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International audience; In my paper at Inventiones 2009, we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf cohomology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.; Dans [2], on détermine les groupes de cohomologie des espaces de Lubin-Tate par voie globale en calculant les fibres des faisceaux de cohomologie du faisceau pervers des cyclesévanescentscyclesévanescents Ψ d'une variété de Shimura de type Kottwitz-Harris-Taylor. L'ingrédient le plus complexe consistè a contrôler lesfì eches de deux suites spectrales calculant l'une les faisceaux de cohomologie des faisceaux pervers d'Harris-Taylor, et l'autre ceux de Ψ. Dans cet article, nous contournons ces difficultés en utilisant la théorie classique des représentations du groupe mirabolique ainsi qu'un argument géométrique simple. Abstract (Mirabolic group, ramified Newton stratification and cohomology of Lubin-Tate spaces) In [2], we determine the cohomology of Lubin-Tate spaces globally using the comparison theorem of Berkovich by computing the fibers at supersingular points of the perverse sheaf of vanishing cycle Ψ of some Shimura variety of Kottwitz-Harris-Taylor type. The most difficult argument deals with the control of maps of the spectral sequences computing the sheaf coho-mology of both Harris-Taylor perverse sheaves and those of Ψ. In this paper, we bypass these difficulties using the classical theory of representations of the mirabolic group and a simple geometric argument.

51 pages, in French, v2: extended introduction, journal version; International audience; The aim of this paper is to prove a generalization of the famous Theorem A of Quillen for strict $\infty$-categories. This result is central to the homotopy theory of strict $\infty$-categories developed by the authors. The proof presented here is of a simplicial nature and uses Steiner's theory of augmented directed complexes. In a subsequent paper, we will prove the same result by purely $\infty$-categorical methods.

This paper is an introduction to the theory of 1d stochastic Burgers equation under periodic boundary conditions and with a stochastic force, sufficiently smooth in the space variable. We prove the classical results on the existence and uniqueness of solutions, study their regularity and discuss their properties when the time goes to infinity or the viscosity goes to zero. The latter limit describes the turbulence in the Burgers equation, named by U. Frish "the burgulence". Our paper may be used as an elementary introduction to the modern methods of stochastic PDE., Comment: in French

Dans un r\'ecent article, Lagarias et Soundararajan \'etudient les solutions friables \`a l'\'equation a+b=c. Sous l'hypoth\`ese de Riemann g\'en\'eralis\'ees aux fonctions L de Dirichlet, ils obtiennent une estimation pour le nombre de solutions pond\'er\'ees par un poids lisse et une minoration pour le nombre de solutions non pond\'er\'ees. Le but de cet article est de pr\'esenter des arguments qui permettent d'une part de pr\'eciser les termes d'erreur et d'\'etendre les domaines de validit\'e de ces estimations afin de faire le lien avec un travail de la Bret\`eche et Granville, d'autre part d'obtenir le comportement asymptotique exact du nombre de solutions non pond\'er\'ees. In a recent paper, Lagarias and Soundararajan study the y-smooth solutions to the equation a+b=c. Under the Generalised Riemann Hypothesis, they obtain an estimate for the number of those solutions weighted by a compactly supported smooth function, as well as a lower bound for the number of bounded unweighted solutions. In this paper, we aim to prove a more precise estimate for the number of weighted solutions that is valid when y is relatively large with respect to x, so as to connect our estimate with the one obtained by La Bret\`eche and Granville in a recent work. We also prove the conjectured upper bound for the number of bounded unweighted solutions, thus obtaining its exact asymptotic behaviour., Comment: 20 pages

International audience; It was proved that Euler-Maxwell systems converge globally-in-time to driftdiffusion systems in a slow time scaling, as the relaxation time goes to zero. The convergence was established to the Cauchy problem with smooth periodic initial data sufficiently close to constant equilibrium states. In this paper, we establish error estimates between smooth periodic solutions of Euler-Maxwell systems and those of drift-diffusion systems. Similar error estimates are also obtained for Euler-Poisson systems in place of Euler-Maxwell systems. The proof of these results uses stream function techniques together with energy estimates. Résumé. Il a été prouvé que des systèmes d'Euler-Maxwell convergent globalement en temps vers des systèmes de dérive-diffusion dans une échelle de temps lent, lorsque le temps de relaxation tend vers zéro. La convergence a éte établie au problème de Cauchy avec données initiales régulières périodiques suffisamment proches d'états d'équilibres constants. Dans cet article, nous établissons des estimations d'erreur entre les solutions régulières périodiques des systèmes d'Euler-Maxwell et celles des systèmes de dérivediffusion. Des estimations d'erreur similaires sont aussi obtenues pour des systèmes d'Euler-Poisson au lieu des systèmes d'Euler-Maxwell. Nous utilisons des techniques de fonctions de courant et des estimations d'energies pour la démonstration de ces résultats.

In his 1983 paper [3], R. F. Gundy introduced a new functional related to the Littlewood-Paley theory, called the density of the area integral. In this paper, we prove that this functional (although highly non-linear) can be expressed as the principal value of an explicit singular integral. This result provides us with a new and precise connection between the density of the area integral and the theory of Calderón-Zygmund operators. It does not seem to be a consequence of the standard Calderón-Zygmund-Cotlar theory, because the sign of a harmonic function in the half-space fails to have, in some appropriate sense, boundary limits.

Version incluant des corrections mineures à la version publiée. Ajout d'une note à la fin.; International audience; This paper is a continuation of the papers "Moduli spaces of decomposable morphims of sheaves and quotients by non-reductive groups" and "Espace abstraits de morphismes et mutations". I show here that the method used in the first paper to construct moduli spaces of morphisms and the one used in the second paper to extend the results of the first are particular cases of a more general method of construction of algebraic quotients. It is based on the notion of quasi-isomorphism. Let G, G' algebraic groups acting on algebraic varieties X, X' respectively. A quasi-morphism is a map $X/G\longrightarrow X'/G'$ that can be everywhere locally lifted to morphisms from open subsets of X to X' (plus some additional technical conditions...). A quasi-isomorphism is a quasi-morphism which is bijective and whose inverse is also a quasi-morphism. If there is a quasi-isomorphism between X and X'it is possible to deduce the existence of algebraic quotients of open subsets of X from the existence of quotients of the corresponding open subsets of X'. The method used to construct moduli spaces of morphisms is extended in this paper to complexes of decomposables sheaves.

Our aim in this paper is to show that the set $G_j(A)\cup \{0\}$, where $G_j(A)$ is the set of all J-invertible elements of a topological Jordan algebra $A$ without J-g.t.d.z, is isomorphic to $\mathbb{C}$ in the complex case. In the real case it is a $p$-normed quadratic J-division subalgebra of $A$. This result extends a well-known result of W. Żelazko ([11]) in the associative case to Jordan algebras.

In this paper, via the study of the modifications of vector bundles on the Fargues-Fontaine curve, we prove a geometric formula relating the Lubin-Tate towers with the simple basic unramified Rapoport-Zink spaces of EL type of signature $ (1, n-1), (p_1, q_1), \cdots, (p_k, q_k) $ where $ p_iq_i = 0 $. In particular, we deduce the computation of the cohomology groups of the latter., in French

Pre-print of a publication in "Annales math\'ematiques du Qu{\'e}bec". Let $k$ be a totally real number field and let $k_\infty$ be its cyclotomic $\mathbb{Z}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement r\'eels" (Annales Math\'ematiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_\infty$ ; indeed, this conjecture (on the nullity of the invariants $\lambda$ et $\mu$ of Iwasawa) depends of images in the torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, thus of Artin symbols in a finite extension $F/k$ obtained by Galois descent of ${\mathcal T}_k$. An assumption of distribution of these norms implies $\lambda=\mu=0$. Several statistics and numerical examples in the quadratic case confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg's conjecture in the sole context of Iwasawa's theory., Comment: in French, Completes with numerical computations and heuristics our previous paper arXiv:1611.09592 on Greenberg's conjecture. Annales math{\'e}matiques du Quebec, A para{\^i}tre