypothesis, we prove the finiteness of the intersection with the special curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication. Furthermore, we show that this large Galois orbits hypothesis holds for curves satisfying a condition on their intersection with the boundary of the Baily--Borel compactification of A_2.
More generally, we show that a Hodge generic curve in an arbitrary Shimura variety has only finitely many intersection points with the generic points of a so-called Hecke--facteur family, again under a large Galois orbits hypothesis.
This is a joint work with Martin Orr (University of Warwick).
+ Bruno Klingler Tame topology and Hodge theory. 15/02/2019 14:15 ENS, Salle W
I will explain how tame topology seems the natural setting for variational Hodge theory. As an instance I will sketch a new proof of the algebraicity of the components of the Hodge locus, a celebrated result of Cattani-Deligne-Kaplan (joint work with Bakker and Tsimerman).
+ Martin Bays Definability in the infinitesimal subgroup of a simple compact Lie group 11/01/2019 14:15 ENS, Salle W
Joint work with Kobi Peterzil.
Let G be a simple compact Lie group, for example G=SO_3(R). We consider the structure of definable sets in the subgroup G^00 of infinitesimal elements. In an aleph_0-saturated elementary extension of the real field, G^00 is the inverse image of the identity under the standard part map, so is definable in the corresponding valued field. We show that the pure group structure on G^00 recovers the valued field, making this a bi-interpretation. Hence the definable sets in the group are as rich as possible.

+ Wouter Castryck Scrollar invariants, resolvents, and syzygies 11/01/2019 11:00 ENS, Salle W
With every cover C -> P^1 of the projective line one can associate its so-called scrollar invariants (also called Maroni invariants) which describe how the push-forward of the structure sheaf of C splits over P^1. They can be viewed as geometric counterparts of the successive minima of the lattice associated with the ring of integers of a number field. In this talk we consider the following problem: how do the scrollar invariants of the Galois closure C' -> P^1 and of its various subcovers (the so-called resolvents of C -> P^1) relate to known invariants of the given cover? This concerns ongoing work with Yongqiang Zhao, in which we put a previous observation for covers of degree 4 due to Casnati in a more general framework. As we will see the answer involves invariants related to syzygies that were introduced by Schreyer. As time permits, we will discuss a number-theoretic manifestation of the phenomena observed.
+ Amador Martin-Pizarro Tame open core and small groups in pairs of topological geometric structures 11/01/2019 16:00 ENS, Salle W
Using the group configuration theorem, Hrushovski and Pillay showed that the law of a group definable in the reals or the p-adics is locally an algebraic group law, up to definable isomorphism. There are some natural expansions of these two theories of fields, by adding a predicate for a dense substructure, for example the algebraic reals or the algebraic p-adics. We will present an overview on some of the features of these expansions, and particularly on the characterisation of open definable sets as well as of groups definable in the pairs.
+ Omar León Sánchez On differentially large fields. 14/12/2018 00:00 ENS, Salle W
Recall that a field K is large if it is existentially closed in K((t)). Examples of such fields are the complex, the real, and the p-adic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galois-theoretic problems. In recent work with M. Tressl we introduced and explored a differential analogue of largeness, that we conveniently call “differentially large”. I will present some properties of such fields, and use a twisted version of the Taylor morphism to characterise them using formal Laurent series and to even construct “natural” examples (which ultimately yield examples of DCFs and CODFs... acronyms that will be explained in the talk).
+ Arthur Forey Uniform bound for points of bounded degree in function fields of positive characteristic 14/12/2018 00:00 ENS, salle W
I will present a bound for the number of 𝔽_q[t]-points of bounded degree in a variety defined over ℤ[t], uniform in q. This generalizes work by Sedunova for fixed q. The proof involves model theory of valued fields with algebraic Skolem functions and uniform non-Archimedean Yomdin-Gromov parametrizations. This is joint work with Raf Cluckers and François Loeser.
+ Guy Casale Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups 14/12/2018 14:15 ENS, Salle W
We prove the Ax-Lindemann-Weierstrass theorem for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory of Schwarzian equations and machinery from the model theory of differentially closed fields. This result generalizes previous work of Pila-Tsimerman on the j function.
Joint work with James Freitag and Joel Nagloo.
+ Antoine Ducros Non-standard analysis and non-archimedean geometry 16/11/2018 11:00 ENS, Salle W
In this talk I will describe a joint work (still in progress) with E. Hrushovski and F. Loeser, in which we explain how the integrals I have defined with Chambert-Loir on Berkovich spaces can be seen (in the t-adic case) as limits of usual integrals on complex algebraic varieties; a crucial step is the development of a non-standard integration theory on a huge real closed field. I plan to devote a lot of time to the precise description of the objects involved, before stating our main theorem and saying a some words about is proof.
+ Philipp Dittman First-order logic in finitely generated fields 16/11/2018 14:15 ENS, Salle W
The expressive power of first-order logic in the class of finitely generated fields, as structures in the language of rings, is relatively poorly understood. For instance, Pop asked in 2002 whether elementarily equivalent finitely generated fields are necessarily isomorphic, and this is still not known in the general case. On the other hand, the related situation of finitely generated rings is much better understood by recent work of Aschenbrenner-Khélif-Naziazeno-Scanlon.
Building on work of Pop and Poonen, and using geometric results due to Kerz-Saito and Gabber, I shall show that every infinite finitely generated field of characteristic not two admits a definable subring which is a finitely generated algebra over a global field. This implies that any such finitely generated field is biinterpretable with arithmetic, and gives a positive answer to the question above in characteristic not two.
+ Jean-Philippe Rolin Oscillatory integrals of subanalytic functions 16/11/2018 16:00 ENS, Salle W
In several papers, R. Cluckers and D. Miller have built and investigated a class of real functions which contains the subanalytic functions and which is closed under parameterized integration. This class does not allow any oscillatory behavior, nor stability under Fourier transform. On the other hand, the behavior of oscillatory integrals, in connection with singularity theory, has been heavily investigated for decades. In this talk, we explain how to build a class of complex functions, which contains the subanalytic functions and their complex exponentials, and which is closed under parameterized integration and under Fourier transform.
Our techniques involve appropriate preparation theorems for subanalytic functions, and some elements of the theory of uniformly distributed families of maps.
(joint work with R. Clucker, G. Comte, D. Miller and T. Servi).
+ Laurent Bartholdi Groups and algebras 18/05/2018 11:00 Jussieu, salle 101 couloir 15-16 (1er étage)
To every group G is associated an associative algebra, namely its group ring kG. Which (geometric) properties are reflected in (algebraic) properties of kG? I will survey some results and conjectures in this area, concentrating on specific examples: growth, amenability, torsion, and filtrations.
+ Omer Friedland Doubling parametrizations and Remez-type inequalities 18/05/2018 16:00 Jussieu, salle 101 couloir 15-16 (1er étage)
A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ : B_1 → Y of the unit ball in ℂ^n, which is extendible to the (say) four times larger concentric ball of B_1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney's ball coverings of a domain W ⊂ ℝ^n.
Our main motivation is that doubling coverings form a special class of “smooth parameterizations”, which are used in bounding entropy type invariants in smooth dynamics on one side, and in bounding density of rational points in diophantine geometry on the other. Complexity of smooth parameterizations is a key issue in some important open problems in both areas.
In this talk we present various estimates on the complexity of these objects. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.
This is a joint work with Yosef Yomdin.
+ Rahim Moosa Isotrivial Mordell-Lang and finite automata 18/05/2018 14:15 Jussieu, salle 101 couloir 15-16 (1er étage)
About fifteen years ago, Thomas Scanlon and I gave a description of sets that arise as the intersection of a subvariety with a finitely generated subgroup inside a semiabelian variety over a finite field. Inspired by later work of Derksen on the positive characteristic Skolem-Mahler-Lech theorem, which turns out to be a special case, Jason Bell and I have recently recast those results in terms of finite automata. I will report on this work, as well as on the work-in-progress it has engendered on an effective version of the isotrivial Mordell-Lang theorem.
+ Tom Scanlon The dynamical Mordell-Lang problem in positive characteristic 23/03/2018 14:15 IHP, amphitheatre Darboux
The dynamical Mordell-Lang conjecture in characteristic zero predicts that if f : X → X is a map of algebraic varieties over a field K of characteristic zero, Y ⊆ X is a closed subvariety and a in X(K) is a K-rational point on X, then the return set n in N : f^n(a) in Y(K) is a finite union of points and arithmetic progressions. For K a field of characteristic p > 0, it is necessary to allow for finite unions with sets of the form a + ∑_i=1^m p^n_i : (n_1, ... , n_m) in N^m and one might conjecture that all return sets are finite unions of points, arithmetic progressions and such p-sets. We studied the special case of the positive characteristic dynamical Mordell-Lang problem on semiabelian varieites and using our earlier results with Moosa on so-called F-sets reduced the problem to that of solving a class of exponential diophantine equations in characteristic zero. In so doing, under the hypothesis that X is a semiabelian variety and either Y has small dimension or f is sufficiently general, we prove the conjecture. However, we also show that our reduction to the exponential diiophantine problems may be reversed so that the positive characteristic dynamical Mordell-Lang conjecture in general is equivalent to a class of hard exponential diophantine problems which the experts consider to be out of reach given our present techniques.
(This is a report on joint work with Pietro Corvaja, Dragos Ghioca and Umberto Zannier available at arXiv:1802.05309.)
+ Florian Pop On a conjecture of Colliot-Thélène 23/03/2018 11:00 IHP, amphitheatre Darboux
Let f be a morphism of projective smooth varieties X, Y defined over the rationals. The conjecture by Colliot-Thélène under discussion gives (conjectural) sufficient conditions which imply that for almost all rational prime numbers p, the map f maps the p-adic points X(ℚ_p) surjectively onto Y(ℚ_p). The aim of the talk is to present some recent results by Denef, Skorobogatov et al; further to report on work in progress on a different method to attack the conjecture under quite relaxed hypotheses.
+ Sergei Starchenko A model theoretic generalization of the one-dimensional case of the Elekes-Szabo theorem 23/03/2018 16:00 IHP, amphitheatre Darboux
(Joint work with A. Chernikov)
Let V ⊆ ℂ^3 be a complex variety of dimension 2.
The Elekes-Szabo Theorem says that if V contains “too many” points on n × n × n Cartesian products then V has a special form: either V contains a cylinder over a curve or V is related to the graph of the multiplication of an algebraic group.
In this talk we generalize the Elekes-Szabo Theorem to relations on strongly minimal sets interpretable in distal structures.
+ Jan Tuitman Effective Chabauty and the Cursed Curve 19/01/2018 14:15 Institut Henri Poincaré, amphi Hermite
The Chabauty method often allows one to find the rational points on curves of genus at least 2 over the rationals, but has a lot of limitations. On a theoretical level, the Mordell-Weil rank of the Jacobian of the curve has to be strictly smaller than its genus. In practice, even when this condition is satisfied, the relevant Coleman integrals can usually only be computed for hyperelliptic curves. We will report on recent work of ours (with different combinations of collaborators) on extending the method to more general curves. In particular, we will show how one can use an extension of the Chabauty method by Kim to find the rational points on the split Cartan modular curve of level 13, which is also known as the cursed curve. The talk will be aimed at non-specialists with an interest in number theory.
+ Jonathan Kirby Blurred Complex Exponentiation 19/01/2018 16:00 IHP, Hermite
Zilber conjectured that the complex field equipped with the exponential function is quasiminimal: every definable subset of the complex numbers is countable or co-countable. If true, it would mean that the geometry of solution sets of complex exponential-polynomial equations and their projections is somewhat like algebraic geometry. If false, it is likely that the real field is definable and there may be no reasonable geometric theory of these definable sets.
I will report on some progress towards the conjecture, including a proof when the exponential function is replaced by the approximate version given by ∃ q,r in Q [y = e^x+q+2πi r]. This set is the graph of the exponential function blurred by the group exp(Q + 2 πi Q). We can also blur by a larger subgroup and prove a stronger version of the theorem. Not only do we get quasiminimality but the resulting structure is isomorphic to the analogous blurring of Zilber's exponential field and to a reduct of a differentially closed field.
Reference: Jonathan Kirby, Blurred Complex Exponentiation, arXiv:1705.04574
+ Isaac Goldbring Spectral gap and definability 19/01/2018 11:00
Originating in the theory of unitary group representations, the notion of spectral gap has played a huge role in many of the deep results in the theory of von Neumann algebras in the last couple of decades. Recently, with my collaborators, we are slowly understanding the model-theoretic significance of spectral gap, in particular its connection with definability. In this talk, I will discuss a few of our recent observations in this direction and speculate on some further possible developments. I will assume no knowledge of von Neumann algebras nor continuous logic. Various parts of this work are joint with Bradd Hart, Thomas Sinclair, and Henry Towsner.
+ Adam Topaz On the conjecture of Ihara/Oda-Matsumoto 15/12/2017 11:00 ENS, salle W
Following the spirit of Grothendieck's Esquisse d'un Programme, the Ihara/Oda-Matsumoto conjecture predicted a combinatorial description of the absolute Galois group of Q based on its action on geometric fundamental groups of varieties. This conjecture was resolved in the 90's by Pop using anabelian techniques. In this talk, I will discuss the proof of stronger variant of this conjecture, using mod-ell two-step nilpotent quotients, while highlighting some connections with model theory.
+ Julien Sebag Géométrie des arcs et singularités 15/12/2017 14:15 ENS, salle W
Soulignée par Nash dans les années 60, l'interaction entre la géométrie des espaces d'arcs et la théorie des singularités s'est fortement amplifiée sous l'influence de la théorie de l'intégration motivique notamment. Dans cet exposé, nous introduirons le schéma des arcs associé à une variété algébrique et donnerons quelques illustrations de cette interaction. Parmi elles, nous parlerons de l'interprétation (possible) du point de vue des singularités d'un théorème de Drinfeld et Grinberg-Kazhdan démontré au début des années 2000. (Cette dernière partie de l'exposé s'appuie sur une collaboration avec David Bourqui.)
+ Martin Bays The geometry of combinatorially extreme algebraic configurations 15/12/2017 16:00 ENS. salle W
Given a system of polynomial equations in m complex variables with solution set of dimension d, if we take finite subsets X_i of C each of size at most N, then the number of solutions to the system whose ith co-ordinate is in X_i is easily seen to be bounded as O(N^d). We ask: when can we improve on the exponent d in this bound?

Hrushovski developed a formalism in which such questions become amenable to the tools of model theory, and in particular observed that incidence bounds of Szemeredi-Trotter type imply modularity of associated geometries. Exploiting this, we answer a (more general form of) our question above. This is part of a joint project with Emmanuel Breuillard.
+ Olivier Benoist Sur les polynômes positifs qui sont sommes de peu de carrés 17/11/2017 11:00 ENS, salle W
Artin a résolu le 17ème problème de Hilbert : un polynôme réel positif en n variables est somme de carrés de fractions rationnelles. Pfister a amélioré ce résultat en démontrant qu'il est somme de 2^n carrés. Décider si la borne 2^n de Pfister est optimale est un problème ouvert si n>2. Nous expliquerons que cette borne peut être améliorée en petit degré et, en deux variables, pour un ensemble dense de polynômes positifs.
+ Alex Wilkie Quasi-minimal expansions of the complex field 17/11/2017 16:00 ENS, Salle W
I discuss a back-and-forth technique for proving that in certain expansions of the complex field every L_∞, ω-definable subset of ℂ is either countable or co-countable. Some successes of the method will also be discussed.
+ Dmitry Sustretov Incidence systems on Cartesian powers of algebraic curves 17/11/2017 14:15 ENS, Salle W
The classical theory of abstract projective geometries establishes an equivalence between axiomatically defined incidence systems of points and lines and projective planes defined over a field. Zilber's Restricted Trichotomy conjecture in dimension one is a generalization of this statement in a sense, with lines replaced by algebraic curves; it implies that a non-locally modular strongly minimal structure with the universe an algebraic curve over an algebraically closed field and basic relations constructible subsets of Cartesian powers of the curve interprets an infinite field. The talk will present the basic structure of the proof of the conjecture, and outline its application, by Zilber, to Torreli-type theorem for curves over finite fields of Bogomolov, Korotiaev and Tschinkel. Joint work with Assaf Hasson.
+ Boris Zilber Approximation, domination and integration 13/10/2017 14:15 ENS, salle W
The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' (arxiv) established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).
+ Immi Halupczok Un nouvel analogue de l'o-minimalité dans des corps valués 13/10/2017 16:00 ENS, salle W
Pour les corps réel clos, la notion d'o-minimalité a eu un énorme succès; il s'agit d'une condition très simple à une expansion du langage des corps, qui implique que les ensembles définissables se comportent très bien d'un point de vue géométrique. Il existe plusieurs adaptations de cette notion aux corps valués (p.ex. p-mininalité, C-minimalité, B-minimalité, v-minimalité), mais la plupart de ces adaptations (a) s'appliquent seulement à une classe de corps valués assez restrictive, (b) elles n'impliquent pas tout ce qu'on voudrait, et/ou (c) elles sont définies de manière nettement plus compliquée. Dans cet exposé, je vais présenter une nouvelle notion qui n'a pas les problèmes (a) et (b) et qui a une définition raisonnablement simple.
+ Itaï Ben Yaacov Corps globalement valués 13/10/2017 11:00 ENS, salle W
Dans un travail en commun avec E Hrushovski, nous étudions les corps globalement valués, qui sont une abstraction des corps de nombres, de fonctions, ou autres dans lesquels la formule du produit est vérifiée. Les questions habituelles de la théorie des modèles, telle que l'existence d'une modèle-compagne ou encore sa stabilité, nous mènent vers de nouvelles questions de nature plutôt géométrique.
Je vais expliquer quelques avancées récentes dans ce sens, où une analyse géométrique locale nous permet de déduire des propriété globales dans un corps globalement valués.
+ Chris Miller Beyond o-minimality, and why 19/05/2017 11:00 ENS, salle W
O-minimal structures on the real field have many desirable properties. As examples:
(a) Hausdorff (and even packing) dimension agrees with topological dimension on locally closed definable sets.
(b) Locally closed definable sets have few rational points (in the sense of the Pila-Wilkie Theorem).
(c) For each positive integer p, every closed definable set is the zero set of a definable C^p function.
(d) Connected components of definable sets are definable.
But to what extent is o-minimality necessary for these properties to hold? I will discuss this question, and illustrate via examples as to why one might care about answers.
+ Ayhan Günaydin Tame Expansions of o-minimal Structures 19/05/2017 14:15 ENS, salle W
Expanding a model theoretically “tame” structure in a way that it stays “tame” has been a theme in the recent years. In the first part of this talk, we present a history of work done in that frame. Then we focus on the case of expansions of o-minimal structures by a unary predicate. There is a dividing line according to whether the predicate is dense or discrete; even though the results obtained are similar, there is an enormous difference in the techniques used. We shall present some of the results obtained in the dense case. Starting from a set of abstract axioms, we obtain a decomposition theorem for definable sets and a local structure theorem for definable groups.
The abstract axioms mentioned above are “smallness”, “o-minimal open core” and “quantifier elimination up to existential formulas”. We shall illustrate a proof of the fact that the first two imply “quantifier elimination up to bounded formulas”, which is a weak form of the last axiom and we give reasons why it is really weaker than that axiom.
(Joint work with P. Eleftheriou and P. Hieronymi)
+ George Comte Zéros et points rationnels des fonctions analytiques ou oscillant. 19/05/2017 16:00 ENS, salle W
Compter les points rationnels de hauteur bornée dans le graphe d'une fonction, ou plus généralement d'une courbe (plane), se ramène à estimer le nombre Z_d de points d'intersection de cette courbe avec un ensemble algébrique de degré d donné. J'expliquerai
- d'une part comment on peut produire des familles de fonctions analytiques sur [0,1] telle que Z_d est polynomialement borné en d, et comment une telle borne assure que le graphe d'une telle fonction recèle moins de logα(T) points rationnels de hauteur < T,
- d'autre part comment on peut traiter le cas de certaines courbes oscillant (ie non o-minimales) pour obtenir encore une borne du type logα(T).
Il s'agit de travaux en commun avec Y. Yomdin d'une part et C. Miller d'autre part.
+ François Loeser Un théorème d'Ax-Lindemann non-archimédien 28/04/2017 11:00 ENS, Salle W
On présentera un résultat de type Ax-Lindemann pour les produits de courbes de Mumford sur un corps p-adique. Notre preuve reprend en l'adaptant les grandes lignes de l'approche de Pila dans le cas archimédien. En particulier nous utilisons un théorème de Pila-Wilkie p-adique obtenu avec R. Cluckers et G. Comte. Il s'agit d'un travail en commun avec A. Chambert-Loir.
+ Luck Darnière Triangulation des ensembles semi-algébriques p-adiques 28/04/2017 14:15 ENS, Salle W
On sait que les ensembles semi-algébriques p-adiques admettent une décomposition cellulaire semblable à celle des semi-algébriques réels (Denef 1984). On sait aussi les classifier à bijection semi-algébrique près (Cluckers 2001), mais pas à homéomorphismes semi-algébriques près. En introduisant une notion appropriée de simplexe sur les corps p-adiquement clos, on peut montrer que tout ensemble semi-algébrique p-adique est semi-algébriquement homéomorphe à un complexe simplicial p-adique, exactement comme dans le cas réel clos. C'est ce résultat récent de `triangulation p-adique' que je tâcherai de présenter, avec ses applications les plus directes (existence de découpages avec contraintes aux bords, existence de rétractions, etc).
+ Omid Amini Séries linéaires limites et applications 28/04/2017 16:00 ENS, Salle W
Je présente un formalisme combinatoire pour l'étude des dégénérescences des séries linéaires dans une famille de courbes algébriques. J'en déduis quelques applications dont notamment l'équirépartition selon la mesure admissible de Zhang des points de ramification des fibrés en droite sur les courbes de Berkovich, un analogue non-archimédien du théorème de Mumford-Neeman. Je discuterai aussi la question de la convergence de la mesure d'Arakelov vers la mesure de Zhang dans une famille de surfaces de Riemann.
+ Evelina Viada Rational points on families of curves 10/03/2017 11:00 ENS, Salle W
The TAC (torsion anomalous conjecture) states that for an irreducible variety V embedded transversaly in an abelian variety A there are only finitely many maximal V-torsion anomalous varieties. It is well know that the TAC implies the Mordell-Lang conjecture. S. Checcole, F. Veneziano and myself were trying to prove some new cases of the TAC. In this process we realised that some methods could be made not only effective but even explicit. So we analysed the implication of this explicit methods on the Mordell Conjeture. Namely: can we make the Mordell Conjecture explicit for some new families of curves and so determine all the rational points on these curves? Of course we started with the easiest situation, that is curves in ExE for E an elliptic curve. We eventually could give some new families of curves of growing genus for which we can determine all the rational points. I will explain the difficulties and the ingredients of this result. I will then discuss the generalisations of the method and also its limits.
+ Anne Moreau Satellites of spherical subgroups and Poincaré polynomials 10/03/2017 16:00 ENS, Salle W
Let G be a connected reductive group over C. One can associate with every spherical homogeneous space G/H its lattice of weights X^*(G/H) and a subset S of M of linearly independent primitive lattice vectors which are called the spherical roots. For any subset I of S we define, up to conjugation, a spherical subgroup H_I in G such that dim H_I = dim H and X^*(G/H_I) = X^*(G/H). We call the subgroups H_I the satellites of the spherical subgroup H. Our interest in satellites H_I is motivated by the space of arcs of the spherical homogeneous space G/H.
We show a close relation between the Poincaré polynomials of the two spherical homogeneous spaces G/H and G/H_I.
All of this is useful for the computation of the stringy E-function of Q-Gorenstein spherical embeddings.
The talk is based on joint works with Victor Batyrev.
+ Patrick Speissegger Quasianalytic Ilyashenko algebras 10/03/2017 14:15 ENS, salle W
In 1923, Dulac published a proof of the claim that every real analytic vector field on the plane has only finitely many limit cycles (now known as Dulac's Problem). In the mid-1990s, Ilyashenko completed Dulac's proof; his completion rests on the construction of a quasianalytic class of functions. Unfortunately, this class has very few known closure properties. For various reasons I will explain, we are interested in constructing a larger quasianalytic class that is also a Hardy field. This can be achieved using Ilyashenko's idea of superexact asymptotic expansion. (Joint work with Zeinab Galal and Tobias Kaiser)
+ Wouter Castryck Geometric invariants that are encoded in the Newton polygon 10/02/2017 11:00 ENS, Salle W
Let k be a field and let P be a lattice polygon, i.e. the convex hull in R^2 of finitely many non-collinear points of Z^2. Let C/k be the algebraic curve defined by a sufficiently generic Laurent polynomial that is supported on P. A result due to Khovanskii states that the geometric genus of C equals the number of Z^2-valued points that are contained in the interior of P. In this talk we will give an overview of various other curve invariants that can be told by looking at the combinatorics of P, such as the gonality, the Clifford index, the Clifford dimension, the scrollar invariants associated to a gonality pencil, and in some special cases the canonical graded Betti numbers. This will cover joint work with Filip Cools, Jeroen Demeyer and Alexander Lemmens.
+ David Evans Determining finite simple images of finitely presented groups 10/02/2017 14:15 ENS, Salle W
I will discuss joint work with Martin Bridson and Martin Liebeck which addresses the question: for which collections of finite simple groups does there exist an algorithm that determines the images of an arbitrary finitely presented group that lie in the collection? We prove both positive and negative results. For a collection of finite simple groups that contains infinitely many alternating groups, or contains classical groups of unbounded dimensions, we prove that there is no such algorithm. On the other hand, for a collection of simple groups of fixed Lie type we obtain positive results by using the model theory of finite fields.

+ Eva Leenknegt Cell Decomposition for P-minimal structures: a story 10/02/2017 16:00 ENS, salle W
P-minimality is a concept that was developed by Haskell and Macpherson as a p-adic equivalent for o-minimality. For o-minimality, the cell decomposition theorem is probably one of the most powerful tools, so it is quite a natural question to ask for a p-adic equivalent of this.
In this talk I would like to give an overview of the development of cell decomposition in the p-adic context, with an emphasis on how questions regarding the existence of definable skolem functions have complicated things. The idea of p-adic cell decomposition was first developed by Denef, for p-adic semi-algebraic structures, as a tool to answer certain questions regarding quantifier elimination, rationality and p-adic integration. This first version eventually resulted in a cell decomposition theorem for P-minimal structures. This theorem, proven by Mourgues, was however dependent on the existence of definable Skolem functions. The second part of the talk will focus a bit more on Skolem functions, and sketch a generalized version of the Denef-Mourgues theorem that does not rely on the existence of such functions, by introducing a notion of clustered cells. We will explain the notion, give an informal sketch of the proof, and compare with other versions of cell decomposition.
+ Gal Binyamini Wilkie's conjecture for restricted elementary functions 13/01/2017 11:00 ENS, Salle W
Let X be a set definable in some o-minimal structure. The Pila-Wilkie theorem (in its basic form) states that the number of rational points in the transcendental part of X grows sub-polynomially with the height of the points. The Wilkie conjecture stipulates that for sets definable in R_exp, one can sharpen this asymptotic to polylogarithmic.
I will describe a complex-analytic approach to the proof of the Pila-Wilkie theorem for subanalytic sets. I will then discuss how this approach leads to a proof of the “restricted Wilkie conjecture”, where we replace R_\exp by the structure generated by the restrictions of exp and sin to the unit interval (both parts are joint work with Dmitry Novikov). If time permits I will discuss possible generalizations and applications.

+ Jonathan Pila Some Zilber-Pink-type problems 13/01/2017 14:15 ENS, salle W
I will discuss some problems which are analogous to, but formally not comprehended within, the Zilber-Pink conjecture, involving collections of “special subvarieties” connected with uniformization maps of suitable domains.
+ Silvain Rideau Imaginaires dans les corps valués avec opérateurs 13/01/2017 16:00 ENS, Salle W
Au début des années 2000, Haskell, Hrushovski and Macpherson ont décrit les ensembles interprétables dans un corps valué algébriquement clos à l'aide d'équivalents en plus grande dimension des boules. Plus précisément, ils ont prouvé l'élimination des imaginaires dans le language géométrique. Pendant la même période, l'intérêt des théoriciens des modèles pour les corps valués avec opérateurs s'est grandement développé. Les questions résolues pour ces structures tournent, pour la plupart, autour de l'élimination des quantificateurs et de la modération. Mais, au vu des résultats de Haskell, Hrushovski and Macpherson, il est tentant de vouloir aussi classifier les ensembles interprétables.
Dans cet exposé, je traiterai des deux exemples les mieux compris: la modèle complétion de Scanlon des corps valués munis d'une dérivation contractive et les corps valués séparablement clos de degré d'imperfection fini. En particulier, je montrerai comment l'élimination des imaginaires dans ces structures est liée à l'existence d'une base canonique pour les types définissables et comment la propriété d'indépendance (ou plutôt son absence) peut aider à contrôler ces bases canoniques.
+ Arthur Forey Densité locale motivique et p-adique uniforme 09/12/2016 11:00 ENS, salle W
Je présenterai un analogue motivique de la densité locale introduite par Kurdyka-Raby dans le cas réel et Cluckers-Comte-Loeser dans le cas p-adique. Celle-ci s'applique aux définissables dans une théorie de corps Henséliens modérée (au sens de Cluckers-Loeser), en caractéristique nulle et caractéristique résiduelle quelconque.
Comme dans les cas sus-cités, il existe un cône tangent distingué sur lequel on peut calculer la densité si on lui attache des multiplicités, qu'on définit en décomposant l'ensemble définissable étudié en graphes de fonctions (localement) 1-Lipschitziennes. Cela implique en particulier une version uniforme du théorème de Cluckers-Comte-Loeser sur la densité p-adique.
+ Martin Hils Théorie des modèles de variétés compactes complexes avec automorphisme 09/12/2016 14:15 ENS, salle W
On peut développer la théorie des modèles des variétés compactes complexes (CCM) avec automorphisme générique en analogie avec ce qui a été fait pour les corps aux différences existentiellement clos, autrement dit pour la théorie ACFA, dans des travaux importants de Chatzidakis et Hrushovski, entre autres. La théorie (du premier ordre) correspondante CCMA est supersimple, et on a la trichotomie de Zilber pour les types “fini-dimensionnels” de rang SU 1.
Dans l'exposé, je vais présenter quelques résultats dans CCMA qui relèvent de la simplicité géométrique, et je vais discuter comment on peut traiter de systèmes dynamiques méromorphes dans ce cadre. Enfin, j'indiquerai pourquoi CCMA n'élimine pas les imaginaires, contrairement à ce qui se passe dans ACFA.
+ Christopher Voll Uniform analytic properties of representation zeta functions of groups 09/12/2016 16:00 ENS, salle W
Representation zeta functions of groups are Dirichlet-type generating functions enumerating the groups' finite-dimensional irreducible complex representations, possibly up to suitable equivalence relations. Under favourable conditions, these zeta functions satisfy Euler products whose factors are indexed by the places of number fields. I will discuss how p-adic integrals can be used to study these Euler products and how this sometimes allows us to capture some key analytic properties of representation zeta functions of groups.

+ Dugald Macpherson Cardinalities of definable sets in finite structures 04/11/2016 16:00 ENS, salle W
I will discuss model-theoretic developments stemming from a theorem of Chatzidakis, van den Dries and Macintyre, which states that given a formula φ(x,y) in the language of rings, there are finitely many pairs (μ ,d) (μ rational, d a natural number) such that for any finite field F_q and parameter a, the definable set φ(F_q,a) has size roughly μ q^d for one of these pairs (μ ,d). A model-theoretic framework suggested by this was developed by Elwes, myself, and Steinhorn, with notions of an `asymptotic class' of finite structures, and `measurable' infinite structure: an ultraproduct of an asymptotic class is measurable (and in particular has supersimple finite rank theory).

I will discuss recent work with Anscombe, Steinhorn and Wolf on `multi-dimensional asymptotic classes' of finite structures and infinite `generalised measurable' structures which greatly extends this framework, to include classes of multi-sorted structures, which may have infinite rank ultraproducts, or even have ultraproducts with non-simple theory (though these ultraproducts can never have the strict order property). The key feature is the fixed bound, for each formula φ(x,y), on the number of approximate sizes of sets φ(M,a) as M ranges through a class of finite structures and the parameter a varies through M. The focus will be on naturally-arising examples.
+ Pierre Dèbes Perspectives sur le probl&egrave;me inverse de Galois 04/11/2016 11:00 ENS, salle W
Les résultats dont je parlerai sont motivés par le Problème Inverse de Galois Régulier (PIGR):

Montrer que tout groupe fini G est le groupe de Galois d'une extension galoisienne F/Q(T) avec Q algébriquement
clos dans F.

Je présenterai deux types de résultats. J'expliquerai d'abord que certaines variantes fortes liées aux
notions d'extensions génériques, d'extensions paramétriques et de type de ramification paramétriques ne sont pas
vraies. Puis, je montrerai une conséquence forte du PIGR liée à une conjecture de Malle sur le nombre
d'extensions galoisiennes de Q de groupe donné et de discriminant borné.
+ Elisabeth Bouscaren Orthogonalité et théorie des modèles des groupes de rang fini dans les preuves de Mordell-Lang pour les corps de fonctions - Orthogonality and model theory of finite rank groups in the proofs of Function Field Mordell-Lang 04/11/2016 14:00 ENS, salle W
Dans cet exposé, nous essayerons d'expliquer l'utilisation de la théorie des modèles des groupes de rang fini et de la notion d'orthogonalité dans les preuves modèles théoriques de la conjecture de Mordell-Lang pour les corps de fonction, à la fois dans la preuve originelle de Hrushovski et dans des travaux plus récents sur le sujet (en commun avec Franck Benoist et Anand Pillay). Nous parlerons en particulier de l'utilisation du “Théorème du Socle” dans ces preuves.

In this talk, we will try to explain the use of the model-theory of finite rank groups and of the notion of orthogonality in the model theoretic proofs of the Mordell-Lang Conjecture for function fields, in Hrushovski's original proof as well as in other more recent work (joint with Franck Benoist and Anand PIllay). In particular we will talk about the use of the “Socle Theorem” in these proofs.

[Début de l'exposé à 14h15]
+ Vincenzo Mantova Non-standard fewnomials 13/05/2016 11:00 ENS, Salle W
Call non-standard fewnomial (or sparse/lacunary polynomial) a non-standard polynomial whose number of non-zero terms is finite. The non-standard translation of a conjecture of Rényi and Erdöt;s, proved by Schinzel and then improved by Zannier, says that if the square of a non-standard polynomial is a fewnomial, then the polynomial itself is a fewnomial. With C. Fuchs and Zannier, we proved the more general statement that the ring of fewnomials is integrally closed in the ring of non-standard polynomials. This can be used to show certain properties of covers of multiplicative groups, such as a kind of Bertini irreducibility theorem. I will discuss both standard and non-standard formulations of the theorem, some of the applications, and give a sketch of a new non-standard proof.
+ Katrin Tent Profinite NIP groups 13/05/2016 14:15 ENS, Salle W
We consider profinite groups as 2-sorted first order structures, with a group sort, and a second sort which acts as an index set for a uniformly definable basis of neighbourhoods of the identity. It is shown that if the basis consists of all open subgroups, then the first order theory of such a structure is NIP (that is, does not have the independence property) precisely if the group has a normal subgroup of finite index which is a direct product of finitely many compact p-adic analytic groups, for distinct primes p. In fact, the condition NIP can here be weakened to NTP2.
We also show that any NIP profinite group, presented as a 2-sorted structure, has an open prosoluble normal subgroup.
(Joint work with Dugald Macpherson)
+ Michel Raibaut Wave front sets of distributions in non-archimedean analysis. 13/05/2016 16:00 ENS, Salle W
In 1969, Sato and Hörmander introduced the notion of wave front set of a distribution in the real context. This concept gives a better understanding of operations on distributions such as product or pullback and it plays an important role in the theory of partial differential equations. In 1981, Howe introduced a notion of wave front set for some Lie group representations and in 1985, Heifetz gave an analogous version in the p-adic context. In this talk, in the t-adic context in characteristic zero, using Cluckers-Loeser motivic integration we will present analogous constructions of test functions, distributions and wave front sets. In particular, we will explain how definability can be used as a substitute for topological compactness of the sphere in the real and p-adic contexts to obtain finiteness.
This a joint work with R. Cluckers, and F. Loeser.
+ Nadja Hempel Enveloppes définissable de sous groupe abélien, nilpotent ou résoluble 08/04/2016 14:00 ENS, salle W.
Étant donné un groupe G, un problème particulier qui nous intéresse est de trouver des enveloppes définissables de sous-groupes abéliens, nilpotents ou résolubles de G qui ayant les mêmes propriétés algébriques.
Au cours des dernières décennies, il y a eu des progrès remarquables pour répondre a cette question pour des groupes qui satisfont certaines propriétés modèle-théoriques (théorie stable, dépendante, simple, etc.), ainsi que pour des groupes dont les centralisateurs satisfont certaines conditions de chaîne sur des centralisateur.
Je présente ces résultats et donne des applications.
+ Jean-Benoit Bost Réseaux euclidiens de rang fini et infini, séries thêta et formalisme thermodynamique 08/04/2016 16:00 ENS, salle W.
Un réseau euclidien est la donnée (E, | . |) d'un ℤ-module E isomorphe à ℤ^r, r in ℕ, et d'une norme euclidienne | . | sur le ℝ-espace vectoriel E_ℝ ≅ ℝ^r qui lui est associé.
En géométrie arithmétique, il s'avère naturel d'associer à un réseau euclidien un invariant dans ℝ_+ défini au moyen d'une série thêta par la formule:
h^0_θ(E, | . |) := log sum_v in E e^-π|v|^2.
Dans cet exposé, je discuterai diverses propriétés, classiques et moins classiques, de cet invariant h^0_θ. Notamment, j'expliquerai comment certaines de ses propriétés se rattachent à la théorie des grandes déviations et au formalisme thermodynamique.
Je présenterai aussi des généralisations de l'invariant h^0_θ attachées à des avatars de rang infini des réseaux euclidiens.
+ Junyi Xie La conjecture de Manin-Mumford dynamique pour les relevés du Frobenius 08/04/2016 11:00 ENS, salle W
La conjecture de Manin-Mumford dynamique est un analogue dynamique de la conjecture de Manin-Mumford. Dans cet exposé, on démontre une version de cette conjecture pour les endomorphismes d'espaces projectifs sur un corps p-adique dont la réduction modulo p est le Frobenius. Notre méthode est de transporter la dynamique p-adique à une dynamique sur un corps de caractéristique p par la théorie des espaces perfectoïdes de Peter Scholze.
+ Géométrie et Théorie des Modèles 04/03/2016 11:00
+ Martin Ziegler Géométrie et Théorie des Modèles 14/11/2014 11:00 Ecole Normale Supérieure, 45 rue d'Ulm, 75005 Paris (RER : Luxembourg)
+ Chris Daw Degrees of strongly special subvarieties and the André-Oort conjecture. 14/11/2014 14:15 Ecole Normale Supérieure, 45 rue d'Ulm, 75005 Paris (RER : Luxembourg)
We give a new proof of the André-Oort conjecture under the generalised Riemann hypothesis. In fact, we generalise the strategy pioneered by Edixhoven, and implemented by Klingler and Yafaev, to all special subvarieties. Thus, we remove ergodic theory from the proof of Klingler, Ullmo and Yafaev and replace it with tools from algebraic geometry. Our key ingredient is a lower bound for the degrees of strongly special subvarieties coming from Prasad's volume formula for S-arithmetic quotients of semisimple groups.