Séminaires : Géométrie et Théorie des Modèles

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.

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.

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.

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!) 

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.

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).

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.

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.

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. 
 

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

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).

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.

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. 

 

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.

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.

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.

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.

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. 
 

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.

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.

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.

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.

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.

A σ-variety over a difference field (K,σ) is a pair (X,φ) consisting of an algebraic variety X over K and φ:X → X^σ is a regular map from X to its transform Xσ under σ. A subvariety Y ⊆ X is skew-invariant if φ(Y) ⊆ Y^σ. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of σ-varieties of the form (𝔸^n,φ) where φ(x_1,...,x_n) = (P_1(x_1),...,P_n(x_n)). The most important case, from which the others may be deduced, is that of n = 2. In the present work we give a sharper description of the skew-invariant curves in the case where P_2 = P_1^τ for some other automorphism of K which commutes with σ. Specifically, if P in K[x] is a polynomial of degree greater than one which is not eventually skew-conjugate to a monomial or ± Chebyshev (i.e. P is “nonexceptional”) then skew-invariant curves in (𝔸^2,(P,P^τ)) are horizontal, vertical, or skew-twists: described by equations of the form y = α^{σ^n} ∘ P^{σ^{n-1}} ∘ ⋅⋅⋅ ∘ P^σ ∘ P(x) or x = β^{σ{-1}}∘ P^{τ σ^{-n-2}}∘ P^{τ σ^{-n-3}}∘ ⋅⋅⋅ ∘ P^τ(y) where P = α ∘ β and P^τ = α^{σ^{n+1}}∘ β^{σ^n}} for some integer n.

To a complex of l-adic sheaves on a quasi-projective variety one associate an integer, its complexity. The main result on the complexity is that it is continuous with tensor product, pullback and pushforward, providing effective version of the constructibility theorems in l-adic cohomology. Another key feature is that the complexity bounds the dimensions of the cohomology groups of the complex. This can be used to prove equidistribution results for exponential sums over finite fields. This is due to Will Sawin, written up in collaboration with Javier Fresán and Emmanuel Kowalski.

Over the last 15 years a remarkable link between o-minimality and algebraic/arithmetic geometry has been unfolding following the discovery of Pila-Wilkie's counting theorem and its applications around unlikely intersections, functional transcendence etc. While the counting theorem is nearly optimal in general, Wilkie has conjectured a much sharper form in the structure R_exp. There is a folklore expectation that such sharper bounds should hold in structures "coming from geometry", but for lack of a general formalism explicit conjectures have been made only for specific structures.
I will describe a refinement of the standard o-minimality theory aimed at capturing the finer "arithmetic tameness" that we expect to see in structures coming from geometry. After presenting the general framework I will discuss my result with Vorobjov showing that the restricted Pfaffian structure is sharply o-minimal, and how this was used in our recent work with Novikov and Zack to prove Wilkie's conjecture for the restricted Pfaffian structure and for Wilkie's original case of R_exp. I will also discuss some conjectures on the construction of larger sharply o-minimal structures, and some partial results in this direction. Finally I will explain the crucial role played by these results in my recent work with Schmidt and Yafaev on Galois orbit lower bounds for CM points in general Shimura varieties, and subsequently in the recent resolution of general André-Oort conjecture by Pila-Shankar-Tsimerman-(Esnault-Groechenig).

The theme of the talk is around the theory of Igusa's local zeta functions, his broader program on local global principles, and recent progress on these via singularity theory and the minimal model program with M. Mustata and K. H. Nguyen. I will also present some new open questions that push Igusa's program further, and partial evidence obtained with K. H. Nguyen.

A field K is called pseudo-algebraically closed (PAC) if every absolutely irreducible variety defined over K has a K-rational point. These fields were introduced by Ax in his characterization of pseudo-finite fields and have since become an important object of study in both model theory and field arithmetic. We will explain how the analysis of a PAC field often reduces to questions about the model theory of the absolute group and describe how these reductions combine with a graph-coding construction of Cherlin, van den Dries, and Macintyre together with to construct PAC fields with prescribed combinatorial properties.

We fix an o-minimal expansion of the real field, M say. Definability notions are with respect to M. Let F = {f_x : x in X} be a definable family of (single valued) complex analytic functions, each one having domain some disk, D_x say, in ℂ, where the parameter space X is a definable subset of ℝ^m for some m. We present some finiteness theorems for such families F which are uniform in parameters and give some applications.
We also speculate on the notion of “definable” Riemann surface.

Determining all rational points on a curve of genus at least 2 can be difficult. Chabauty's method (1941) is to intersect, for a prime number p, in the p-adic Lie group of p-adic points of the jacobian, the closure of the Mordell-Weil group with the p-adic points of the curve. If the Mordell-Weil rank is less than the genus then this method has never failed. Minhyong Kim's non-abelian Chabauty programme aims to remove the condition on the rank. The simplest case, called quadratic Chabauty, was developed by Balakrishnan, Dogra, Mueller, Tuitman and Vonk, and applied in a tour de force to the so-called cursed curve (rank and genus both 3). This article aims to make the quadratic Chabauty method small and geometric again, by describing it in terms of only `simple algebraic geometry' (line bundles over the jacobian and models over the integers).

Using subresultants, we modify a recent real-algebraic proof due to Eisermann of the Fundamental Theorem of Algebra ([FTA]) to obtain the following quantitative information: in order to prove the [FTA] for polynomials of degree d, the Intermediate Value Theorem ([IVT]) is requested to hold for real polynomials of degree at most d^2. We also explain that the classical algebraic proof due to Laplace requires [IVT] for real polynomials of exponential degree. These quantitative results highlight the difference in nature of these two proofs.

(travail en commun avec Silvain Rideau-Kikuchi)
Les imaginaires (c'est-à-dire les quotients définissables) dans la théorie ACVF des corps algébriquement clos non-trivialement valués sont classifiés par les sortes “géométriques”. Ceci est un résultat fondamental dû à Haskell, Hrushovski et Macpherson. En utilisant l'approche via la densité des types définissables/invariants, nous donnons une réduction des imaginaires dans des corps valués henséliens, sous des hypothèses assez générales, aux sortes géométriques et à des imaginaires de RV avec des sortes pour certains espaces vectoriels de dimension finie sur le corps résiduel.
Dans l'exposé, je vais principalement parler d'une application qui a été à l'origine de notre travail: Les imaginaires de la théorie VFA des corps algébriquement clos valués non-trivialement de caractéristique 0, munis d'un Frobenius non-standard, sont classifiés par les sortes géométriques. Entre autre, notre preuve passe par une étude fine des imaginaires dans une suite exacte courte (pure) ainsi que par un résultat clé du papier de Hrushovski sur les groupoïdes.

The famous Mumford-Tate conjecture asserts that, for every prime number l, Hodge cycles are &Qopf;_l linear combinations of Tate cycles, through Artin's comparisons theorems between Betti and étale cohomology. The algebraic Sato-Tate conjecture, introduced by Serre and developed by Banaszak and Kedlaya, is a powerful tool in order to prove new instances of the generalized Sato-Tate conjecture. This previous conjecture is related with the equidistribution of Frobenius traces.
Our main goal is to prove that the Mumford-Tate conjecture for an abelian variety A implies the algebraic Sato-Tate conjecture for A. The relevance of this result lies mainly in the fact that the list of known cases of the Mumford-Tate conjecture was up to now a lot longer than the list of known cases of the algebraic Sato-Tate conjecture. This is a joint work with Johan Commelin.

In this talk, we characterize NIP henselian valued fields modulo the theory of their residue field. Assuming the conjecture that every infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.

Étant donné un anneau de valuation V de corps résiduel F et contenant un corps k, et une extension k' de k, on cherche à construire une extension V' de V contenant k', d'idéal maximal engendré par celui de V, et de corps résiduel composé de F et k'. On y parvient notamment si F ou k' est séparable sur k.

Lipschitz geometry is a branch of singularity theory that studies the metric data of a germ of a complex analytic space.
I will discuss a new approach to the study of such metric germs, and in particular of an invariant called Lipschitz inner rate, based on the combinatorics of a space of valuations, the so-called non-archimedean link of the singularity. I will describe completely the inner metric structure of a complex surface germ showing that its inner rates both determine and are determined by global geometric data: the topology of the germ, its hyperplane sections, and its generic polar curves.
This is a joint work with André Belotto and Anne Pichon.

My goal, in this talk, is to explain a new notion of minimality for (characteristic zero) Henselian fields, which generalizes C-minimality, P-minimality and V-minimality and puts no restriction on the residue field or valued group contrary to these previous notions. This new notion, h-minimality, can be defined, analogously to other minimality notions, by asking that 1-types, over algebraically closed sets, are entirely determined by their reduct to some sublanguage - in that case the pure language of valued fields. However, contrary to what happens with other minimality notions, particular care has to be taken with regards to the parameters. In fact, we define a family of notions: l-h-min for l a natural number or omega. My second goal in this talk will be to explain the various geometric properties that follow form h-minimality, among which the well-known Jacobian property, but also higher degree and higher dimensional versions of that property.

I will present a work on flattening by blow-ups in the context of Berkovich geometry (inspired by Raynaud and Gruson's paper on the same topic in the scheme-theoretic setting), and explain how it gives rise to the description of the image of an arbitrary analytic map between two compact Berkovich spaces, and why this description is (very likely) related to quantifier elimination in the Lipshitz-Cluckers variant of Lipshitz-Robinson's analytic language. (I plan to spend most of the talk discussing the results rather than their proofs.)