Géométrie et Théorie des Modèles

Année 2017 - 2018

Vendredi 13 octobre, ENS, salle W. Orateurs :

11h : Itaï Ben Yaacov (Lyon 1), Corps globalement valués
Dans un travail en commun avec E Hrushovski, nous étudions les corps globalement valués, qui sont une abstraction des corps de nombres, de fonctions, ou autres dans lesquels la formule du produit est vérifiée. Les questions habituelles de la théorie des modèles, telle que l'existence d'une modèle-compagne ou encore sa stabilité, nous mènent vers de nouvelles questions de nature plutôt géométrique.
Je vais expliquer quelques avancées récentes dans ce sens, où une analyse géométrique locale nous permet de déduire des propriété globales dans un corps globalement valués.

14h15 : Boris Zilber (Oxford), Approximation, domination and integration
The talk will focus on results of two related strands of research undertaken by the speaker. The first is a model of quantum mechanics based on the idea of 'structural approximation'. The earlier paper 'The semantics of the canonical commutation relations' (arxiv) established a method of calculation, essentially integration, for quantum mechanics with quadratic Hamiltonians. Currently, we worked out a (model-theoretic) formalism for the method, which allows us to perform more subtle calculations, in particular, we prove that our path integral calculation produce correct formula for quadratic Hamiltonians avoiding non-conventional limits used by physicists. Then we focus on the model-theoretic analysis of the notion of structural approximation and show that it can be seen as a positive model theory version of the theory of measurable structures, compact domination and integration (p-adic and adelic).
Notes de l'exposé

16h : Immi Halupczok (Düsseldorf), Un nouvel analogue de l'o-minimalité dans des corps valués
Pour les corps réel clos, la notion d'o-minimalité a eu un énorme succès; il s'agit d'une condition très simple à une expansion du langage des corps, qui implique que les ensembles définissables se comportent très bien d'un point de vue géométrique. Il existe plusieurs adaptations de cette notion aux corps valués (p.ex. p-mininalité, C-minimalité, B-minimalité, v-minimalité), mais la plupart de ces adaptations (a) s'appliquent seulement à une classe de corps valués assez restrictive, (b) elles n'impliquent pas tout ce qu'on voudrait, et/ou (c) elles sont définies de manière nettement plus compliquée. Dans cet exposé, je vais présenter une nouvelle notion qui n'a pas les problèmes (a) et (b) et qui a une définition raisonnablement simple.

Vendredi 17 novembre, ENS, salle W. Orateurs :

11h : Olivier Benoist (Strasbourg), Sur les polynômes positifs qui sont sommes de peu de carrés
Artin a résolu le 17ème problème de Hilbert : un polynôme réel positif en n variables est somme de carrés de fractions rationnelles. Pfister a amélioré ce résultat en démontrant qu'il est somme de 2^n carrés. Décider si la borne 2^n de Pfister est optimale est un problème ouvert si n>2. Nous expliquerons que cette borne peut être améliorée en petit degré et, en deux variables, pour un ensemble dense de polynômes positifs.

14h15 : Dmitry Sustretov (MPIM), Incidence systems on Cartesian powers of algebraic curves
The classical theory of abstract projective geometries establishes an equivalence between axiomatically defined incidence systems of points and lines and projective planes defined over a field. Zilber's Restricted Trichotomy conjecture in dimension one is a generalization of this statement in a sense, with lines replaced by algebraic curves; it implies that a non-locally modular strongly minimal structure with the universe an algebraic curve over an algebraically closed field and basic relations constructible subsets of Cartesian powers of the curve interprets an infinite field. The talk will present the basic structure of the proof of the conjecture, and outline its application, by Zilber, to Torreli-type theorem for curves over finite fields of Bogomolov, Korotiaev and Tschinkel. Joint work with Assaf Hasson.

16h : Alex Wilkie (Oxford), Quasi-minimal expansions of the complex field
I discuss a back-and-forth technique for proving that in certain expansions of the complex field every L_{∞, ω}-definable subset of ℂ is either countable or co-countable. Some successes of the method will also be discussed.

Vendredi 15 décembre, ENS, salle W. Orateurs :

11h : Adam Topaz (Oxford), On the conjecture of Ihara/Oda-Matsumoto
Following the spirit of Grothendieck's Esquisse d'un Programme, the Ihara/Oda-Matsumoto conjecture predicted a combinatorial description of the absolute Galois group of Q based on its action on geometric fundamental groups of varieties. This conjecture was resolved in the 90's by Pop using anabelian techniques. In this talk, I will discuss the proof of stronger variant of this conjecture, using mod-ell two-step nilpotent quotients, while highlighting some connections with model theory.

14h15 : Julien Sebag (Rennes), Géométrie des arcs et singularités
Soulignée par Nash dans les années 60, l'interaction entre la géométrie des espaces d'arcs et la théorie des singularités s'est fortement amplifiée sous l'influence de la théorie de l'intégration motivique notamment. Dans cet exposé, nous introduirons le schéma des arcs associé à une variété algébrique et donnerons quelques illustrations de cette interaction. Parmi elles, nous parlerons de l'interprétation (possible) du point de vue des singularités d'un théorème de Drinfeld et Grinberg-Kazhdan démontré au début des années 2000. (Cette dernière partie de l'exposé s'appuie sur une collaboration avec David Bourqui.)

16h : Martin Bays (Münster), The geometry of combinatorially extreme algebraic configurations
Given a system of polynomial equations in m complex variables with solution set of dimension d, if we take finite subsets X_i of C each of size at most N, then the number of solutions to the system whose ith co-ordinate is in X_i is easily seen to be bounded as O(N^d). We ask: when can we improve on the exponent d in this bound?
Hrushovski developed a formalism in which such questions become amenable to the tools of model theory, and in particular observed that incidence bounds of Szemeredi-Trotter type imply modularity of associated geometries. Exploiting this, we answer a (more general form of) our question above. This is part of a joint project with Emmanuel Breuillard.

Vendredi 19 janvier 2018, à l'IHP, amphithéâtre Hermite. (Attention au changement de lieu). Orateurs :

11h : Isaac Goldbring (UC Irvine), Spectral gap and definability
Originating in the theory of unitary group representations, the notion of spectral gap has played a huge role in many of the deep results in the theory of von Neumann algebras in the last couple of decades. Recently, with my collaborators, we are slowly understanding the model-theoretic significance of spectral gap, in particular its connection with definability. In this talk, I will discuss a few of our recent observations in this direction and speculate on some further possible developments. I will assume no knowledge of von Neumann algebras nor continuous logic. Various parts of this work are joint with Bradd Hart, Thomas Sinclair, and Henry Towsner.

14h15 : Jan Tuitman (Leuven), Effective Chabauty and the Cursed Curve
The Chabauty method often allows one to find the rational points on curves of genus at least 2 over the rationals, but has a lot of limitations. On a theoretical level, the Mordell-Weil rank of the Jacobian of the curve has to be strictly smaller than its genus. In practice, even when this condition is satisfied, the relevant Coleman integrals can usually only be computed for hyperelliptic curves. We will report on recent work of ours (with different combinations of collaborators) on extending the method to more general curves. In particular, we will show how one can use an extension of the Chabauty method by Kim to find the rational points on the split Cartan modular curve of level 13, which is also known as the cursed curve. The talk will be aimed at non-specialists with an interest in number theory.
Notes de l'exposé.

16h : Jonathan Kirby (East Anglia), Blurred Complex Exponentiation
Zilber conjectured that the complex field equipped with the exponential function is quasiminimal: every definable subset of the complex numbers is countable or co-countable. If true, it would mean that the geometry of solution sets of complex exponential-polynomial equations and their projections is somewhat like algebraic geometry. If false, it is likely that the real field is definable and there may be no reasonable geometric theory of these definable sets.
I will report on some progress towards the conjecture, including a proof when the exponential function is replaced by the approximate version given by ∃ q,r in Q [y = e^{x+q+2πi r}]. This set is the graph of the exponential function blurred by the group exp(Q + 2 πi Q). We can also blur by a larger subgroup and prove a stronger version of the theorem. Not only do we get quasiminimality but the resulting structure is isomorphic to the analogous blurring of Zilber's exponential field and to a reduct of a differentially closed field.
Reference: Jonathan Kirby, Blurred Complex Exponentiation, arXiv:1705.04574

Vendredi 23 mars 2018, amphithéatre Darboux, IHP. Orateurs prévus :

11h : Florian Pop (U. of Pennsylvania), On a conjecture of Colliot-Thélène
Let f be a morphism of projective smooth varieties X, Y defined over the rationals. The conjecture by Colliot-Thélène under discussion gives (conjectural) sufficient conditions which imply that for almost all rational prime numbers p, the map f maps the p-adic points X(ℚ_p) surjectively onto Y(ℚ_p). The aim of the talk is to present some recent results by Denef, Skorobogatov et al; further to report on work in progress on a different method to attack the conjecture under quite relaxed hypotheses.

14h15 : Thomas Scanlon (UC Berkeley), The dynamical Mordell-Lang problem in positive characteristic
The dynamical Mordell-Lang conjecture in characteristic zero predicts that if f : X → X is a map of algebraic varieties over a field K of characteristic zero, Y ⊆ X is a closed subvariety and a in X(K) is a K-rational point on X, then the return set { n in N : f^n(a) in Y(K) } is a finite union of points and arithmetic progressions. For K a field of characteristic p > 0, it is necessary to allow for finite unions with sets of the form { a + ∑_{i=1}^m p^{n_i} : (n_1, ... , n_m) in N^m } and one might conjecture that all return sets are finite unions of points, arithmetic progressions and such p-sets. We studied the special case of the positive characteristic dynamical Mordell-Lang problem on semiabelian varieites and using our earlier results with Moosa on so-called F-sets reduced the problem to that of solving a class of exponential diophantine equations in characteristic zero. In so doing, under the hypothesis that X is a semiabelian variety and either Y has small dimension or f is sufficiently general, we prove the conjecture. However, we also show that our reduction to the exponential diiophantine problems may be reversed so that the positive characteristic dynamical Mordell-Lang conjecture in general is equivalent to a class of hard exponential diophantine problems which the experts consider to be out of reach given our present techniques.
(This is a report on joint work with Pietro Corvaja, Dragos Ghioca and Umberto Zannier available at arXiv:1802.05309.)
Notes de l'exposé

16h : Sergei Starchenko (U. Notre Dame), A model theoretic generalization of the one-dimensional case of the Elekes-Szabo theorem
(Joint work with A. Chernikov)
Let V ⊆ ℂ^3 be a complex variety of dimension 2.
The Elekes-Szabo Theorem says that if V contains “too many” points on n × n × n Cartesian products then V has a special form: either V contains a cylinder over a curve or V is related to the graph of the multiplication of an algebraic group.
In this talk we generalize the Elekes-Szabo Theorem to relations on strongly minimal sets interpretable in distal structures.

Vendredi 18 mai 2018, à Jussieu, salle 101, couloir 15-16. (Jussieu = 4 place Jussieu, 75005 Paris. Métro Jussieu).
Programme :

11h : Laurent Bartholdi (ENS), Groups and algebras
To every group G is associated an associative algebra, namely its group ring kG. Which (geometric) properties are reflected in (algebraic) properties of kG? I will survey some results and conjectures in this area, concentrating on specific examples: growth, amenability, torsion, and filtrations.

14h15 : Rahim Moosa (Waterloo), Isotrivial Mordell-Lang and finite automata
About fifteen years ago, Thomas Scanlon and I gave a description of sets that arise as the intersection of a subvariety with a finitely generated subgroup inside a semiabelian variety over a finite field. Inspired by later work of Derksen on the positive characteristic Skolem-Mahler-Lech theorem, which turns out to be a special case, Jason Bell and I have recently recast those results in terms of finite automata. I will report on this work, as well as on the work-in-progress it has engendered on an effective version of the isotrivial Mordell-Lang theorem.

16h : Omer Friedland (IMJ-PRG), Doubling parametrizations and Remez-type inequalities
A doubling chart on an n-dimensional complex manifold Y is a univalent analytic mapping ψ : B_1 → Y of the unit ball in ℂ^n, which is extendible to the (say) four times larger concentric ball of B_1. A doubling covering of a compact set G in Y is its covering with images of doubling charts on Y. A doubling chain is a series of doubling charts with non-empty subsequent intersections. Doubling coverings (and doubling chains) provide, essentially, a conformally invariant version of Whitney's ball coverings of a domain W ⊂ ℝ^n.
Our main motivation is that doubling coverings form a special class of “smooth parameterizations”, which are used in bounding entropy type invariants in smooth dynamics on one side, and in bounding density of rational points in diophantine geometry on the other. Complexity of smooth parameterizations is a key issue in some important open problems in both areas.
In this talk we present various estimates on the complexity of these objects. As a consequence, we obtain an upper bound on the Kobayashi distance in Y, and an upper bound for the constant in a doubling inequality for regular algebraic functions on Y. We also provide the corresponding lower bounds for the length of the doubling chains, through the doubling constant of specific functions on Y.
This is a joint work with Yosef Yomdin.

Slides.

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.
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