Séminaire de Logique, Algèbre, Arithmétique et Géométrie
- Le vendredi 14 novembre 2025 à partir de 13h30
Frodo Moonen (Leuven): Grothendieck Rings of Ordered Subgroups of the Rationals
Giovanni Bosco: TBA
Séances passées
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Le mercredi 30 avril 2025 (Bâtiment De Vinci, 1er Etage, salle Mirzakhani) – Après-midi.
13h30-14h30: Vincent Bagayoko (Paris), Taylor expansions over generalised power series
Résumé:
In real or complex analysis, the Taylor expansion of a function at a given point contains all the local information of that function around that point, and Taylor series can be used both to study and to define analytic functions. In o-minimal geometry, it is usual to embed algebras of real-valued regular functions into algebras of functions defined on ordered fields of generalised power series, such as transseries or generalisations thereof. The Taylor expandableness of o-minimal real-valued functions should translate into the existence of formal Taylor expansions of their formal avatars.
I will show how to define Taylor expansions for functions over generalised power series, and show that composition laws on fields of generalised transseries can be understood and defined using such expansions. We will also see that Taylor expansions give an instance of the Lie-type correspondence between derivations and automorphisms on algebras of generalised power series, as described with Krapp, Kuhlmann, Panazzolo and Serra. This is based on joint work with Vincenzo Mantova.
14h50-15h50: Elliot Kaplan (Bonn), Generic derivations on o-minimal structures
Résumé:
Let T be a model complete o-minimal theory that extends the theory of real closed ordered fields (RCF). We introduce T-derivations: derivations on models of T which cooperate with T-definable functions. The theory of models of T expanded by a T-derivation has a model completion in which the derivation acts « generically. » If T=RCF, then this model completion is the theory of closed ordered differential fields (CODF) as introduced by Singer. We can recover many of the known facts about CODF (open core, distality) in our setting. Time permitting, I will also discuss some more recent work on this theory (thorn-rank, Kolchin polynomials). This is joint work with Antongiulio Fornasiero.
16h10-17h10: Mathias Stout (Leuven), Integration in Hensel minimal fields
Résumé:
An important theme in the model theory of valued fields is reducing questions about a valued field to ones about its residue field and value group. Model-theoretic frameworks for motivic integration such as the ones developed by Cluckers-Loeser and Hrushovski-Kazhdan achieve a similar reduction on the level of integrals. Such results require a certain tameness of the first order structure under consideration. For example, Hrushovski and Kazhdan work with V-minimal fields.
- Le mardi 6 mai 2025 (Pentagone, salle 0A11)
15h45: Valentin Ramlot, On finite subgroups of semisimple algebras – Introduction
Résumé:
The structure of semisimple algebras is well-known and is characterized by their Wedderburn decomposition. The finite multiplicative subgroups of such algebras are diverse. We begin a classification of such subgroups up to isomorphism assuming that they are abelian, and therefore products of cyclic groups. We mainly use standard tools coming from field theory and linear algebra.
17h00: Gabriel Ng, A brief introduction to differentially large fields
Résumé:
Differentially large fields are an analogue for large fields in the context of differential algebra, introduced by Leon Sanchez and Tressl. These are large fields which are equipped with a derivation which is in some sense “generic”. Many model-theoretically interesting differential fields are examples of these objects, for instance, differentially closed fields and closed ordered differential fields. In this talk, we will give a gentle introduction to the topic, introducing the necessary concepts and providing motivating examples as necessary.
- Le mercredi 14 mai 2025 (Pentagone, salle 0A11) – suite des exposés de la séance précédente.
13h30: Valentin Ramlot, On finite subgroups of semisimple algebras
14h45: Gabriel Ng, Abstract Taylor Morphisms
Résumé:
The Taylor morphism is a natural construction which generalises the notion of Taylor series from analysis to the algebraic context. In essence, the Taylor morphism turns ring homomorphisms into differential ring homomorphisms into the ring of formal power series. In studying differentially large fields, Leon Sanchez and Tressl introduce a “twisted” Taylor morphism, which allows us to introduce derivations in the target field. We introduce a generalisation of this notion, and show that all such abstract Taylor morphisms (over fields of arbitrary characteristic) must have a certain concrete form.
- Le mercredi 27 août 2025 (Bâtiment De Vinci, 1er Etage, salle Mirzakhani) – Journée estivale
Matinée: Présentation des stages d’initiation à la recherche (réalisés à l’issue du bachelier)
10h-11h: Vanille Zilmia, La théorie des corps valués algébriquement clos
11h15-12h15: Chelsea Brohet, Sur la classification des groupes simples : le théorème de Burnside
12h30-13h30: Lunch, au restaurant universitaire, bâtiment 9
Après-midi: Exposés par des chercheurs
14h00-15h00: Justin Vast (Louvain-la-Neuve), Groupes BMW, automates, fractales
Résumé:
Soient T et T’, deux arbres réguliers de degrés finis.
Un groupe BMW est un sous-groupe Γ ⩽ Aut(T) x Aut(T’) agissant librement et transitivement sur les sommets du produit cartésien TxT’.
Un groupe BMW Γ est dit réductible s’il existe un sous-groupe d’indice fini de la forme F x F’ ⩽ Γ , où F ⩽ Aut(T) et F’ ⩽ Aut(T’) sont des groupes libres.
Une façon d’identifier un groupe BMW irréductible est d’y montrer l’existence d’un anti-tore.
À un groupe BMW peut être associé un automate de Mealey bireversible.
Comme nous le verrons, il est partiellement possible de transférer la notion d’anti-tore aux automates de Mealy généraux.
Un célèbre automate de Mayley est celui associé au groupe d’allumeur de réverbères et un « anti-tore » bien choisi de l’automate laisse apparaître le fameux triangle de Sierpiński.
Certains anti-tores de groupes BMW irréductibles mettent en évidence d’autres fractales pour le moins surprenantes…
Résumé:
In 1900, David Hilbert poses a question at an international mathematics conference in Paris: Is there an algorithm that can determine whether a given polynomial equation with integer coefficients has an integer solution? The question became known as Hilbert’s 10th Problem. Several decades later, it became increasingly clear that such an algorithm may never exist. This marked the start of a research area on the intersection of logic, algebra, and number theory: to determine which classes of problems from number theory, algebra and geometry are decidable (i.e. solvable by an algorithm) and which are undecidable.
In this talk, I will give a gentle introduction to some of the algebraic and arithmetic ideas underlying approaches to variations of Hilbert’s 10th Problem in number theory and field theory. I will in particular give an own interpretation of some ideas of Julia Robinson to study number fields, and of Jan Denef to study function fields. Both Robinson’s and Denef’s approaches relied, in some way, on the theory of quadratic forms over fields.
Insofar time allows, I will talk about how recent developments on solvability of equations over function fields (like local-global principles for rational points on varieties) can be applied to obtain new results on definability and decidability of equations over function fields of curves. This part will discuss some results of joint research with Karim Johannes Becher and Philip Dittmann, as well as open questions for the future.