Géométrie et Théorie
des Modèles
Année 2023  2024
Organisateurs :
Zoé
Chatzidakis, Raf
Cluckers et George Comte.
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 24 novembre 2023, de 11h à 17h45. IHP (salle Yvette Cauchois, Bâtiment
Perrin). Orateurs :
11h : Jonathan Pila (Oxford/IHES), AxSchanuel and
exceptional integrability
In joint work with Jacob Tsimerman we study when the primitive of a given algebraic function
can be constructed using primitives from some given finite set of algebraic functions, their inverses,
algebraic functions, and composition. When the given finite set is just {1/x} this is the classical problem
of “elementary integrability” (of algebraic functions). I will discuss some results, including a decision procedure
for this question, and further problems and conjectures.
14h15 : George Comte (Chambéry), Inequalities for some
metric motivic invariants
In a joint work with Immanuel Halupczok we introduce, on one hand, a partial preorder on the set of motivic constructible functions,
extending the one originally defined by Cluckers and Loeser, and, on the other hand, a notion of number of connected components, still in the definable
nonarchimedean context. For the last one we use the existence special canonical stratifications. Those two notions meet, for instance, in a nonarchimedean
version of a real inequality involving the metric entropy and integralgeometric invariants, called Vitushkin invariants. I will try to explain how.
16h30 : Floris
Vermeulen (KU Leuven), Dimension growth for affine varieties.
Given a projective algebraic variety X over Q, the dimension growth conjecture predicts general upper bounds for the number of points of bounded height on X. It was originally conjectured by Serre, and independently in a uniform way by HeathBrown. By work of Browning, HeathBrown and Salberger, uniform dimension growth is now a theorem.
I will give a general overview of dimension growth and explain some ideas of the proof. The main ingredient is the socalled determinant method, which goes back to Bombieri and Pila, and has been successfully applied to many counting problems. I will then turn to dimension growth for affine varieties, and report on recent work with Raf Cluckers, Pierre Dèbes, Yotam Hendel, and Kien Nguyen.
Vendredi 26 janvier,
Bâtiment Perrin, IHP, Amphithéâtre Yvonne
ChoquetBruhat (le matin) et Salle Yvette Cauchois (l'aprèsmidi). Orateurs prévus :
11h, Amphithéâtre ChoquetBruhat: Alex
Wilkie (Oxford), Analytic Continuation and
Zilber's Quasiminimality Conjecture
This is the title of a paper that has recently been accepted for the
volume of the journal “Model Theory” dedicated to Boris Zilber
on the occasion of his 75th birthday. (The paper can be found on the GTM
preprint server or on arXiv.) The conjecture asserts that every
definable subset of the complex field expanded by the complex
exponential function is either countable or cocountable. In the paper I
propose a conjecture concerning the analytic continuation of ominimally
defined complex analytic functions which implies Zilber's conjecture
(and much more) and in this talk I will give an outline of the main
argument in the paper as well as some further remarks. (I was going to
write “as well as some recent progress”, but that would be too
strong!)
14h15 Salle Yvette Cauchois: Antoine Ducros
(IMJPRG), Stratification of the image of a map between analytic spaces
Let f : Y → X be a morphism between compact Berkovich spaces over an arbitrary nonArchimedean field. In general, the structure of the image f(Y) appears to be rather mysterious, unless one makes strong assumption on f
(like flatness, or properness). Nevertheless, I will explain how recent flattening results in nonArchimedean geometry allow to exhibit, under very weak assumptions on f
(automatically fulfilled if Y is irreducible, for example) a finite stratification of f(Y) with reasonable pieces (each of them is a Zariskiclosed subset of an analytic domain of X).
16h15 Salle Yvette Cauchois et Zoom: Gabriel Conant (Ohio State), Group compactifications in continuous logic, with applications to multiplicative combinatorics
I will discuss recent work on the general theme of continuous logic as
an environment wellsuited for certain methods in multiplicative
combinatorics (i.e., the extension of additive combinatorics to
noncommutative groups). The starting point is Pillay's result that the
connected component of a definable compactification of a pseudofinite
group is abelian. In joint work with Hrushovski and Pillay, we give a
short proof of this using only classical tools, including a result of
A. Turing on finitely approximated Lie groups. Using a connection
between Turing's theorem and a (relatively) more recent result of
Kazhdan on approximate homomorphisms, one obtains a generalization of
Pillay's theorem to ultraproducts of amenable torsion groups. In
previous work on “tame arithmetic regularity”, the results of
Pillay and of Kazhdan were instrumental for introducing classical Bohr
neighborhoods into the setting of noncommutative groups. However, the
execution of this approach was quite complicated due to certain
drawbacks of classical firstorder logic. In the paper with Hrushovski
and Pillay, we build Kazhdan's result into continuous logic in order to
remove these complications. As an illustration of the method, we use
the stabilizer theorem to extend a fundamental result from additive
combinatorics (called Bogolyubov's Lemma) to arbitrary amenable groups.
More recently, in work with Pillay, we combine this continuous setting
with local stability to prove a regularity lemma for “stable
functions” on amenable groups. This result is an analytic analogue
of the arithmetic regularity lemma for stable subsets of finite groups,
proved first in the abelian case by Terry and Wolf, and then
generalized by myself, Pillay, and Terry. As a consequence of stability
of Hilbert spaces, the analytic stable arithmetic regularity lemma
applies to convolutions of arbitrary functions on amenable groups. This
allows one to deduce the previous generalization of Bogolyubov's Lemma
as a quick corollary of analytic stable arithmetic regularity.
Notes and Video
Vendredi 15 mars 2024 (Lieu à préciser). Les
exposés auront lieu à Jussieu, dans des salles
différentes. Les horaires sont inhabituels à cause des problèmes de salle. Donc ouvrez l'oeil. Orateurs
prévus :
10h30  12h, salle 1516, 413 : Ehud Hrushovski
(Oxford), Definable model equivalence relations and
their invariants. An interpretation between theories can be presented as a composition of the construction of imaginary sorts, and the
taking of reducts. In this work with Michael Benedikt, we consider more general ways
of reducing structure, using definable equivalence relations on models with a given universe or, equivalently as it turns out, definable groupoids extending the groupoid
of models and isomorphisms. We characterize the simplest ones from several points of view; continuous logic turns out surprisingly to play an intrinsic role.
Examples seem to hint at a possibility of contact with categories that are usually inaccessible to definability considerations, notably from differential geometry.
This is a preliminary investigation, and I hope to be able to give complete proofs of the main results.
14h30  16h, salle 1626, 113 : Yohan Brunebarbe
(Bordeaux), Algebraicity of Shafarevich
morphisms. For a normal complex algebraic variety
X equipped with a semisimple complex local system V, a
Shafarevich morphism X → Y is a map which contracts precisely those algebraic subvarieties on which V has finite monodromy. The existence of such maps has interesting consequences on the geometry of universal covers of complex algebraic varieties. Shafarevich morphisms were constructed for projective X by Eyssidieux, and recently have been constructed analytically in the quasiprojective case independently by DengYamanoi and myself using techniques from nonabelian Hodge theory. In joint work with B. Bakker and J. Tsimerman, we show that these maps are algebraic, and that in fact Y is quasiprojective. This is a generalization of the Griffiths conjecture on the quasiprojectivity of images of period maps, and the proof critically uses ominimal geometry.
16h15  17h45, salle 1516, 101: Lou van den Dries
(UIUC), Analytic Hardy fields.
Joint work of Matthias Aschenbrenner, Joris van der Hoeven, and me led to the following two theorems about maximal Hardy fields:
(1) they are all elementarily equivalent to the ordered differential field of transseries;
(2) they are η_1 in the sense of Hausdorff.
This happened several years ago. As to (1), the proof goes through with “Hardy field” replaced by “analytic Hardy field” (with corresponding notion of “maximal”). This was not the case for (2), where we used gluing constructions and partitions of unity unavailable in the analytic context. Last year, Aschenbrenner and I did establish (2) also in the analytic case by reduction to the nonanalytic setting, using Whitney's powerful approximation theorem. I will give an overview of this, recalling also the background about transseries and asymptotic differential algebra.
There are further things to say about analytic Hardy fields that have no obvious analogue for arbitrary
Hardy fields, such as analytic continuation to the complex plane. The second part of my talk will be about that. Some of this, in particular possible connections to ominimality, will be partly speculative.
Vendredi 24 mai, Jussieu, salle 1516 101. Orateurs
:
11h : Emmanuel Peyre (Grenoble), Espaces de modules
asymptotiques / Asymptotic Moduli spaces
La version géométrique du programme de Manin
conduit à des prédictions très précises sur le comportement
asymptotique des espaces de modules de morphismes d'une courbe
dans une variété qui fait intervenir un produit eulérien motivique
dû à M. Bilu.
Après la description d'un cas concret élémentaire, le but de cet
exposé est d'expliquer la formule heuristique conjecturée ainsi que
le principe d'équidistribution qui lui est associé.
14h15 : Tom Scanlon (UC Berkeley), Manin
maps. differential algebra, ominimality, and intermediate
KodairaSpencer rank
In 1963, Manin used a construction related to the work of Fuchs, Gauss, and Picard on linear differential operators to prove a function field version of the Mordell conjecture about rational points on algebraic curves. Over the years, Manin's construction has been reinterpreted in various ways , most notably using differential algebra. Indeed, the analytic presentation is generally regarded as a heuristic, as it was even in Manin's proofs. I will describe some work (joint with T. Dupuy and J. Freitag) using ominimality and differential algebra exploiting the analytic presentation to explain some known properties of the Manin maps (e.g. Manin's theorem of the kernel and the one basedness of Manin kernels of nonisotrivial simple abelian varieties), to produce examples of simple abelian varieties with Manin kernels of intermediate rank, and to show why such examples cannot come from low dimensional families of abelian varieties.
16h : (Zoom) James Freitag (UI Chicago), Variations on the degree of nonminimality.
A few years ago, Rahim Moosa and I introduced the degree of
nonminimality, which was designed for proving transcendence
statements for solutions of differential equations. We'll
spend the first half of the talk developing the notion and
describing the applications. The degree of nonminimality is
just one way to make quantitative the general model theoretic
fact that dividing of any partial type can always be detected
by some indiscernible sequence in the parameters of the
formulas. In the second half of the talk, we will describe
several variations on the notion which have recent
applications beyond transcendence results.
Video
Programme des séances
passées : 200607,
200708,
200809,
200910,
201011,
201112,
201213,
201314,
201415,
201516,
201617,
201718,
201819,
201920,
202021,
202122,
202223.
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