Géométrie et Théorie des Modèles

Année 2013 - 2014

11h : David Madore (ENST), Church-Turing computability of the étale cohomology mod l
[Work in common with Fabrice Orgogozo]
The dimension of the étale cohomology groups, with coefficients in Z/lZ, of a scheme of finite type over an algebraically closed field of characteristic different from l, is computable in the sense of Church-Turing. To prove this, we construct a hypercovering of X by schemes (analogous to Artin's “good neighborhoods”) having algorithmically testable geometric properties which allow to reduce the computation of the cohomology of X to that of their completed fundamental group.

14h15 : Leonard Lipshitz (Purdue), Strictly convergent rigid subanalytic sets
Let K be a non-archimedean complete normed field, K_alg the algebraic closure of K and let L be the language of normed fields augmented with symbols for the strictly convergent powerseries over K. Strictly convergent rigid subanalytic sets over K are the subsets of (K_alg)^n definable in L. I will survey what is known about these sets, including recent joint results with Raf Cluckers.

16h : Artem Chernikov (Paris 7) Tame definable topological dynamics
(Joint work with Pierre Simon) I will present some new results on definably amenable groups in NIP theories (typical examples of which are definably amenable groups in o-minimal theories, algebraically closed valued fields and p-adics). In particular I will demonstrate that in this context various notions of genericity coincide (answering some questions of Newelski and Petrykowski) and a characterization of ergodic measures will be given. Arguments rely on the theory of forking for types and measures in NIP theories and the so-called (p,q)-theorem from combinatorics.
If time permits, I will describe how these results generalize to homogeneous spaces, ind-definable groups and action of the group of automorphisms, and how these developments can be viewed as a study of the definable case of abstract tame dynamical systems introduced by Glasner.

Vendredi 15 novembre, salle W (ENS). Programme :

11h : Lou van den Dries (UI Urbana), Valued differential fields
We consider valued fields of equicharacteristic zero equipped with a continuous derivation. This class of structures is rather diverse, including both monotone differential fields and asymptotic differential fields. (These terms will be defined.) Nevertheless, some results can be established uniformly for the entire class: algebraic extensions, construction of residue field extensions, the Equalizer Theorem, construction of immediate extensions, differential-henselianity. Next I will revisit Scanlon's thesis on the model theory of differential-henselian monotone differential fields with enough constants. Time permitting I will add some remarks on the case of asymptotic differential fields.
The above is (a small part of) ongoing joint work with Matthias Aschenbrenner and Joris van der Hoeven focused on developing a model theory for differential fields of transseries.
Transparents

14h15 : Daniel Bertrand (Paris 6), Du théorème du noyau de Manin aux équations de Pell-Fermat.
Je décrirai tout d'abord le rôle du théorème de Manin pour l'indépendence algébrique de logarithmes sur les D-groupes algébriques. Je résumerai ensuite un travail en commun avec D. Masser, A. Pillay et U. Zannier, où nous l'appliquons à la conjecture de Manin-Mumford relative pour des schémas semi-abéliens et, en corollaire, à certaines équations de Pell-Fermat sur les anneaux de polynômes.

16h : Jean-Benoît Bost (Orsay), Autour de l'algébrisation des fibrés en droites
Cet exposé décrira quelques questions d'algébrisation concernant des fibrés analytiques ou formels: ces questions trouvent leur origine dans des problèmes classiques de géométrie algébrique et arithmétique, mais s'avèrent entretenir des liens étroits avec des objets mathématiques et des techniques de preuve qui ont retenu l'intérêt des théoriciens des modèles au cours des dernières années. Je discuterai notamment (1) une nouvelle preuve du théorème d'existence de SGA2 et (2) le rôle des D-groupes algébriques en géométrie diophantienne.

Vendredi 13 décembre, salle W à l'ENS. Programme :

11h : Pantelis Eleftheriou (Konstanz), Pregeometries and definable groups
We describe a recent program for analyzing definable sets and groups in certain model theoretic settings. Those settings include:
(a) o-minimal structures (M, P), where M is an ordered group and P is a real closed field defined on a bounded interval (joint work with Peterzil),
(b) tame expansions (M, P) of a real closed field M by a predicate P, such as expansions with o-minimal open core (work in progress with Gunaydin and Hieronymi).
The analysis of definable groups first goes through a local level, where a pertinent notion of a pregeometry and generic elements is each time introduced.

14h15 : Cyrille Corpet (Toulouse), Galois equations on torsion points and the Tate-Voloch conjecture on p-adic fields
The Tate-Voloch conjecture is a statement about p-adic distance from torsion points to subvarieties in a semi-abelian variety defined over C_p. The use of Galois equations on torsion points by Pink and Rossler to prove the Manin-Mumford conjecture can be adapted to prove that conjecture in the case where both the semi-abelian variety and its subvariety are defined over a finite extension of Q_p.
In this talk, we will present such a proof, and try to give an insight on how this proof differs from the model-theoretic one given by Scanlon.

16h : Raf Cluckers (Lille/KU Leuven), Non-archimedean Yomdin-Gromov parametrizations and points of bounded height
In the spirit of work by Pila-Wilkie (2006) and by Pila (2009), we will present bounds on the number of points of bounded height in the non-archimedean context. An important tool to make the determinant method work is provided by a non-archimedean version of the Yomdin - Gromov parameterizing lemma. We will explain these results, obtained in joint work with G. Comte and F. Loeser.

Vendredi 10 janvier, salle W à l'ENS. Orateurs prévus :

11h : Julien Sebag (Rennes), Les fonctions Zêta de Denef et Loeser sont-elles motiviques ?
Dans cet exposé, nous discuterons de la question posée.

14h15 : Emmanuel Ullmo (IHES), La conjecture d'Ax-Lindemann hyperbolique
La conjecture d'Ax-Lindemann hyperbolique est un énoncé de transcendance fonctionnelle concernant les morphismes d'uniformisation des variétés de Shimura par des espaces symétriques hermitiens. Ces derniers sont munis d'une structure semi-algébrique naturelle et la conjecture d'Ax-Lindemann hyperbolique décrit l'adhérence de Zariski des “flots algébriques” dans la variété de Shimura. Nous expliquerons la preuve récente de cette conjecture obtenue dans un travail en commun avec Bruno Klingler et Andrei Yafaev. Un point important de cette preuve consiste à montrer que la restriction de l'application d'uniformisation à un domaine fondamental convenable est définissable dans la structure o-minimale R_{an,exp} en généralisant un énoncé de Peterzil et Starchenko.
Si le temps le permet nous expliquerons aussi la place de cet énoncé dans la stratégie de Pila-Zannier pour une preuve inconditionnelle de la conjecture d'André-Oort. Nous montrerons en particulier comment on obtient la preuve de la conjecture d'André-Oort pour une puissance arbitraire du module des variétés abéliennes principalement polarisées de dimension 6 et une nouvelle preuve de la conjecture d'André-Oort sous GRH (au moins) pour le module des variétés abéliennes principalement polarisées de dimension arbitraire.

16h : Enrique Casanovas (Barcelone), Stable types and nonforking extensions
We prove that a nonforking restriction of a stable type is stable. The main tool in the proof is the use of generically stable types. There are open problems related to similar statements for simple and NIP types. It is a joint work with Hans Adler and Anand Pillay.

Vendredi 20 juin 2014. Institut Henri Poincaré, salle 314. Orateurs prévus :

11h : Thomas Hales (U Pittsburgh), The transfer principle for the fundamental lemma for Hecke algebras.
The transfer principle in motivic integration allows identities of integrals to be transferred between local fields of characteristic zero and positive characteristic. The fundamental lemma is an identity of integrals arising in the stabilization of the trace formula for automorphic representations. Previous work showed how the transfer principle applies to the unit element of the Hecke algebra. This work, joint with Jorge Cely, extends the transfer principle to the fundamental lemma for the full Hecke algebra.

14h15 : Maryanthe Malliaris (U Chicago), Comparing the complexity of unstable theories
In 1967 Keisler posed the problem of Keisler's order, a suggested program for comparing the complexity of classes of mathematical structures using an asymptotic (ultrapower) point of view. The talk will be about recent results in this area, due to Malliaris and to Malliaris and Shelah, which advance this program by developing a sort of fine structure theory for pseudofinite behavior in model theory. In particular, the focus will be on simple theories, a key model-theoretic class which includes the random graph and pseudofinite fields.

16h : Thomas Scanlon (UC Berkeley), Classification theory and the j-function
Using a functional transcendence theorem of Pila proven using o-minimal methods, we show that the differnential equation satisfied by the j-function (as well as for some related functions) defines a strongly minimal set with trivial though not omega-categorical forking geometry. Using this fact and an effective finiteness theorem of Hrushovski and Pilllay for differentially closed fields, we answer a question of Mazur about finding explicit bounds for the number of points in man isogeny class satisfying a given algebraic equation. (This is a report on joint work with Freitag.)

Programme des séances passées : 2006-07, 2007-08, 2008-09, 2009-10, 2010-11, 2011-12, 2012-13.
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