Géométrie et Théorie des Modèles

Année 2022 - 2023

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 25 novembre 2022, à l'IHP, salle 01, et/ou Zoom. Programme :

11h : Franziska Jahnke (Münster), Taming perfectoid fields
Tilting perfectoid fields, developed by Scholze, allows to transfer results between certain henselian fields of mixed characteristic and their positive characteristic counterparts and vice versa. We present a model-theoretic approach to tilting via ultraproducts, which allows to transfer many first-order properties between a perfectoid field and its tilt (and conversely). In particular, our method yields a simple proof of the Fontaine-Wintenberger Theorem which states that the absolute Galois group of a perfectoid field and its tilt are canonically isomorphic. A key ingredient in our approach is an Ax-Kochen/Ershov principle for perfectoid fields (and generalizations thereof).
This is joint work with Konstantinos Kartas.

14h15 : Rémi Jaoui (Lyon), Abundance of strongly minimal autonomous differential equations
In several classical families of differential equations such as the Painlevé families (Nagloo, Pillay) or finite dimensional families of Schwarzian differential equations (Blazquez-Sanz, Casale, Freitag, Nagloo), the following picture has been obtained regarding the transcendence properties of their solutions:

- (Strong minimality): outside of an exceptional set of parameters, the corresponding differential equations are strongly minimal,
- (Geometric triviality): algebraic independence of several solutions is controlled by pairwise algebraic independence outside of this exceptional set of parameters,
- (Multidimensionality): the differential equations defined by generic independent parameters are orthogonal.

Are the families of differential equations satisfying such transcendence properties scarce or abundant in the universe of algebraic differential equations?

I will describe an abundance result for families of autonomous differential equations satisfying the first two properties. The model-theoretic side of the proof uses a fine understanding of the structure of autonomous differential equations internal to the constants that we have recently obtained in a joint work with Rahim Moosa. The geometric side of the proof uses a series of papers of S.C. Coutinho and J.V. Pereira on the dynamical properties of a generic foliation.


16h15 : Hector Pasten (PUC Santiago), On non-Diophantine sets in rings of functions (Zoom)
For a ring R, a subset of a cartesian power of R is said to be Diophantine if it is positive existentially definable over R with parameters from R. In general, Diophantine sets over rings are not well-understood even in very natural situations; for instance, we do not know if the ring of integers Z is Diophantine in the field of rational numbers. To show that a set is Diophantine requires to produce a particular existential formula that defines it. However, to show that a set is not Diophantine is a more subtle task; in lack of a good description of Diophantine sets it requires to find at least a property shared by all of them. I will give an outline of some recent joint work with Garcia-Fritz and Pheidas on showing that several sets and relations over rings of polynomials and rational functions that are not Diophantine.

Notes de l'exposé et vidéo.

Vendredi 9 décembre 2022, à l'ENS, salle W. Orateurs :

14h15 : François Loeser (IMJ-PRG), Un théorème de finitude pour les fonctions tropicales sur les squelettes
Les squelettes sont des sous-ensembles linéaires par morceaux d'espaces analytiques non-archimédiens apparaissant naturellement dans nombre de situations. Nous présenterons un résultat général de finitude, obtenu en collaboration avec A. Ducros, E. Hrushovski et J. Ye, concernant le groupe abélien ordonné des fonctions tropicales sur les squelettes des analytifiés de Berkovich de variétés algébriques. Notre approche utilise la version modèle théorique de l'analytification (la complétion stable) développée dans un travail antérieur avec E. Hrushovski.


16h : Alex Wilkie (U. of Oxford), Integer points on analytic sets
In 2004 I proved that that if C is a transcendental curve definable in the structure R_{an}, then the number of points on C with integer coordinates of modulus less than H, is bounded by k loglog H for some constsnt k depending only on C. (The situation is vastly different for rational points.) The proof used the fact that such sets C are, in fact, semi-analytic everywhere-including infinity-and so the crux of the matter was to bound the number of solutions to equations of the form

(*)    F(1/n) = 1/m
for n, m integers bounded in modulus by (large) H, and where F is a non-algebraic, analytic function defined on an open interval containing 0.
It turns out that there is probably no generalization of the 2004 result for arbitrary R_{an}-definable sets (which need not be globally, or even locally, semi-analytic) but inspired by observations of Gareth Jones and Gal Binyamini, the three of us began looking at equations of the form (*) in many variables and I shall be reporting on our results.


Vendredi 24 mars 2023, à l'IHP, salle 314. Programme :

11h : Blaise Boissonneau (KU Leuven), Defining valuations using this one weird trick
In this talk, we present classical methods to define valuations and use them to derive conditions on the residue fields and value groups guaranteeing definability, and discuss how close these conditions are to being optimal.


14h15 : Arno Fehm (TU Dresden), Axiomatizing the existential theory of F_p((t))
From a model theoretic point of view, local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are much less well understood than their characteristic zero counterparts - the fields of real, complex and p-adic numbers. I will discuss different approaches to axiomatize and decide at least their existential theory in various languages and under various forms of resolution of singularities. From a geometric point of view, deciding the existential theory essentially means to determine algorithmically which algebraic varieties have rational points over these fields. Joint work with Sylvy Anscombe and Philip Dittmann.

16h : Sebastian Eterovic (Leeds), Generic solutions to systems of equations involving functions from arithmetic geometry
In arithmetic geometry one encounters many important transcendental functions exhibiting interesting algebraic properties. Perhaps the most famous example of this is the complex exponential function, which is well-known to satisfy the definition of a group homomorphism. When studying these algebraic properties, a very natural question that arises is something known as the “existential closedness problem”: when does an algebraic variety intersect the graph of the function in a Zariski dense set?
In this talk I will introduce the existential closedness problem, we will review what is known about it, and I will present results about a strengthening of the problem where we seek to find a point in the intersection of the algebraic variety and the graph of the function which is generic in the algebraic variety.


Vendredi 21 avril 2023, à l'IHP, amphithéâtre Hermite. Programme :

11h : Margaret Bilu (IMB, Bordeaux), A motivic circle method
The Hardy-Littlewood circle method is a well-known technique of analytic number theory that has successfully solved several major number theory problems. In particular, it has been instrumental in the study of rational points on hypersurfaces of low degree. More recently, a version of the method over function fields, combined with spreading out techniques, has led to new results about the geometry of moduli spaces of rational curves on hypersurfaces of low degree. In this talk I will show how to implement a circle method with an even more geometric flavour, where the computations take place in a suitable Grothendieck ring of varieties, and explain how this leads to a more precise description of the geometry of the above moduli spaces. This is joint work with Tim Browning.

14h15 : Philipp Hieronymi (Bonn), Fractals and Model Theory
This talk is motivated by the following fundamental question: What is the logical/model-theoretic complexity generated by fractal objects?
Here I will focus on fractal objects defined in first-order expansions of the ordered real additive group. The main problem I want to address here is: If such an expansion defines a fractal object, what can be said about its logical complexity in the sense of Shelah-style combinatorial tameness notions such as NIP and NTP2?
The main results I will mention are joint work with Erik Walsberg.

16h : Martin Hils (Münster), Spaces of definable types and beautiful pairs in unstable theories
By classical results of Poizat, the theory of beautiful pairs of models of a stable theory T is “meaningful” precisely when the set of all definable types in T is strict pro-definable, which is the case if and only if T is nfcp.
We transfer the notion of beautiful pairs to unstable theories and study them in particular in henselian valued fields, establishing Ax-Kochen-Ershov principles for various questions in this context. Using this, we show that the theory of beautiful pairs of models of ACVF is “meaningful” and infer the strict pro-definability of various spaces of definable types in ACVF, e.g., the model theoretic analogue of the Huber analytification of an algebraic variety.
This is joint work with Pablo Cubides Kovacsics and Jinhe Ye.

Vendredi 16 juin 2023. Jussieu, salle 101, couloir 15-16. Programme :

11h : Neer Bhardwaj (Weizman), Approximate Pila-Wilkie type counting for complex analytic sets
We develop a variation of the Pila-Wilkie counting theorem, where we count rational points that approximate bounded complex analytic sets. A unique aspect of our result is that it does not depend on the analytic set (or family) in question. We apply this approximate counting to obtain an effective Pila-Wilkie type statement for analytic sets cut out by computable functions. This is joint work with Gal Binyamini.

14h15 : Sylvy Anscombe (IMJ-PRG), Interpretations of fragments of theories of fields
In previous work with Fehm, and then Dittmann and Fehm, we found that the existential theory of an equicharacteristic henselian valued field is axiomatised using the existential theory of its residue field, conditionally, similar to an earlier theorem of Denef and Schoutens -- giving a transfer of decidability for existential theories. In this talk I'll describe parts of ongoing work with Fehm (in the main different to those discussed recently at CIRM) in which we use an “abstract” framework for interpreting families of incomplete theories in others in order to find transfers of decidability in various settings. I will discuss consequences for theories of PAC fields and parts of the universal-existential theory of equicharacteristic henselian valued fields.

16h : Tom Scanlon (Berkeley), (Un)likely intersections and definable complex quotient spaces
The Zilber-Pink conjectures predict that if S is a special variety, X ⊆ S is an irreducible subvariety of S which is not contained in a proper special subvariety, then the union of the unlikely intersections of X with special subvarieties of S is not Zariski dense in X, where here, an intersection between subvarieties X and Y of S is unlikely if dim X + dim Y < dim S. To make this precise, we need to specify what is meant by “special subvariety”. We will do so through the theory of definable complex quotient spaces, modeled on those introduced by Bakker, Klingler, and Tsimerman. Using this formalism we will prove a complement to the Zilber-Pink conjecture to the effect that under some natural geometric conditions likely intersections will be Zariski dense in X (joint work with Sebastian Eterović) and in the other direction that a function field version of the Zilber-Pink conjecture holds effectively (joint work with Jonathan Pila).

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.
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