Géométrie et Théorie des Modèles

Année 2020 - 2021

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 16 octobre, Zoom.

9h - 10h20 : Dmitry Novikov (Weizmann Institute), Complex Cellular Structures

Real semialgebraic sets admit so-called cellular decomposition, i.e. representation as a union of convenient semialgebraic images of standard cubes.
The Gromov-Yomdin Lemma (later generalized by Pila and Wilkie) proves that the maps could be chosen of C^r-smooth norm at most one, and the number of such maps is uniformly bounded for finite-dimensional families. This number was not effectively bounded by Yomdin or Gromov, but it necessarily grows as r → ∞.
It turns out there is a natural obstruction to a naive holomorphic complexification of this result related to the natural hyperbolic metric of complex holomorphic sets.
We prove a lemma about holomorphic functions in annulii, a quantitative version of the great Picard theorem. This lemma allowed us to construct an effective holomorphic version of the cellular decomposition results in all dimensions, with explicit polynomial bounds on complexity for families of complex (sub)analytic and semialgebraic sets.
As the first corollary we get an effective version of Yomdin-Gromov Lemma with polynomial bounds on the complexity, thus proving a long-standing Yomdin conjecture about tail entropy of analytic maps. Further connection to diophantine applications will be explained in Gal's talk.

Notes de l'exposé et vidéo.

10h30 - 11h50 : Gal Binayamini (Weizmann Institute), Tame geometry and diophantine approximation

Tame geometry is the study of structures where the definable sets admit finite complexity. Around 15 years ago Pila and Wilkie discovered a deep connection between tame geometry and diophantine approximation, in the form of asymptotic estimates on the number of rational points in a tame set (as a function of height). This later led to deep applications in diophantine geometry, functional transcendence and Hodge theory.
I will describe some conjectures and a long-term project around a more effective form of tame geometry, suited for improving the quality of the diophantine approximation results and their applications. I will try to outline some of the pieces that are already available, and how they should conjecturally fit together. Finally I will survey some applications of the existing results around the Manin-Mumford conjecture, the Andre-Oort conjecture, Galois-orbit lower bounds in Shimura varieties, unlikely intersections in group schemes, and some other directions (time permitting).

Notes de l'exposé et vidéo (les deux premières minutes manquent).

Vendredi 13 novembre :

9h - 10h20 : Will Johnson (Fudan U), The étale-open topology
Fix an abstract field K. For each K-variety V, we will define an “étale-open” topology on the set V(K) of rational points of V. This notion uniformly recovers (1) the Zariski topology on V(K) when K is algebraically closed, (2) the analytic topology on V(K) when K is the real numbers, (3) the valuation topology on V(K) when K is almost any henselian field. On pseudo-finite fields, the étale-open topology seems to be new, and has some interesting properties.
The étale-open topology is mostly of interest when K is large (also known as ample). On non-large fields, the étale-open topology is discrete. In fact, this property characterizes largeness. Using this, one can recover some well-known facts about large fields, and classify the model-theoretically stable large fields. It may be possible to push these arguments towards a classification of NIP large fields. Joint work with Chieu-Minh Tran, Erik Walsberg, and Jinhe Ye.

Notes de l'exposé et vidéo (les cinq premières minutes manquent).
Vendredi 27 novembre, 9h : suite et fin de l'exposé. Notes et vidéo de l'exposé.

10h30 - 11h50 : Jinhe (Vincent) Ye (Sorbonne Université), Belles paires of valued fields and analytification
In their work, Hrushovski and Loeser proposed the space V̂ of generically stable types concentrating on V to study the homotopy type of the Berkovich analytification of V. An important feature of V̂ is that it is canonically identified as a projective limit of definable sets in ACVF, which grants them tools from model theory. In this talk, we will give a brief introduction to this object and present an alternative approach to internalize various spaces of definable types, motivated by Poizat's work on belles paires of stable theories. Several results of interest to model theorists will also be discussed. Particularly, we recover the space V̂ is strict pro-definable and we propose a model-theoretic counterpart Ṽ of Huber's analytification. Time permitting, we will discuss some comparison and lifting results between V̂ and Ṽ. This is a joint project with Pablo Cubides Kovacsics and Martin Hils.

Vidéo et notes.

Vendredi 11 décembre :

9h : Annalisa Conversano (Massey University), Groups definable in o-minimal structures and algebraic groups
Groups definable in o-minimal structures have been studied by many authors in the last 30 years and include algebraic groups over algebraically closed fields of characteristic 0, semi-algebraic groups over real closed fields, important classes of real Lie groups such as abelian groups, compact groups and linear semisimple groups. In this talk I will present results on groups definable in o-minimal structures, demonstrating a strong analogy with topological decompositions of linear algebraic groups. Limitations of this analogy will be shown through several examples.

Vidéo (les deux premières minutes manquent) et Notes

10h30 : Pablo Cubides-Kovacsics (Düsseldorf), Cohomology of algebraic varieties over non-archimedean fields
I will report on a joint work with Mário Edmundo and Jinhe Ye in which we introduced a sheaf cohomology theory for algebraic varieties over non-archimedean fields based on Hrushovski-Loeser spaces. After informally framing our main results with respect to classical statements, I will discuss some details of our construction and the main difficulties arising in this new context. If time allows, I will further explain how our results allow us to recover results of V. Berkovich on the sheaf cohomology of the analytification of an algebraic variety over a rank 1 complete non-archimedean field.

Vidéo et Notes

Vendredi 15 janvier 2021 :

9h30 : Alessandro Berarducci (Pisa), An application of surreal numbers to the asymptotic analysis of certain exponential functions
Skolem (1956) studied the germs at infinity of the smallest class of real valued functions on the positive real line containing the constant 1, the identity function x, and such that whenever f and g are in the set, f+g, fg and f^g are also in the set. This set of germs is well ordered and Skolem conjectured that its order type is epsilon-zero. Van den Dries and Levitz (1984) computed the order type of the fragment below 2^(2^x). They did so by studying the possible limits at infinity of the quotient f(x)/g(x) of two functions in the fragment: if g is kept fixed and f varies, the possible limits form a discrete set of real numbers of order type omega.
Using the surreal numbers, we extend the latter result to the whole class of Skolem functions and we discuss some additional progress towards the conjecture of Skolem. This is joint work with Marcello Mamino (http://arxiv.org/abs/1911.07576, to appear in the JSL).
Vidéo et Notes.

11h10 : Daniel Palacin (Freiburg/Complutense), Solving equations in finite groups and complete amalgamation
Roth's theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length 3, that is, the equation x+z=2y has a solution in A.
Finitary versions of Roth's theorem study subsets A of {0, ... , N}, and ask whether the same holds for sufficiently large N, for a fixed lower bound on the density. In a similar way, concerning finite groups, one may study whether or not sufficiently large sets of a finite group contain solutions of an equation, or even a system of equations. For instance, for the equation xy=z, Gowers (2008) showed that any subset of a finite simple non-abelian group will contain many solutions to this equation, provided it has sufficiently large density.
We will report on recent work with Amador Martin-Pizarro on how to find solutions to the above equations in the context of pseudo-finite groups, using techniques from model theory which resonate with (a group version of) the independence theorem in simple theories due to Pillay, Scanlon and Wagner. In this talk, we will not discuss the technical aspects of the proof, but present the main ideas to a general audience.

Vendredi 12 février. Orateurs : 9h : Yatir Halevi (Ben Gurion), et 10h30 : Franziska Jahnke (Münster), On dp-finite fields

Shelah's conjecture predicts that any infinite NIP field is either separably closed, real closed or admits a non-trivial henselian valuation. Recently, Johnson proved that Shelah's conjecture holds for fields of finite dp-rank, also known as dp-finite fields. The aim of these two talks is to give an introduction to dp-rank in some algebraic structures and an overview of Johnson's work.
In the first talk, we define dp-rank (which is a notion of rank in NIP theories) and give examples of dp-finite structures. In particular, we discuss the dp-rank of ordered abelian groups and use them to construct multitude of examples of dp-finite fields. We also prove that every dp-finite field is perfect and sketch a proof that any valued field of dp-rank 1 is henselian.
In the second talk, we give an overview of Johnson's proof that every infinite dp-finite field is either algebraically closed, real closed or admits a non-trivial henselian valuation. Crucially, this relies on the notion of a W-topology, a natural generalization of topologies arising from valuations, and the construction of a definable W-topology on a sufficiently saturated unstable dp-finite field.
Vidéos : Halevi, Jahnke. Notes du 2e exposé.

Vendredi 26 mars, 15 h - 18h

15h : Jason Bell (U. of Waterloo), Effective isotrivial Mordell-Lang in positive characteristic.
The Mordell-Lang conjecture (now a theorem, proved by Faltings, Vojta, McQuillan,...) asserts that if G is a semiabelian variety G defined over an algebraically closed field of characteristic zero, X is a subvariety of G, and Γ is a finite rank subgroup of G, then X ∩ Γ is a finite union of cosets of Γ. In positive characteristic, the naive translation of this theorem does not hold, however Hrushovski, using model theoretic techniques, showed that in some sense all counterexamples arise from semiabelian varieties defined over finite fields (the isotrivial case). This was later refined by Moosa and Scanlon, who showed in the isotrivial case that the intersection of a subvariety of a semiabelian variety G with a finitely generated subgroup Γ of G that is invariant under the Frobenius endomorphism F: G → G is a finite union of sets of the form S+A, where A is a subgroup of Γ and S is a sum of orbits under the map F. We show how how one can use finite-state automata to give a concrete description of these intersections Γ ∩ X in the isotrivial setting, by constructing a finite machine that identifies all points in the intersection. In particular, this allows us to give decision procedures for answering questions such as: is X ∩ Γ empty? finite? does it contain a coset of an infinite subgroup? In addition, we are able to read off coarse asymptotic estimates for the number of points of height ≤ H in the intersection from the machine. This is joint work with Dragos Ghioca and Rahim Moosa.
Vidé et Notes.

16h30 : Rémi Jaoui (U. of Notre Dame), Linearization procedures in the semi-minimal analysis of algebraic differential equations
It is well-known that certain algebraic differential equations restrain in an essential way the algebraic relations that their solutions share. For example, the solutions of the first equation of Painlevé y'' = 6y^2 + t are “new” transcendental functions of order two which whenever distinct are algebraically independent (together with their derivatives).
I will first describe an account of such phenomena using the language of geometric stability theory in a differentially closed field. I will then explain how linearization procedures and geometric stability theory fit together to study such transcendence results in practice.
Vidéo et Notes.

Vendredi 23 avril 2021. 15h à 18h. Orateurs :

15h : Gabriel Conant (Cambridge), VC-dimension in model theory, discrete geometry, and combinatorics
In statistical learning theory, the notion of VC-dimension was developed by Vapnik and Chervonenkis in the context of approximating probabilities of events by the relative frequency of random test points. This notion has been widely used in combinatorics and computer science, and is also directly connected to model theory through the study of NIP theories. This talk will start with an overview of VC-dimension, with examples motivated by discrete geometry and additive combinatorics. I will then present several model theoretic applications of VC-dimension. The selection of topics will focus on the use of finitely approximable Keisler measures to analyze the structure of algebraic and combinatorial objects with bounded VC-dimension.
Vidéo et Notes.

16h30 : Artem Chernikov (UCLA), Recognizing groups and fields in Erdős geometry and model theory
Assume that Q is a relation on R^s of arity s definable in an o-minimal expansion of R. I will discuss how certain extremal asymptotic behaviors of the sizes of the intersections of Q with finite n × ... × n grids, for growing n, can only occur if Q is closely connected to a certain algebraic structure.
On the one hand, if the projection of Q onto any s-1 coordinates is finite-to-one but Q has maximal size intersections with some grids (of size >n^(s-1 - ε)), then Q restricted to some open set is, up to coordinatewise homeomorphisms, of the form x_1+...+x_s=0. This is a special case of the recent generalization of the Elekes-Szabó theorem to any arity and dimension in which general abelian Lie groups arise (joint work with Kobi Peterzil and Sergei Starchenko). On the other hand, if Q omits a finite complete s-partite hypergraph but can intersect finite grids in more that than n^(s-1 + ε) points, then the real field can be definably recovered from Q (joint work with Abdul Basit, Sergei Starchenko, Terence Tao and Chieu-Minh Tran).
I will explain how these results are connected to the model-theoretic trichotomy principle and discuss variants for higher dimensions, and for stable structures with distal expansions.
Vidéo et Notes.

Vendredi 21 mai 2021, 9h - 12h. Orateurs :

9h - 10h20 : Will Johnson (Fudan U.), Curve-excluding fields
Let T be the theory of fields K of characteristic 0 such that the equation x^4 + y^4 = 1 has only four solutions in K. We show that T has a model companion. More generally, if K_0 is a field of characteristic 0 and C is a curve (affine or projective) of genus ≥ 2 with C(K_0) = ∅, then there is a model companion CXF of the theory of fields K extending K_0 with C(K) = ∅.
We can use this theory to construct a field K with an interesting combination of properties. On the model-theoretic side, the theory of K is complete, decidable, model-complete, and algebraically bounded, and K is a “geometric structure” in the sense of Hrushovski and Pillay. Additionally, some classification-theoretic properties might hold in K. On the field-theoretic side, K is non-large---there is a smooth curve C such that C(K) is finite and non-empty. This is unusual; the vast majority of model-theoretically tractable fields are large or finite. On the other hand, K is “virtually large”---it has a finite extension which is large. In fact, every proper algebraic extension of K is pseudo algebraically closed (PAC). The absolute Galois group of K is an ω-free profinite group. This negatively answers a question of Junker and Koenigsmann (is every model-complete infinite field large?) and a question of Macintyre (does every model-complete field have a small Galois group?).
This is based on joint work with Erik Walsberg and Vincent Ye.
Vidéo et Notes.

10h30 - 11h50 : Silvain Rideau (IMJ-PRG), Pseudo-T-closed fields, approximations and NTP2
Joint work with Samaria Montenegro
The striking resemblance between the behaviour of pseudo-algebraically closed, pseudo real closed and pseudo p-adically fields has lead to numerous attempts at describing their properties in a unified manner. In this talk I will present another of these attempts: the class of pseudo-T-closed fields, where T is an enriched theory of fields. These fields verify a “local-global” principle with respect to models of T for the existence of points on varieties. Although it very much resembles previous such attempts, our approach is more model theoretic in flavour, both in its presentation and in the results we aim for.
The first result I would like to present is an approximation result, generalising a result of Kollar on PAC fields, respectively Johnson on henselian fields. This result can be rephrased as the fact that existential closeness in certain topological enrichments come for free from existential closeness as a field. The second result is a (model theoretic) classification result for bounded pseudo-T-closed fields, in the guise of the computation of their burden. One of the striking consequence of these two results is that a bounded perfect PAC field with n independent valuations has burden n and, in particular, is NTP2.
Vidéo et Notes.

Vendredi 18 juin. Orateurs :

15h : Pierre Simon (Berkeley), Monadically NIP ordered graphs and bounded twin-width
An open problem in theoretical computer science asks to characterize tameness for hereditary classes of finite structures. The notion of bounded twin-width was proposed and studied recently by Bonnet, Geniet, Kim, Thommasé and Watrignant. Classes of graphs of bounded twin-width have many desirable properties. In particular, they are monadically NIP (remain NIP after naming arbitrary unary predicates). In joint work with Szymon Torunczyk we show the converse for classes of ordered graphs. We then obtain a very clear dichotomy between tame (slow growth, monadically NIP, algorithmically simple ...) and wild hereditary classes of ordered graphs. Those results were also obtained by Bonnet, Giocanti, Ossona de Mendez and Thomassé. In this talk, I will focus on the model theoretic input.
Vidéo et Notes.

16h40 : André Belotto da Silva (Aix-Marseille), Real perspectives on monomialization.
I will discuss recent work in collaboration with Edward Bierstone on transformation of a mapping to monomial form (with respect to local coordinates) by simple modifications of the source and target. Our techniques apply in a uniform way to the algebraic and analytic categories, as well as to classes of infinitely differentiable real functions that are quasianalytic or definable in an o-minimal structure. Our results in the real cases are best possible. The talk will focus on real phenomena and on an application to quantifier elimination of certain o-minimal polynomially bounded structure.
Vidéo et Notes.

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.
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