Géométrie et Théorie des Modèles

Année 2019 - 2020

Organisateurs : Zoé Chatzidakis, Raf Cluckers.
Pour recevoir le programme par e-mail, écrivez à : zchatzid_at_dma.ens.fr.
Pour les personnes ne connaissant pas du tout de théorie des modèles, des notes introduisant les notions de base (formules, ensembles définissables, théorème de compacité, etc.) sont disponibles ici. Elles peuvent aussi consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998.
Les notes de quelques-uns des exposés sont disponibles.

Vendredi 11 octobre 2019, ENS, Salle W. Orateurs :

11h : Lorenzo Fantini (Frankfurt), A valuative approach to the inner geometry of surfaces
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.

14h15 : Antoine Ducros (IMJ-PRG),Quantifier elimination in algebraically closed valued fields in the analytic language: a geometric approach
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.)

16h : Silvain Rideau (IMJ-PRG), H-minimality
(joint with Raf Cluckers and Immi Halupczok)
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.

Vendredi 8 novembre 2019. Salle W, ENS. Orateurs :

11h : Franziska Jahnke (Münster), Characterizing NIP henselian fields
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.

14h15 : Victoria Cantoral Farfan (Leuven), The Mumford-Tate conjecture implies the algebraic Sato-Tate conjecture
The famous Mumford-Tate 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 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.

16h15 : Laurent Moret-Bailly (Rennes), Une construction d'extensions faiblement non ramifiées d'un anneau de valuation
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.

Vendredi 31 janvier 2020, salle W. Programme :

11h : Martin Hils (Münster), Classification des imaginaires dans VFA
(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.

14h15 : Marie-Françoise Roy (Rennes), Quantitative Fundamental Theorem of Algebra
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.

16h : Bas Edixhoven (Leiden), Geometric quadratic Chabauty.
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).

Vendredi 6 mars (ENS, Salle W). Orateurs :

11h : Alex Wilkie (Manchester/Oxford), Some remarks on complex analytic functions in a definable context
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.

14h15 : Raf Cluckers (Lille), Exponential sums modulo powers of primes, singularity theory, and local global principles
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.

16h : Nick Ramsey (ENS), Constructing pseudo-algebraically closed fields
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.

Programme des séances passées : 2006-07, 2007-08, 2008-09, 2009-10, 2010-11, 2011-12, 2012-13, 2013-14, 2014-15, 2015-16, 2016-17, 2017-18, 2018-19.
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