Vendredi 27 janvier 2012, ENS,
amphithéatre Rataud. Programme et horaire
:
11h : Jakob Stix
(Heidelberg), On the p-adic section
conjecture
The section conjecture of Grothendieck
suggests that a section of the
fundamental group extension
of a proper hyperbolic curve over a number field k necessarily comes from a
k-rational point.
In valuation theoretic terms, this can be reformulated and simultaneously
extended to affine hyperbolic curves U/k
by saying that the image of a section must be contained in the
decomposition subgroup of a k-rational place of the
Galoisgroup of the maximal extension of the function field that is
unramified over U.
If we replace the number field by a p-adic field, then the valuation
theoretic analogue suggests a natural extension to more complicated
valuations of the function field, or, with a more geometric viewpoint to
certain Berkovich points of the p-adic curve. This analogue is now known
due to joint work with Florian Pop and the talk aims to survey this result.
14h : Amador Martin-Pizarro (Lyon
I), On variants of CM-triviality
In 2003, Pillay and Ziegler reproved the function field case of
Mordell-Lang in all characteristics inspired by Hrushovski's original proof
but avoiding the use of the so-called Zariski Geometries. Instead,
motivated on Campana's work on algebraic coreductions, they showed that
given an (irreducible) definable set X of bounded differential degree in a
universal differential closed field, the field of definition of the
constructible set determined by X can be understood (i.e. it is internal)
over a generic realisation of X in terms of a finite set of elements coming
from the constant field. In model-theoretical terms, DCF_0 has the CBP for
types of finite Morley rank with respect to the type of the constants. The
CBP is a generalisation of 1-basedness, which has many structural
consequences for the definability of groups and fields, in particular,
every definable group in a 1-based theory is virtualy abelian.
Another possible generalisation of 1-basedness is called CM-triviality,
which prohibits the existence of a particular point-line-plane
configuration, present in Euclidean Geometry. In particular, a CM-trivial
theory has no infinite definable fields and in the finite rank context, all
definable groups are virtually nilpotent.
We will present an overview of the aforementioned concepts, aimed to a
general audience, without assuming a strong model theoretical background,
and present some variants of CM-triviality and discuss definability of
fields and groups in this context.
16h : Jean-Marie Lion
(Rennes), Théorème du complémentaire pour
les sous-pfaffiens emboîtés
Soit A une structure o-minimale qui admet la propriété de
stratification analytique: tout élément de A est la réunion
d'un nombre fini de sous-variétés analytiques définissables dans
A. Un sous-ensemble V de R^n est un pfaffien
emboîté associé à A si c'est une sous-variété analytique
connexe qui est définissable dans A ou s'il existe un
pfaffien emboîté (associé à A) V' et une 1-forme
différentielle analytique ω définissable dans A tels
que V\subset V', V sépare V' en deux composantes connexes et c'est
la feuille d'un feuilletage de codimension un induit par ω sur
V'. On déduit de résultats de Khovanskii et de Wilkie que les
pfaffiens emboîtés associés à A engendrent une structure o-minimale
P(A). On explique dans l'exposé pourquoi tout
élément de P(A) est la projection d'une combinaison
booléenne d'éléments de A et de pfaffiens emboîtés associés à
A (travail en commun avec Patrick Speissegger).
Vendredi 6 avril 2012, ENS,
salle W. Orateurs prévus :
11h
: Artem
Chernikov (Lyon I), Generalizations of
stability and NTP2
In his work on classification, Shelah had developed a number of
tools for analyzing models of stable first-order theories, with
the so-called forking calculus as one of the essential ingredients.
Later on in the work of Hrushovski, Kim and Pillay it was demonstrated
that forking calculus can be generalized to the class of simple theories
(with pseudo-finite fields and certain difference fields as
typical examples). More recently it was realized that forking remains
an important tool in the study of NIP theories (examples are
o-minimal theories, algebraically closed valued fields and p-adics).
Early on Shelah had defined a class of theories containing both simple and
NIP theories, called NTP2 (No Tree Property of the 2nd kind). In the recent
work with Itay Kaplan, and then with Itai Ben Yaacov, I had demonstrated
that a large part of the forking calculus can be generalized to the class
of NTP2 theories. Apart from the pure model theory reasons, understanding
the structure of NTP2 theories is motivated by particular examples such as
power series over pseudo-finite fields (e.g. the ultraproduct of p-adics)
and certain
valued difference fields. In particular, NTP2 appears to be the
right context for developing simple domination (generalizing the
machinery of stable domination of Haskell, Hrushovski and Macpherson used
in for the analysis of algebraically closed valued fields).
Transparents.
14h : George
Comte (Chambéry) Fibres de Milnor
motiviques en géométrie semi-algébrique
réelle.
J'expliquerai un travail en commun avec Goulwen Fichou, qui consiste
à mettre en place un anneau de Grothendieck
K_0(BSA_R) des formules semi-algébriques grâce auquel on
peut définir, sur le modèle complexe, des fonctions
zêta motiviques de singularités réelles. On montre
que ces fonctions zêtas sont rationnelles et que leur expression
rationnelle définit des fibres de
Milnor motiviques des singularités réelles. Il s'agit
d'éléments de l'anneau K_0(BSA_R)\otimes Z[1/2] dont on
montre qu'ils se
réalisent, via le morphisme caractéristique d'Euler, sur
la caractéristique d'Euler des fibres de Milnor ensemblistes
correspondantes.
16h : Raf
Cluckers
(Lille/Leuven), Motivic Lipschitz
continuity
In a general set-up for non-archimedean geometry, we show how local
Lipschitz continuity implies piecewise Lipschitz continuity (globally
on the whole piece) for definable functions. This is joint work with
G. Comte and F. Loeser which generalizes previous work by the same
three authors for a fixed p-adic field in [GAFA, 2010] and which fits
in a broader program at the interplay of arithmetic and non-archimedean
geometry.
Programme des séances
passées : 2006-07,
2007-08,
2008-09,
2009-10,
2010-11.
Retour
à la page principale.