Géométrie et Théorie
des Modèles
Année 2018  2019
Organisateurs :
Zoé
Chatzidakis, Raf Cluckers.
Pour recevoir le programme par email, 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 quelquesuns des exposés sont disponibles.
Vendredi 16 novembre 2018. ENS, salle W. Programme :
11h : Antoine Ducros
(IMJPRG), Nonstandard analysis and nonarchimedean
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 ChambertLoir on Berkovich spaces can be
seen (in the tadic case) as limits of usual integrals on complex algebraic varieties; a crucial step is the development of a nonstandard 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), Firstorder logic in finitely generated fields The expressive power of firstorder 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 AschenbrennerKhélifNaziazenoScanlon.
Building on work of Pop and Poonen, and using geometric results due to
KerzSaito 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 : JeanPhilippe 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 nonArchimedean YomdinGromov parametrizations. This is joint work with Raf Cluckers and François Loeser.
14h15 : Guy Casale (Rennes 1), AxLindemannWeierstrass with derivatives and the genus 0 Fuchsian groups
We prove the AxLindemannWeierstrass 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
PilaTsimerman 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
padic numbers. This class of fields has been exploited significantly by
F. Pop and others in inverse Galoistheoretic 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
socalled scrollar invariants (also called Maroni invariants) which
describe how the pushforward 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 socalled 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
numbertheoretic 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_0saturated 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
biinterpretation. 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 padics 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 padics. 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 ZilberPink 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 BailyBorel 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 socalled Heckefacteur 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 CattaniDeligneKaplan (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 nontrivially 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 HrushovskiLoeser about the
isodefinability 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 (..., BuiumPoonen, 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 ZilberPink
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 realanalytic version of the method due to Bombieri and Pila and a padic version due to HeathBrown. The aim of our talk is to describe a global refinement of the padic method and some applications like a uniform bound for nonsingular cubic curves which improves upon earlier bounds of EllenbergVenkatesh and HeathBrown.
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 localglobal 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 localglobal principle for function
fields of curves that generalizes that of the above mentioned
authors.
Vendredi 10 mai, ENS, Salle W. Orateurs
:
11h : Katrin Tent
(Münster), Almost strongly minimal
ample geometries
The notion of ampleness captures essential properties of projective
spaces over fields. It is natural to ask whether any sufficiently
ample strongly minimal set arises from an algebraically closed
field. In this talk I will explain the question and present recent
results on ample strongly minimal structures.
14h15 : Avraham Aizenbud
(Weizman), Pointwise surjective
presentations of stacks, or why I am not afraid of (infinity)
stacks anymore. Any algebraic stack X can be represented by a groupoid object in the category of schemes: that is, a pair of schemes Ob, Mor and morphisms
source, target: Mor → Ob, inversion: Mor → Mor, composition: Mor ×_{Ob} Mor → Mor and identity: Ob → Mor that satisfy certain axioms. Yet this description of the stack X might be misleading.
Namely, given a field F which is not algebraically closed, we have a natural functor between the groupoid (Ob(F),Mor(F)) and the groupoid X(F). While this functor is fully faithful, it is often not essentially surjective.
In joint work with Nir Avni (in progress) we show that any algebraic groupoid has a presentation such that this functor will be essentially surjective for many fields (and under some assumptions on the stack, for any field). The results are also extended to Henselian rings.
Despite the title, the talk will be about usual stacks and not infinitystacks, yet in some of the proofs it is more convenient to use the language of higher categories and I'll try to explain why.
No prior knowledge of infinity stacks will be assumed, but a superficial acquaintance with usual stacks will be helpful.
16h : Dimitri
Wyss (IMJPRG), Nonarchimedean and motivic integrals on the Hitchin fibration
Based on mirror symmetry considerations, Hausel and Thaddeus conjectured an equality between `stringy' Hodge numbers for moduli spaces of SL_n/PGL_n Higgs bundles. With Michael Groechenig and Paul Ziegler we prove this conjecture using nonarchimedean integrals on these moduli spaces, building on work of DenefLoeser and Batyrev. Similar ideas also lead to a new proof of the geometric stabilization theorem for anisotropic Hitchin fibers, a key ingredient in the proof of the fundamental lemma by Ngô.
In my talk I will outline the main arguments of the proofs and discuss the adjustments needed, in order to replace nonarchimedean integrals by motivic ones. The latter is joint work with François Loeser.
Programme des séances
passées : 200607,
200708,
200809,
200910,
201011,
201112,
201213,
201314,
201415,
201516,
201617,
201718.
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