Géométrie et Théorie des Modèles

Année 2023 - 2024

Organisateurs : Zoé Chatzidakis, Raf Cluckers et George Comte.
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 24 novembre 2023, de 11h à 17h45. IHP (salle Yvette Cauchois, Bâtiment Perrin). Orateurs :

11h : Jonathan Pila (Oxford/IHES), Ax-Schanuel 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 integral-geometric 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 Heath-Brown. By work of Browning, Heath-Brown 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 so-called 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 Choquet-Bruhat (le matin) et Salle Yvette Cauchois (l'après-midi). Orateurs prévus :

11h, Amphithéâtre Choquet-Bruhat: 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 o-minimally 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 (IMJ-PRG), Stratification of the image of a map between analytic spaces
Let f : Y → X be a morphism between compact Berkovich spaces over an arbitrary non-Archimedean 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 non-Archimedean 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 Zariski-closed 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 well-suited 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 first-order 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 15-16, 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 16-26, 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 Deng--Yamanoi and myself using techniques from non-abelian 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 o-minimal geometry.

16h15 - 17h45, salle 15-16, 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 non-analytic 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 o-minimality, will be partly speculative.

Vendredi 24 mai, Jussieu, salle 15-16 101. 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, o-minimality, and intermediate Kodaira-Spencer 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 o-minimality 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 non-isotrivial 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.


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, 2018-19, 2019-20, 2020-21, 2021-22, 2022-23.
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