For 𝒢 some groupoid whose objects are the sub-vector spaces of a 𝔽 p -vector space W , we define H * ( W ) 𝒢 a n i l -closed, noetherian, unstable sub-algebra of H * ( W ) over the Steenrod algebra. The application on the appropriate ordered set of groupoids, that maps 𝒢 to H * ( W ) 𝒢 defines an isomorphism of posets to the set of noetherian, n i l -closed, unstable sub-algebras of H * ( W ) of transcendence degree dim ( W ) , ordered by inclusion. Since any noetherian and integral unstable algebra of transcendence degree dim ( W ) admits an injection into H * ( W ) , any such n i l -closed unstable algebra is isomorphic to some H * ( W ) 𝒢 . We prove that 𝒢 encodes the centre, in the sense of Heard, of H * ( W ) 𝒢 . Also, there is a H * ( C ) -comodule structure on K that is associated with the centre of K . For K = H * ( W ) 𝒢 , we explain how the sub-algebra of primitive elements of H * ( W ) 𝒢 for this comodule structure is also encoded in 𝒢 . Along the way, we prove that this algebra of primitive elements is also noetherian.
A marked surface is a compact oriented surface equipped with some pairwise disjoint arcs embedded in its boundary. In this paper, we extend the notion of character varieties to marked surfaces, in such a way that they have a nice behaviour for the operation of gluing two boundary arcs together. These stated character varieties are affine Poisson varieties which coincide with the Culler-Shalen character varieties when the surface is unmarked and are closely related to the Fock-Rosly and Alekseev-Kosmann-Malkin-Meinrenken constructions in the marked case. These Poisson varieties are the classical moduli spaces underlying stated skein algebras and share similar properties. In particular, stated character varieties admit triangular decompositions, associated to triangulations of the surface. We identify the Zariski tangent spaces of these varieties with some twisted groupoid cohomological groups and provide a generalization of Goldman's formula for the Poisson bracket of curve functions in terms of intersection form in homology.
We consider several coupled systems of one-dimensional linear parabolic equations where only one equation is controlled with a distributed control. For these systems we study the minimal null-control time that is the minimal time needed to drive any initial condition to zero. In a previous work [ Comptes Rendus. Mathématique , 361:1191–1248, 2023] we extended the block moment method to obtain a complete characterization of the minimal null-control time in an abstract setting encompassing such non-scalar controls. In this paper, we push forward the application of this general approach to some classes of 1D parabolic systems with distributed controls whose analysis is out of reach by the usual approaches in the literature like Carleman-based methods, fictitious control and algebraic resolubility, or standard moment method. To achieve this goal, we need to prove refined spectral estimates for Sturm–Liouville operators that have their own interest.
We consider the existence problem of conformal metrics with prescribed fractional curvature on the standard sphere S n ,n≥2. It is equivalent to solving a fractional nonlinear variational equation involving a critical nonlinearity. By studying the lack of compactness of the associated variational problem, we extend the existence results of [2] and [3] to any fractional order σ∈(0,n 2) and prove a general existence and multiplicity Theorem under an Euler–Hopf type criterion.
Dans cet article, on va s’intéresser à la classification de certains entiers naturels reliés à la combinatoire des sous-groupes de congruence du groupe modulaire. Plus précisément, on va s’intéresser ici à la notion de solutions monomiales minimales. Celles-ci sont les solutions d’une équation matricielle (apparaissant également lors de l’étude des frises de Coxeter), modulo un entier N, dont toutes les composantes sont identiques et minimales pour cette propriété. Notre objectif ici est d’étudier les entiers N pour lesquels les solutions monomiales minimales vérifiant certaines conditions fixées possèdent une propriété d’irréductibilité. En particulier, on effectuera la classification des entiers monomialement irréductibles qui sont les entiers pour lesquels toutes les solutions monomiales minimales non nulles sont irréductibles.
