# Séminaires : Géométrie et Théorie des Modèles

Logique Mathématique
Zoé Chatzidakis, Raf Cluckers.
Zoom'

http://www.logique.jussieu.fr/ zoe/GTM/

Pour recevoir le programme par e-mail, écrivez à : zchatzid_at_dma.ens.fr.
Pour les personnes ne connaissant pas du tout de théorie des modèles, des notes introduisant les notions de base (formules, ensembles définissables, théorème de compacité, etc.) sont disponibles ici : http://www.logique.jussieu.fr/~zoe/papiers/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.
Les notes de quelques-uns des exposés sont disponibles.

## Séances à suivre

+ Séances antérieures

### Séances antérieures

+ 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.