La suite d’Oldenburger-Kolakoski est l’unique mot infini sur l’alphabet {1,2} qui commence par un “1” et est point fixe de l’opérateur de dérivation. En 1991, M.S. Keane conjecture que cette suite admet une fréquence d’1/2 pour la lettre “1”.

Les suites dites “auto-descriptives” sont une généralisation du mot d’Oldenburger-Kolakoski. Ces suites sont en bijection naturelle avec l’ensemble de toutes les suites sur l’alphabet {1,2} : une suite auto-descriptive est dite “dirigée” par son homologue naturelle sur {1,2}. Est-il possible d’inférer les fréquences de lettres de l’une à partir de l’autre ?

Je présenterai dans cet exposé deux approches à cette question : l’une probabiliste (Boisson, Jamet, Marcovici — 2024), l’autre analytique (Akiyama, Jamet, Marcovici, T.C. — 2024).

Many arguments in mathematics rely on the introduction of an auxiliary function that is compactly supported. Sometimes this compactly supported function is required to satisfy some additional regularity, but this cannot be pushed too far. It can for instance not be analytic as follows by the identity principle.

In this talk, we wish to present a technique that may bypass this obstruction. Namely, instead of considering a single function, we shall construct a sequence of function that are supported in the same compact, and satisfy some good uniform bounds on their derivatives.

This method is a powerful tool as it can, in some circumstances, turn a heuristic argument relying on a compactly supported analytic auxiliary function, into a rigorous proof. As an application, we shall discuss how this technique leads to improvements in some quantified Tauberian theorems.

We consider Lie groups $G$ endowed with a pair of anticommuting left-invariant abelian complex structures $(J_1,J_2)$ and a left-invariant, possibly indefinite, metric $g$ such that $(G,J_1,J_2,g)$ results to be a hyperkähler manifold. We study the algebraic structure and geometric properties of such Lie groups with an abelian hyperkähler structure. It results that such groups are always 3-step nilpotent and there is a correspondence between the associated hyperkähler Lie algebras and certain triples $(V,Omega, J_s)$ defined for a complex (associative) commutative algebra $V$ such that $V^3={0}$. This correspondence allows us to compute the Riemannian curvature of the pseudo-metric, describe the holonomy algebra and show that hyperkähler Lie groups with abelian complex structures are complete and locally symmetric. This clearly implies that every simply-connected Lie group endowed with an abelian hyperkähler structure is actually a symmetric space. In constrast to the definite case, there exist non-flat examples of abelian hyperkähler Lie groups; they cannot be 2-step nilpotent and their dimension is always equal to or greater than 16. Moreover, using the triple description, we classify up to Lie algebra isomorphism all Lie algebras $g$ admitting an abelian hyperkähler structure for $dimgle 12$. Some remarks on their classification up to triholomorphic symplectomorphism will also be mentioned. [BS_HK] I. Bajo, E. Sanmart'{i}n, “Indefinite hyperkähler metrics on Lie groups with abelian complex structures”, 2019, to appear in Transformation Groups.

Résumé

www.iecl.univ-lorraine.fr/~Khalid.Koufany/SL2R2020/programme.html

https://dev-iecl.univ-lorraine.fr/Les-Seminaires/Theorie-Des-Nombres/wolfcms/seminaire.html

https://dev-iecl.univ-lorraine.fr/Les-Seminaires/Theorie-Des-Nombres/wolfcms/seminaire.html

Résumé

Résumé