Théorie des Modèles et Groupes

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 […]

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 […]

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.

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.

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.

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 […]

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 […]

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, […]

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 […]

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 […]

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 […]

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 […]