Equipe(s)  Responsable(s)  Salle  Adresse 

Logique Mathématique 
Zoé Chatzidakis, Raf Cluckers. 
Zoom' 
http://www.logique.jussieu.fr/ zoe/GTM/
Pour recevoir le programme par email, 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 quelquesuns des exposés sont disponibles.
Orateur(s)  Titre  Date  Début  Salle  Adresse  Diffusion  

+  Gabriel Conant  VCdimension in model theory, discrete geometry, and combinatorics  23/04/2021  15:00  Zoom'  
In statistical learning theory, the notion of VCdimension 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 VCdimension, with examples motivated by discrete geometry and additive combinatorics. I will then present several model theoretic applications of VCdimension. 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 VCdimension.  
+  Artem Chernikov  Recognizing groups and fields in Erdős geometry and model theory  23/04/2021  16:30  Zoom'  
Assume that Q is a relation on R^s of arity s definable in an ominimal 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 s1 coordinates is finitetoone but Q has maximal size intersections with some grids (of size >n^(s1  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 ElekesSzabó 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 spartite hypergraph but can intersect finite grids in more that than n^(s1 + ε) points, then the real field can be definably recovered from Q (joint work with Abdul Basit, Sergei Starchenko, Terence Tao and ChieuMinh Tran). I will explain how these results are connected to the modeltheoretic trichotomy principle and discuss variants for higher dimensions, and for stable structures with distal expansions. 
Orateur(s)  Titre  Date  Début  Salle  Adresse  

+  Jason Bell  Effective isotrivial MordellLang in positive characteristic  26/03/2021  15:00  
The MordellLang 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 finitestate 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 semiminimal analysis of algebraic differential equations  26/03/2021  16:30  
It is wellknown 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 dpfinite fields  12/02/2021  09:00  
Shelah's conjecture predicts that any infinite NIP field is
either separably closed, real closed or admits a nontrivial henselian
valuation. Recently, Johnson proved that Shelah's conjecture holds for
fields of finite dprank, also known as dpfinite fields. The aim of these two talks is to give an introduction to dprank in some algebraic structures and an overview of Johnson's work. In the first talk, we define dprank (which is a notion of rank in NIP theories) and give examples of dpfinite structures. In particular, we discuss the dprank of ordered abelian groups and use them to construct multitude of examples of dpfinite fields. We also prove that every dpfinite field is perfect and sketch a proof that any valued field of dprank 1 is henselian. In the second talk, we give an overview of Johnson's proof that every infinite dpfinite field is either algebraically closed, real closed or admits a nontrivial henselian valuation. Crucially, this relies on the notion of a Wtopology, a natural generalization of topologies arising from valuations, and the construction of a definable Wtopology on a sufficiently saturated unstable dpfinite 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 epsilonzero. 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 nonabelian group will contain many solutions to this equation, provided it has sufficiently large density. We will report on recent work with Amador MartinPizarro on how to find solutions to the above equations in the context of pseudofinite 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 ominimal structures and algebraic groups  11/12/2020  09:00  Zoom  
Groups definable in ominimal structures have been studied by many authors in the last 30 years and include algebraic groups over algebraically closed fields of characteristic 0, semialgebraic 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 ominimal 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 nonarchimedean 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 nonarchimedean fields based on HrushovskiLoeser 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 nonarchimedean field.  
+  Will Johnson  The étaleopen topology (suite)  27/11/2020  09:00  Zoom  
Fix an abstract field K. For each Kvariety V, we will define an “étaleopen” 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 pseudofinite fields, the étaleopen topology seems to be new, and has some interesting properties. The étaleopen topology is mostly of interest when K is large (also known as ample). On nonlarge fields, the étaleopen topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some wellknown facts about large fields, and classify the modeltheoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with ChieuMinh Tran, Erik Walsberg, and Jinhe Ye. 

+  Will Johnson  The étaleopen topology  13/11/2020  09:00  Zoom  
Fix an abstract field K. For each Kvariety V, we will define an étaleopen 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 pseudofinite fields, the étaleopen topology seems to be new, and has some interesting properties. The étaleopen topology is mostly of interest when K is large (also known as ample). On nonlarge fields, the étaleopen topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some wellknown facts about large fields, and classify the modeltheoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with ChieuMinh 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 prodefinable and we propose a modeltheoretic 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 socalled cellular decomposition, i.e. representation as a union of convenient semialgebraic images of standard cubes. The GromovYomdin Lemma (later generalized by Pila and Wilkie) proves that the maps could be chosen of C^rsmooth norm at most one, and the number of such maps is uniformly bounded for finitedimensional 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 YomdinGromov Lemma with polynomial bounds on the complexity, thus proving a longstanding 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 longterm 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 ManinMumford conjecture, the AndreOort conjecture, Galoisorbit 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 pseudoalgebraically closed fields  06/03/2020  16:00  ENS, Salle W  
A field K is called pseudoalgebraically closed (PAC) if every absolutely irreducible variety defined over K has a Krational point. These fields were introduced by Ax in his characterization of pseudofinite 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 graphcoding 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 ominimal 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.


+  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 padic Lie group of padic points of the jacobian, the closure of the MordellWeil group with the padic points of the curve. If the MordellWeil rank is less than the genus then this method has never failed. Minhyong Kim's nonabelian 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 socalled 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 realalgebraic 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 RideauKikuchi) 

+  Victoria Cantoral Farfan  The MumfordTate conjecture implies the algebraic SatoTate conjecture  08/11/2019  14:15  ENS, Salle W  
The famous MumfordTate conjecture asserts that, for every prime number l, Hodge cycles are ℚ_l linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic SatoTate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized SatoTate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces. 

+  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 nontrivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields. 

+  Laurent MoretBailly  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.


+  Silvain RIdeau  Hminimality  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 Cminimality, Pminimality and Vminimality and puts no restriction on the residue field or valued group contrary to these previous notions. This new notion, hminimality, can be defined, analogously to other minimality notions, by asking that 1types, 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: lhmin for l a natural number or omega. My second goal in this talk will be to explain the various geometric properties that follow form hminimality, among which the wellknown 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 blowups in the context of Berkovich geometry (inspired by Raynaud and Gruson's paper on the same topic in the schemetheoretic 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 LipshitzCluckers variant of LipshitzRobinson's analytic language. (I plan to spend most of the talk discussing the results rather than their proofs.) 

+  Dimitri Wyss  Nonarchimedean 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 nonarchimedean integrals on these moduli spaces, building on work of DenefLoeser 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 nonarchimedean integrals by motivic ones. The latter is joint work with François Loeser. 

+  Avraham Aizenbud  Pointwise 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 infinitystacks, 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 (..., BuiumPoonen, 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 ZilberPink 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 realanalytic version of the method due to Bombieri and Pila and a padic version due to HeathBrown. The aim of our talk is to describe a global refinement of the padic method and some applications like a uniform bound for nonsingular cubic curves which improves upon earlier bounds of EllenbergVenkatesh and HeathBrown.  
+  Vlerë 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 localglobal 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 localglobal 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 nontrivially 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 HrushovskiLoeser about the isodefinability 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 ZilberPink 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 BailyBorel 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 socalled Heckefacteur 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 CattaniDeligneKaplan (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_0saturated 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 biinterpretation. 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 socalled scrollar invariants (also called Maroni invariants) which describe how the pushforward 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 socalled 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 numbertheoretic manifestation of the phenomena observed.  
+  Amador MartinPizarro  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 padics 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 padics. 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 padic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galoistheoretic 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 nonArchimedean YomdinGromov parametrizations. This is joint work with Raf Cluckers and François Loeser.  
+  Guy Casale  AxLindemannWeierstrass with derivatives and the genus 0 Fuchsian groups  14/12/2018  14:15  ENS, Salle W  
We prove the AxLindemannWeierstrass 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 PilaTsimerman on the j function.
Joint work with James Freitag and Joel Nagloo. 

+  Antoine Ducros  Nonstandard analysis and nonarchimedean 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 ChambertLoir on Berkovich spaces can be seen (in the tadic case) as limits of usual integrals on complex algebraic varieties; a crucial step is the development of a nonstandard 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  Firstorder logic in finitely generated fields  16/11/2018  14:15  ENS, Salle W  
The expressive power of firstorder 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 AschenbrennerKhélifNaziazenoScanlon.
Building on work of Pop and Poonen, and using geometric results due to KerzSaito 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. 

+  JeanPhilippe 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 1516 (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 Remeztype inequalities  18/05/2018  16:00  Jussieu, salle 101 couloir 1516 (1er étage)  
A doubling chart on an ndimensional 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 nonempty 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 MordellLang and finite automata  18/05/2018  14:15  Jussieu, salle 101 couloir 1516 (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 SkolemMahlerLech 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 workinprogress it has engendered on an effective version of the isotrivial MordellLang theorem.  
+  Tom Scanlon  The dynamical MordellLang problem in positive characteristic  23/03/2018  14:15  IHP, amphitheatre Darboux  
The dynamical MordellLang 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 Krational 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 psets. We studied the special case of the positive characteristic dynamical MordellLang problem on semiabelian varieites and using our earlier results with Moosa on socalled Fsets 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 MordellLang 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 ColliotThélène  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 ColliotThélène under discussion gives (conjectural) sufficient conditions which imply that for almost all rational prime numbers p, the map f maps the padic 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 onedimensional case of the ElekesSzabo 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 ElekesSzabo 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 ElekesSzabo 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 MordellWeil 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 nonspecialists 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 cocountable. If true, it would mean that the geometry of solution sets of complex exponentialpolynomial 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 modeltheoretic 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/OdaMatsumoto  15/12/2017  11:00  ENS, salle W  
Following the spirit of Grothendieck's Esquisse d'un Programme, the Ihara/OdaMatsumoto 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 modell twostep 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 GrinbergKazhdan 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 coordinate 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 SzemerediTrotter 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  Quasiminimal expansions of the complex field  17/11/2017  16:00  ENS, Salle W  
I discuss a backandforth technique for proving that in certain expansions of the complex field every L_∞, ωdefinable subset of ℂ is either countable or cocountable. 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 nonlocally 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 Torrelitype 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 (modeltheoretic) 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 nonconventional limits used by physicists. Then we focus on the modeltheoretic 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 (padic and adelic).  
+  Immi Halupczok  Un nouvel analogue de l'ominimalité dans des corps valués  13/10/2017  16:00  ENS, salle W  
Pour les corps réel clos, la notion d'ominimalité 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. pmininalité, Cminimalité, Bminimalité, vminimalité), 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 modèlecompagne 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 ominimality, and why  19/05/2017  11:00  ENS, salle W  
Ominimal 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 PilaWilkie 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 ominimality 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 ominimal 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 ominimal 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”, “ominimal 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 ominimales) 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'AxLindemann nonarchimédien  28/04/2017  11:00  ENS, Salle W  
On présentera un résultat de type AxLindemann pour les produits de courbes de Mumford sur un corps padique. 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 PilaWilkie padique obtenu avec R. Cluckers et G. Comte. Il s'agit d'un travail en commun avec A. ChambertLoir.  
+  Luck Darnière  Triangulation des ensembles semialgébriques padiques  28/04/2017  14:15  ENS, Salle W  
On sait que les ensembles semialgébriques padiques admettent une décomposition cellulaire semblable à celle des semialgébriques réels (Denef 1984). On sait aussi les classifier à bijection semialgébrique près (Cluckers 2001), mais pas à homéomorphismes semialgébriques près. En introduisant une notion appropriée de simplexe sur les corps padiquement clos, on peut montrer que tout ensemble semialgébrique padique est semialgébriquement homéomorphe à un complexe simplicial padique, exactement comme dans le cas réel clos. C'est ce résultat récent de `triangulation padique' 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 nonarchimédien du théorème de MumfordNeeman. 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 Vtorsion anomalous varieties. It is well know that the TAC implies the MordellLang 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 Efunction of QGorenstein 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 mid1990s, 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 noncollinear 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^2valued 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 Pminimal structures: a story  10/02/2017  16:00  ENS, salle W  
Pminimality is a concept that was developed by Haskell and Macpherson as a padic equivalent for ominimality. For ominimality, the cell decomposition theorem is probably one of the most powerful tools, so it is quite a natural question to ask for a padic equivalent of this.
In this talk I would like to give an overview of the development of cell decomposition in the padic context, with an emphasis on how questions regarding the existence of definable skolem functions have complicated things. The idea of padic cell decomposition was first developed by Denef, for padic semialgebraic structures, as a tool to answer certain questions regarding quantifier elimination, rationality and padic integration. This first version eventually resulted in a cell decomposition theorem for Pminimal 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 DenefMourgues 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 ominimal structure. The PilaWilkie theorem (in its basic form) states that the number of rational points in the transcendental part of X grows subpolynomially 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 complexanalytic approach to the proof of the PilaWilkie 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 ZilberPinktype problems  13/01/2017  14:15  ENS, salle W  
I will discuss some problems which are analogous to, but formally not comprehended within, the ZilberPink 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 padique uniforme  09/12/2016  11:00  ENS, salle W  
Je présenterai un analogue motivique de la densité locale introduite par KurdykaRaby dans le cas réel et CluckersComteLoeser dans le cas padique. Celleci s'applique aux définissables dans une théorie de corps Henséliens modérée (au sens de CluckersLoeser), en caractéristique nulle et caractéristique résiduelle quelconque.
Comme dans les cas suscités, 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) 1Lipschitziennes. Cela implique en particulier une version uniforme du théorème de CluckersComteLoeser sur la densité padique. 

+  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 “finidimensionnels” 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 Dirichlettype generating functions enumerating the groups' finitedimensional 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 padic 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 modeltheoretic 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 modeltheoretic 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 `multidimensional asymptotic classes' of finite structures and infinite `generalised measurable' structures which greatly extends this framework, to include classes of multisorted structures, which may have infinite rank ultraproducts, or even have ultraproducts with nonsimple 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 naturallyarising examples. 

+  Pierre Dèbes  Perspectives sur le problème 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 MordellLang pour les corps de fonctions  Orthogonality and model theory of finite rank groups in the proofs of Function Field MordellLang  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 MordellLang 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 modeltheory of finite rank groups and of the notion of orthogonality in the model theoretic proofs of the MordellLang 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  Nonstandard fewnomials  13/05/2016  11:00  ENS, Salle W  
Call nonstandard fewnomial (or sparse/lacunary polynomial) a nonstandard polynomial whose number of nonzero terms is finite. The nonstandard translation of a conjecture of Rényi and Erdöt;s, proved by Schinzel and then improved by Zannier, says that if the square of a nonstandard 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 nonstandard 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 nonstandard formulations of the theorem, some of the applications, and give a sketch of a new nonstandard proof.  
+  Katrin Tent  Profinite NIP groups  13/05/2016  14:15  ENS, Salle W  
We consider profinite groups as 2sorted 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 padic 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 2sorted structure, has an open prosoluble normal subgroup. (Joint work with Dugald Macpherson) 

+  Michel Raibaut  Wave front sets of distributions in nonarchimedean 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 padic context. In this talk, in the tadic context in characteristic zero, using CluckersLoeser 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 padic 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 sousgroupes 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 modèlethéoriques (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. 

+  JeanBenoit 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 ℤmodule E isomorphe à ℤ^r, r in ℕ, et d'une norme euclidienne  .  sur le ℝespace 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^πv^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 ManinMumford dynamique pour les relevés du Frobenius  08/04/2016  11:00  ENS, salle W  
La conjecture de ManinMumford dynamique est un analogue dynamique de la conjecture de ManinMumford. Dans cet exposé, on démontre une version de cette conjecture pour les endomorphismes d'espaces projectifs sur un corps padique dont la réduction modulo p est le Frobenius. Notre méthode est de transporter la dynamique padique 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 AndréOort conjecture.  14/11/2014  14:15  Ecole Normale Supérieure, 45 rue d'Ulm, 75005 Paris (RER : Luxembourg)  
We give a new proof of the AndréOort 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 Sarithmetic quotients of semisimple groups. 