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

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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