IMJ-PRG
Logo IMJ-PRG
Sorbonne Université CNRS Paris Diderot

Géométrie et Théorie des Modèles


2014 2016 2017 2018 2019

Année 2018- 2019

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

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 : http://www.logique.jussieu.fr/~zoe/papiers/MTluminy.dvi. Ces personnes peuvent aussi consulter les premiers chapitres du livre Model Theory and Algebraic Geometry, E. Bouscaren ed., Springer Verlag, Lecture Notes in Mathematics 1696, Berlin 1998.
Les notes de quelques-uns des exposés sont disponibles.


Amador Martin-Pizarro - Freiburg

Tame open core and small groups in pairs of topological geometric structures

vendredi 11 janvier 2019 à 16:00 : ENS, Salle W

Using the group configuration theorem, Hrushovski and Pillay showed that the law of a group definable in the reals or the p-adics is locally an algebraic group law, up to definable isomorphism. There are some natural expansions of these two theories of fields, by adding a predicate for a dense substructure, for example the algebraic reals or the algebraic p-adics. We will present an overview on some of the features of these expansions, and particularly on the characterisation of open definable sets as well as of groups definable in the pairs.


Martin Bays - Münster

Definability in the infinitesimal subgroup of a simple compact Lie group

vendredi 11 janvier 2019 à 14:15 : ENS, Salle W

Joint work with Kobi Peterzil.
Let G be a simple compact Lie group, for example G=SO_3(R). We consider the structure of definable sets in the subgroup G^00 of infinitesimal elements. In an aleph_0-saturated elementary extension of the real field, G^00 is the inverse image of the identity under the standard part map, so is definable in the corresponding valued field. We show that the pure group structure on G^00 recovers the valued field, making this a bi-interpretation. Hence the definable sets in the group are as rich as possible.


Wouter Castryck - Leuven

Scrollar invariants, resolvents, and syzygies

vendredi 11 janvier 2019 à 11:00 : ENS, Salle W

With every cover C -> P^1 of the projective line one can associate its so-called scrollar invariants (also called Maroni invariants) which describe how the push-forward of the structure sheaf of C splits over P^1. They can be viewed as geometric counterparts of the successive minima of the lattice associated with the ring of integers of a number field. In this talk we consider the following problem : how do the scrollar invariants of the Galois closure C’ -> P^1 and of its various subcovers (the so-called resolvents of C -> P^1) relate to known invariants of the given cover ? This concerns ongoing work with Yongqiang Zhao, in which we put a previous observation for covers of degree 4 due to Casnati in a more general framework. As we will see the answer involves invariants related to syzygies that were introduced by Schreyer. As time permits, we will discuss a number-theoretic manifestation of the phenomena observed.


Guy Casale - Rennes 1

Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups

vendredi 14 décembre 2018 à 14:15 : ENS, Salle W

We prove the Ax-Lindemann-Weierstrass theorem for the uniformizing functions of genus zero Fuchsian groups of the first kind. Our proof relies on differential Galois theory of Schwarzian equations and machinery from the model theory of differentially closed fields. This result generalizes previous work of Pila-Tsimerman on the j function.
Joint work with James Freitag and Joel Nagloo.


Omar León Sánchez - Manchester

On differentially large fields.

vendredi 14 décembre 2018 à 00:00 : ENS, Salle W

Recall that a field K is large if it is existentially closed in K((t)). Examples of such fields are the complex, the real, and the p-adic numbers. This class of fields has been exploited significantly by F. Pop and others in inverse Galois-theoretic problems. In recent work with M. Tressl we introduced and explored a differential analogue of largeness, that we conveniently call “differentially large”. I will present some properties of such fields, and use a twisted version of the Taylor morphism to characterise them using formal Laurent series and to even construct “natural” examples (which ultimately yield examples of DCFs and CODFs... acronyms that will be explained in the talk).


Arthur Forey - ETH Zürich

Uniform bound for points of bounded degree in function fields of positive characteristic

vendredi 14 décembre 2018 à 00:00 : ENS, salle W

I will present a bound for the number of 𝔽_q[t]-points of bounded degree in a variety defined over ℤ[t], uniform in q. This generalizes work by Sedunova for fixed q. The proof involves model theory of valued fields with algebraic Skolem functions and uniform non-Archimedean Yomdin-Gromov parametrizations. This is joint work with Raf Cluckers and François Loeser.


Jean-Philippe Rolin - Dijon

Oscillatory integrals of subanalytic functions

vendredi 16 novembre 2018 à 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).


Philipp Dittman - Leuven

First-order logic in finitely generated fields

vendredi 16 novembre 2018 à 14:15 : ENS, Salle W

The expressive power of first-order logic in the class of finitely generated fields, as structures in the language of rings, is relatively poorly understood. For instance, Pop asked in 2002 whether elementarily equivalent finitely generated fields are necessarily isomorphic, and this is still not known in the general case. On the other hand, the related situation of finitely generated rings is much better understood by recent work of Aschenbrenner-Khélif-Naziazeno-Scanlon.
Building on work of Pop and Poonen, and using geometric results due to Kerz-Saito and Gabber, I shall show that every infinite finitely generated field of characteristic not two admits a definable subring which is a finitely generated algebra over a global field. This implies that any such finitely generated field is biinterpretable with arithmetic, and gives a positive answer to the question above in characteristic not two.


Antoine Ducros - IMJ-PRG

Non-standard analysis and non-archimedean geometry

vendredi 16 novembre 2018 à 11:00 : ENS, Salle W

In this talk I will describe a joint work (still in progress) with E. Hrushovski and F. Loeser, in which we explain how the integrals I have defined with Chambert-Loir on Berkovich spaces can be seen (in the t-adic case) as limits of usual integrals on complex algebraic varieties ; a crucial step is the development of a non-standard integration theory on a huge real closed field. I plan to devote a lot of time to the precise description of the objects involved, before stating our main theorem and saying a some words about is proof.