Responsables: Z. Chatzidakis, F. Oger, F. Point.
Tous les mardis ouvrables: à 16h00. Bâtiment Sophie Germain. (Exceptionnellement, sur annonce, pourra avoir lieu à 14h).
Pour recevoir le programme par email : oger_at_math.univ-paris-diderot.fr

Mardi 13 novembre : Frank Wagner (ICJ - Lyon 1), Groupes et anneaux oméga-catégoriques de fardeau fini

Les groupes de fardeau fini sont les groupes NTP_2 qui correspondent aux groupes stables ou simples de rang fini. Or, le fardeau est plus difficile à manipuler car il n'est pas forcément additif par fibration. Nous montrons que ces groupes sont virtuellement abélien-par-fini, et les anneaux sont virtuellement fini-par-nuls. Ceci améliore un résultat de Kaplan, Levi et Simon qui avaient démontré qu'un groupe dp-minimal est virtuellement nilpotent.
Travail en commun avec Jan Dobrowolski

Mardi 20 novembre : Tingxiang Zou (ICJ - Lyon 1), Counting in pseudofinite structures

In pseudofinite structures, the non-standard size of definable sets often reveals important algebraic or model theoretic properties of the corresponding theories. In this talk, we will give two new examples of this correlation. One is between the coarse dimension and the transformal transcendental degree in certain class of pseudofinite difference fields. The other example is that in pseudofinite H-strucures which are built from one-dimensional asymptotic classes, the coarse dimension of a tuple corresponds to the coefficient of the leading term of SU-rank of this tuple. This is the first step to show that they are examples of multidimensional asymptotic classes (mac).
Transparents

Mardi 4 décembre : Christian d'Elbée (ICJ - Lyon 1), Sous-groupe additif générique d'un corps algébriquement clos de caractéristique positive.

La théorie d'un corps algébriquement clos de caractéristique positive p muni d'un prédicat pour un sous-groupe additif admet une modèle-compagne ACF_pG. On se propose de décrire ce nouvel exemple de théorie NSOP_1, en décrivant les imaginaires, le Kim-forking et le forking. On parlera aussi de la généralisation de cette construction afin de présenter de nouveaux exemples de théories NSOP_1.
Transparents

Mardi 18 décembre : Simon André (Rennes 1), Les groupes virtuellement libres sont presque homogènes

Perin et Sklinos, et indépendamment Ould Houcine, ont démontré en 2011 que les groupes libres sont homogènes : deux éléments qui ont le même type sont dans la même orbite sous l'action du groupe d'automorphismes. Dans cet exposé, j'expliquerai que ce résultat reste presque vrai pour les groupes virtuellement libres, au sens suivant : l'ensemble des éléments ayant le même type qu'un élément donné contient un nombre fini d'orbites sous le groupe d'automorphismes, et ce nombre ne dépend pas de l'élément considéré. J'expliquerai également pourquoi je pense que ce résultat est optimal, en donnant un exemple de groupe virtuellement libre dont je conjecture qu'il n'est pas homogène (travail en cours).

Mardi 15 janvier : Margarita Otero (UAM), Density of the union of Cartan subgroups of o-minimal groups.

Let G be a group. A subgroup H of G is a Cartan subgroup of G if H is a maximal nilpotent subgroup of G, and for every normal finite index subgroup X of H, X has finite index in its normalizer in G.

We consider Cartan subgroups of definably connect groups definable in an o-minimal structure. In [BJ0] we proved that, in this context, Cartan subgroups of G exist, they are definable and they fall in finitely many conjugacy classes.

In this talk I will prove that the union of the Cartan subgroups is dense in the group, which was the main question left open in [BBO].
(Joint work with Elías Baro and Alessandro Berarducci.)

[BJ0] E.Baro, E. Jaligot and M.Otero. Cartan subgroups of groups definable in o-minimal structures, J. Inst. Math. Juissieu 13 no. 4 (2014) 849 - 893.

Mardi 29 janvier en salle 1016 (changement de salle) : Paola D'Aquino (University of Campania “Luigi Vanvitelli”, Spectrum of the profinite completion of the integers.

Using ultraproducts, I will describe the spectrum of the profinite completion of the integers and of the finite adeles over the rationals.
The final aim is to describe the structure sheaf of these structures.
Joint work with Margarita Otero and Angus Macintyre.

Mardi 5 février : Tomas Ibarlucia (Paris-Diderot), Groupes d'automorphismes et Propriété (T)

Nous présenterons une preuve de la Propriété (T) de Kazhdan pour les groupes d'automorphismes de structures métriques aleph_0-catégoriques. Ceci généralise des résultats précédents de Bekka (pour le groupe unitaire) et de Evans et Tsankov (pour les groupes pro-oligomorphes), sans besoin de faire appel à des résultats de classification de représentations unitaires. En effet, l'argument est purement modèle-théorique et basé sur des principes de la stabilité locale.

Mardi 19 février : Hector Pasten (PUC Chile), On the theory of rigid meromorphic functions in positive characteristic

There is a well-known analogy between the arithmetic of rational numbers and the theory of meromorphic functions over a normed field. It is a classical result of Julia Robinson that the first order theory of the field of rational numbers is undecidable, and one would expect such a result in the meromorphic setting. In this talk I'll give an outline of the proof of undecidability for rigid meromorphic functions in positive characteristic; the cases of characteristic zero remain open.

Mardi 26 février : Ward Henson (UIUC), Uncountable categoricity of structures based on Banach spaces

A continuous theory T of bounded metric structures is said to be kappa-categorical if T has a unique model of density kappa. Work of Ben Yaacov and Shelah+Usvyatsov shows that Morley's Theorem holds in this context: if T has a countable signature and is kappa-categorical for some uncountable kappa, then T is kappa-categorical for all uncountable kappa. In classical (discrete) model theory, there are several characterizations of uncountable categoricity. For example, there is a structure theorem for uncountably categorical theories T, due to Baldwin+Lachlan: there is a strongly minimal set D defined over the prime model of T such that every uncountable model M of T is minimal and prime over D(M). Moreover (and easier), if T has such a strongly minimal set, then T is uncountably categorical.

In the more general metric structure setting, nothing remotely like this is known. Indeed, the metric analog of a strongly minimal set is nowhere to be seen, at the moment. If one restricts attention to metric structures based on (unit balls) of Banach structures, more is known. The appropriate analog of strongly minimal sets seems to be the unit balls of Hilbert spaces. After the speaker called attention to this phenomenon in some examples from functional analysis, Shelah and Usvyatsov investigated it and proved a remarkable result (arxiv 1402.6513; to appear in Adv. in Math.): if M is a nonseparable Banach structure (with countable signature) whose theory is uncountably categorical, then M is prime over a Morley sequence that is an orthonormal Hilbert basis of length equal to the density of M. There is a wide gap between this result and what is true of verified examples of uncountably categorical Banach structures , which leads to the question: can a stronger such result be proved, which gives a characterization of uncountable categoricity for Banach structures and in which the connection to Hilbert space structure is clearly expressed in the geometric language of functional analysis?

In addition to the above background, we will discuss some new examples of uncountably categorical Banach spaces (of which there have been very few previously known). This is joint work with Yves Raynaud (Paris 6); we have a 2016 paper in Comment. Math. (now freely available on their website) and the examples to be discussed here are more recent.

Slides

Mardi 12 mars, à 14h, en salle 1016 : Salma Kuhlmann (Konstanz), Strongly NIP almost real closed fields

The following conjecture is due to Shelah--Hasson: Any infinite strongly NIP field is either real closed, algebraically closed, or admits a non-trivial definable henselian valuation, in the language of rings. We specialise this conjecture to ordered fields in the language of ordered rings, which leads towards a systematic study of the class of strongly NIP almost real closed fields. As a result, we obtain a complete characterisation of this class.

Mardi 12 mars à 16h, salle 2015 : Martin Bays (Münster), Density of compressiblity in NIP theories

Joint with Itay Kaplan and Pierre Simon.

Distal theories are NIP theories which are “wholly unstable”. Chernikov and Simon's “strong honest definitions” characterise distal theories as those in which every type is compressible. Adapting recent work in machine learning of Chen, Cheng, and Tang on bounds on the “recursive teaching dimension” of a finite concept class, we find that compressibility is dense in NIP structures, i.e. any formula can be completed to a compressible type in S(A). Considering compressibility as an isolation notion (which specialises to l-isolation in stable theories), we obtain consequences on the existence of models with certain properties.

Mardi 19 mars : Armin Darbinyan (ENS), The word and conjugacy problems in finitely generated groups

The word and conjugacy problems are central decision problems associated with finitely generated groups. In particular, there are deep results which bridge some of the main concepts of the theories of computability and computational complexity with group theoretical invariants through the word problem in groups. In this talk I will recall some of the well-known facts about the word and conjugacy problems in groups as well as discuss new results concerning the relationship between them.

Mardi 7 mai : Juan Pablo Acosta (U. Los Andes), Groups definable in Presburger arithmetic

I will give a complete description of all groups definable in Presburger arithmetic, up to finite index subgroups. This builds on previous work on bounded groups in Presburger arithmetic by Mariana Vicaria and Alf Onshuus.

Mardi 4 juin : Arman Darbinyan (ENS), Computability, orders, and groups

Orderable groups are extensively studied by logicians and group theorists. In my talk I will address aspects of left- or bi-orderable groups that are connected with computability theory. In particular, I will talk about constructions of bi-orderable computable groups that cannot be embedded into groups with computable bi-order. I will also discuss our recent work in progress with M. Steenbock about simplicity and computably left-orderability.

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