Géométrie et Théorie des Modèles

Année 2018 - 2019

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 16 novembre 2018. ENS, salle W. Programme :

11h : Antoine Ducros (IMJ-PRG), Non-standard analysis and non-archimedean geometry
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.

14h15 : Philipp Dittman (Leuven), First-order logic in finitely generated fields
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.

16h : Jean-Philippe Rolin (Dijon), Oscillatory integrals of subanalytic functions
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).

Vendredi 14 décembre 2018. ENS, salle W. Programme :

11h : Arthur Forey (ETH Zürich), Uniform bound for points of bounded degree in function fields of positive characteristic
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.

14h15 : Guy Casale (Rennes 1), Ax-Lindemann-Weierstrass with derivatives and the genus 0 Fuchsian groups
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.

16h : Omar León Sánchez (Manchester), On differentially large fields.
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).

Vendredi 11 janvier 2019, ENS Salle W. Orateurs prévus :

11h : Wouter Castryck (Leuven), Scrollar invariants, resolvents, and syzygies
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.

14h15 : Martin Bays (Münster), Definability in the infinitesimal subgroup of a simple compact Lie group
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 ℵ_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.

16h : Amador Martin Pizarro (Freiburg), Tame open core and small groups in pairs of topological geometric structures
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.

Vendredi 15 février, ENS, Salle W. Orateurs :

11h : Chris Daw (Reading), Unlikely intersections with E×CM curves in 𝒜_2
The Zilber-Pink 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 Baily--Borel 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 so-called Hecke--facteur family, again under a large Galois orbits hypothesis.
This is a joint work with Martin Orr (University of Warwick).

14h15 : Bruno Klingler (HU Berlin), Tame topology and Hodge theory.
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 Cattani-Deligne-Kaplan (joint work with Bakker and Tsimerman).

16h : Pablo Cubides Kovacsics (Dresden), Definable subsets of a Berkovich curve
Let k be an algebraically closed complete rank 1 non-trivially 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 Hrushovski-Loeser about the iso-definability 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.

Vendredi 22 mars, ENS, Salle W. Programme :

11h : Jonathan Pila (Oxford), Independence of CM points in elliptic curves
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 (..., Buium-Poonen, 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 Zilber-Pink conjecture and prove a result which improves previous results in some aspects.

14h15 : Per Salberger (Gothenburg), Counting rational points with the determinant method.
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 real-analytic version of the method due to Bombieri and Pila and a p-adic version due to Heath-Brown. The aim of our talk is to describe a global refinement of the p-adic method and some applications like a uniform bound for non-singular cubic curves which improves upon earlier bounds of Ellenberg-Venkatesh and Heath-Brown.

16h : Vlerë Mehmeti (Caen) Patching over Berkovich Curves
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 local-global 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 local-global principle for function fields of curves that generalizes that of the above mentioned authors.

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.
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