*** Théorie des Modèles et Groupes ***

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

Année 2021- - 2022
Liste des exposés précédents et résumés

Mardi 15 février : Samaria Montenegro (U. Costa Rica), Groups definable in partial differential fields with an automorphism

This is a joint work with Ronald Bustamante Medina and Zoé Chatzidakis.
In this talk we are interested in differential and difference fields from the model-theoretic point of view. A differential field is a field with a set of commuting derivations and a difference-differential field is a differential field equipped with an automorphism which commutes with the derivations.
Cassidy studied definable groups in differentially closed fields, in particular she studied Zariski dense definable subgroups of simple algebraic groups and showed that they are isomorphic to the rational points of an algebraic group over some definable field. In this talk we study groups definable in existentially closed difference-differential fields. In particular, we study Zariski dense definable subgroups of simple algebraic groups, and show an analogue of Phyllis Cassidy's result for partial differential fields.

Mardi 22 février : Blaise Boissonneau (Münster), NIPn fields part 2: random hypergraphs and NIPn CHIPS transfer . Salle 1016 et Zoom.

[La partie 1 aura lieu au séminaire général de logique lundi à 15h15 ; cependant chaque exposé sera self-contained.]

A core question in the model theory of fields is to understand how combinatorial patterns and algebraic properties interact. The study of NIPn fields, which can't express the edge relation of random n-hypergraph, is linked to henselianity. In this talk, we use Chernikov and Hils conditions to obtain transfer in some situations, that is, under some algebraic assumptions, it is enough to know that the residue field of a henselian valued field is NIPn in order to known that it is itself NIPn, and we discuss consequences on hypothetical strictly NIPn fields.

Mardi 1er mars : Tomas Ibarlucia (IMJ-PRG), Existentially closed measure-preserving actions of free groups

I will discuss a joint work with Alexander Berenstein and Ward Henson, in which we show that the theory of probability algebras with two automorphisms has a model completion, which moreover has quantifier elimination and is stable. We also exhibit two non-isomorphic (but approximately isomorphic) models of the model completion.
More generally, we give a sufficient set of conditions for the axiomatizability (in continuous logic) of the existentially closed actions of a free group on a separably categorical, stable structure.
I will also mention a number of open questions.

Mardi 15 mars : Vincent Ye (IMJ-PRG), Curve-excluding fields

Consider the class of fields with Char(K)=0 and x^4+y^4=1 has only 4 solutions in K, we show that this class has a model companion, which we denote by curve-excluding fields. Curve-excluding fields provides (counter)examples to various questions. Model theoretically, they are model complete and TP_2. Field theoretically, they are not large and unbounded. We will discuss other aspects such as decidability of such fields. This is joint work with Will Johnson and Erik Walsberg.

Mardi 22 mars : Alf Onshuus (U. Andes), Lie groups definable in o-minimal theories

In this talk we will work out a complete characterization of which Lie groups admit a “definable copy”. This is, characterize for which Lie groups G one can find a group H definable in an o-minimal expansion of the real field, and such that G and H are isomorphic.
When the answer is positive, the definable copy H that we find is definable in the language of exponential ordered fields, and it is such that any Lie automorphism of H is definable.

Mardi 5 avril 14h15 salle 1016 : Alexis Chevalier (Oxford), Piecewise Interpretable Hilbert Spaces (II)

We continue the discussion of piecewise interpretable Hilbert spaces from the Monday seminar. We will prove the main structure theorem of `Piecewise Interpretable Hilbert Spaces' (C., Hrushovski) which analyses a scattered piecewise interpretable Hilbert space into asymptotically free subspaces. We will clarify the model theoretic content of this theorem, highlighting the roles of one-basedness and strong minimality. We will also study its representation theoretic content, establishing a connection with induced represetnations. We will see that this theorem generalises a theorem of Tsankov about unitary representations of oligomorphic groups. This is joint work with Ehud Hrushovski.

Mardi 5 avril 16h : Samuel Zamour (Lyon 1), Quasi-groupes de Frobenius dimensionnels

Dans cet exposé, nous présenterons une généralisation des groupes de Frobenius : les quasi-groupes de Frobenius. On dit qu'une paire de groupes C < G est un quasi-groupe de Frobenius si C est d'indice fini dans son normalisateur (dans G) et s'il satisfait la propriété TI, i.e, deux conjugués distincts de C s'intersectent trivialement.
Du point de vue de la théorie des modèles, nous travaillerons dans un contexte où l'existence d'une bonne notion de dimension (finie) sur les ensembles définissables est assurée (ce qui englobe les univers rangés et les structures o-minimales).
En s'inscrivant dans le prolongement des travaux classiques de l'école de Bachmann et d'un article plus récent de A. Deloro et J. Wiscons, nous examinerons dans quelle mesure l'étude des géométries d'incidence induites par les involutions au sein des quasi-groupes de Frobenius permet d'identifier dans un cadre dimensionnel les groupes classiques GA_1(C), PGL_2(C) et SO_3(R).

Mardi 19 avril : Stefan Ludwig (ENS), Metric valued fields in continuous logic

By work of Itaï Ben Yaacov complete valued fields with value groups embedded in the real numbers can be viewed as metric structures in continuous logic. For technical reasons one has to consider the projective line over such a field rather than the field itself.
In this talk we introduce the above setting and give a classification of the complete theories of metric valued fields in equicharacteristic 0 in terms of their residue field and value group. This can also be seen as an approximate Ax-Kochen-Ershov principle. If time permits, as a second result we give a negative answer to a question of Ben Yaacov on the existence of a model companion for metric valued fields enriched with an isometric automorphism. This is joint work with Martin Hils.

Mardi 10 mai : Sylvy Anscombe (IMJ-PRG), Existential theories of henselian fields, parameters welcome

The first-order theories of local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are far less well understood than their characteristic zero analogues: the fields of real, complex and p-adic numbers. On the other hand, the existential theory of an equicharacteristic henselian valued field in the language of valued fields is controlled by the existential theory of its residue field. One is decidable if and only if the other is decidable. When we add a parameter to the language, things get more complicated. Denef and Schoutens gave an algorithm, assuming resolution of singularities, to decide the existential theory of rings like Fp[[t]], with the parameter t in the language. I will discuss their algorithm and present a new result (from ongoing work, with Dittmann and Fehm) that weakens the hypothesis to a form of local uniformization, and which works in greater generality.

Mardi 31 mai : Adrien Deloro (IMJ-PRG), Le théorème du corps gauche de Zilber
Le théorème du corps est l'observation qu'un groupe de rang de Morley fini connexe, résoluble, et non nilpotent, interprète un corps infini. Par d'autres résultats classiques, le corps est commutatif et même algébriquement clos.
Le théorème du corps est souvent vu comme corollaire du «théorème d'engendrement par des indécomposables» mais c'est une erreur car il en est indépendant. Il a quelques variantes, des théorèmes de linéarisation d'actions de groupes.
Je donnerai un énoncé qui généralise naturellement tous les résultats «à la Zilber». C'est un résultat de linéarisation de bimodules, dans un contexte plus général que les théories de rang de Morley fini. En général on interprète un corps gauche.
Prérequis : notion de définissabilité ; «lemme de Schur» en théorie des représentations (l'anneau des endomorphismes qui commutent avec une représentation irréductible est en fait un corps gauche).

Zilber's Skew-Field Theorem (joint with Frank Wagner)
Zilber's Field Theorem ZFT is the observation that a connected, soluble, non-nilpotent group of finite Morley rank interprets an infinite field. By other classical results, the field is commutative indeed, and even algebraically closed.
The ZFT is often seen as a corollary to Zilber's `indecomposable generation theorem'; but it actually is independent from it. The ZFT has a couple of variants, linearisation results for definable group actions.
I shall give a theorem which generalises naturally all results `à la Zilber'. It is a tool that can linearise bimodule actions, in a broader context than theories of finite Morley rank. In general it produces a definable skew-field.
Prerequisites: definable sets; `Schur's lemma' from representation theory (the ring of endomorphisms commuting with an irreducible representation, actually is a skew-field).

Mardi 7 juin : Nima Hoda (ENS), Cercles isométriques mais contractiles dans les cônes asymptotiques des groupes

La contractilité de tous les cercles dans les cônes asymptotiques d'un groupe G de type fini implique que G est de présentation finie avec fonction de Dehn au plus polynomiale. Le distorsion métrique de tous ces cercles est une propriété plus forte qui implique que G est fortement raccourci (“strongly shortcut”). La propriété fortement raccourci est satisfaite par diverses familles de groupes de courbure négative ou nulle, notamment les groupes hyperboliques, CAT(0), Helly, et systoliques, mais elle est aussi satisfaite par le groupe de Heisenberg discret.
Je discuterai d'un récent travail en commun avec Cashen et Woodhouse où on a montré qu'une famille infinie de groupes flocon de neige (“snowflake groups”) ont des cônes asymptotiques simplement connexes mais des graphes de Cayley qui ne sont pas fortement raccourcis. Ce sont les premiers exemples de groupes qui ont des cônes asymptotiques dans lesquels il existe des cercles isométriques mais contractiles.

Mardi 14 juin : Chris Laskowski (U. Maryland), On the Borel complexity of modules

We prove that among all countable, commutative rings R (with unit) the theory of R-modules is not Borel complete if and only if there are only countably many non-isomorphic countable R-modules. From the proof, we obtain a succinct proof that the class of torsion free abelian groups is Borel complete.
The results above follow from some general machinery that we expect to have applications in other algebraic settings. Here, we also show that for an arbitrary countable ring R, the class of left R-modules equipped with an endomorphism is Borel complete; as is the class of left R-modules equipped with predicates for four submodules. This is joint work with D. Ulrich.

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