Publications

2008

• An explicit stationary phase formula for the local formal Fourier-Laplace transform
  
  • Sabbah Claude
  , 2008, pp.309-330. We give an explicit formula (i.e., a formal stationary phase formula) for the local Fourier-Laplace transform of a formal germ of meromorphic connection of one complex variable with a possibly irregular singularity. This is a complex analogue of the formulas in the preprint math/0702436v1.
• A Proof of Gromov's Algebraic Lemma
  
  • Burguet David
  Israel Journal of Mathematics, Springer, 2008, 168 (1), pp.291-316. Following the analysis of differentiable mappings of Y. Yomdin, M. Gromov has stated a very elegant "Algebraic Lemma" which says that the "differentiable size" of an algebraic subset may be bounded in terms only of its dimension, degree and diameter. We give a complete and elementary proof of Gromov's result using the ideas presented in his Bourbaki talk as well as other necessary ingredients. (10.1007/s11856-008-1069-z)
  DOI : 10.1007/s11856-008-1069-z
• Entropy methods for the Boltzman Equation
  
  • Olla Stefano
  • Golse François
  • Villani Cédric
  • Rezakhanlou Fraydoun
  , 2008, pp.107. (10.1007/978-3-540-73705-6)
  DOI : 10.1007/978-3-540-73705-6
• The six operations for sheaves on Artin stacks I: Finite Coefficients
  
  • Laszlo Yves
  • Olsson Martin
  Publications mathematiques de l' IHES, IHES, 2008, 107, pp.109-168. In this paper we develop a theory of Grothendieck's six operations of lisse-étale constructible sheaves on Artin stacks of finite type over $S$, or more generally for a slightly more general class of stacks, called \emph{nice} stacks, which are not necessarily quasi--compact.
• Stochastic 3D Navier-Stokes equations in a thin domain and its α -approximation
  
  • Chueshov Igor
  • Kuksin Sergei
  Physica D: Nonlinear Phenomena, Elsevier, 2008, 237 (10-12), pp.1352-1367. In the thin domain Oε=T2×(0,ε), where T2 is a two-dimensional torus, we consider the 3D Navier-Stokes equations, perturbed by a white in time random force, and the Leray αα-approximation for this system. We study ergodic properties of these models and their connection with the corresponding 2D models in the limit ε→0. In particular, under natural conditions concerning the noise we show that in some rigorous sense the 2D stationary measure μμ comprises asymptotical in time statistical properties of solutions for the 3D Navier-Stokes equations in Oε, when ε≪1. (10.1016/j.physd.2008.03.012)
  DOI : 10.1016/j.physd.2008.03.012
• Random Kick-Forced 3D Navier-Stokes Equations in a Thin Domain
  
  • Chueshov Igor
  • Kuksin Sergei
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2008, 188 (1), pp.117-153. We consider the Navier-Stokes equations in the thin 3D domain T2×(0,ϵ) , where T2 is a two-dimensional torus. The equation is perturbed by a non-degenerate random kick force. We establish that, firstly, when ε ≪ 1, the equation has a unique stationary measure and, secondly, after averaging in the thin direction this measure converges (as ε → 0) to a unique stationary measure for the Navier-Stokes equation on T2 . Thus, the 2D Navier-Stokes equations on surfaces describe asymptotic in time, and limiting in ε, statistical properties of 3D solutions in thin 3D domains. (10.1007/s00205-007-0068-2)
  DOI : 10.1007/s00205-007-0068-2
• Khasminskii--Whitham averaging for randomly perturbed KdV equation
  
  • Piatnitski Andrey L.
  • Kuksin Sergei
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2008, 89 (4), pp.400-428. We consider the damped-driven KdV equation $$ \dot u-\nu{u_{xx}}+u_{xxx}-6uu_x=\sqrt\nu \eta(t,x), x\in S^1, \int u dx\equiv \int\eta dx\equiv0, $$ where $0<\nu\le1$ and the random process $\eta$ is smooth in $x$ and white in $t$. For any periodic function $u(x)$ let $ I=(I_1,I_2,...) $ be the vector, formed by the KdV integrals of motion, calculated for the potential $u(x)$. We prove that if $u(t,x)$ is a solution of the equation above, then for $0\le t\lesssim\nu^{-1}$ and $\nu\to0$ the vector $ I(t)=(I_1(u(t,\cdot)),I_2(u(t,\cdot)),...) $ satisfies the (Whitham) averaged equation. (10.1016/j.matpur.2007.12.003)
  DOI : 10.1016/j.matpur.2007.12.003
• Infinite Töplitz-Lipschitz matrices and operators
  
  • Eliasson Hakan
  • Kuksin Sergei
  Zeitschrift für Angewandte Mathematik und Physik = Journal of Applied mathematics and physics = Journal de mathématiques et de physique appliquées, Springer Verlag, 2008, 59 (1), pp.24-50. We introduce a class of infinite matrices (Ass′,s,s′∈Zd) , which are asymptotically (as |s| + |s′| → ∞) close to Hankel-Töplitz matrices. We prove that this class forms an algebra, and that flow-maps of nonautonomous linear equations with coefficients from the class also belong to it. (10.1007/s00033-006-6030-6)
  DOI : 10.1007/s00033-006-6030-6
• Histoire d'un vecteur tricentenaire.
  
  • Guichardet Alain
  Gazette des Mathématiciens, Société Mathématique de France, 2008, 117, pp.23-33.
• Suites de Sturm, indice de Maslov et périodicité de Bott
  
  • Barge Jean
  • Lannes Jean
  , 2008, pp.199. (10.1007/978-3-7643-8710-5)
  DOI : 10.1007/978-3-7643-8710-5
• The Abelian/Nonabelian Correspondence and Frobenius Manifolds
  
  • Ciocan-Fontanine Ionut
  • Kim Bumsig
  • Sabbah Claude
  Inventiones Mathematicae, Springer Verlag, 2008, pp.301-343. We propose an approach via Frobenius manifolds to the study (began in math.AG/0407254) of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding ''abelianized'' quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonabelian quotients are partial flag manifolds.
• Classes d'homotopie de fractions rationnelles
  
  • Cazanave Christophe
  Comptes Rendus. Mathématique, Académie des sciences (Paris), 2008, 346 (3-4), pp.129-133. Soient k un corps de caractéristique différente de 2 et n⩾1 un entier ; on munit l'ensemble des classes d'homotopie " algébrique " de fractions rationnelles pointées de degré n à coefficients dans k d'une structure de monoïde gradué par n et l'on construit un isomorphisme entre ce monoïde et celui des orbites sous l'action de SLn(k) de formes bilinéaires symétriques non dégénérées sur kn, muni de la somme orthogonale. (10.1016/j.crma.2008.01.004)
  DOI : 10.1016/j.crma.2008.01.004
• Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances.
  
  • Sjostrand Johannes
  Open Mathematics Journal, Bentham Open, 2008, pp.35-81.
• Inverse zero-sum problems and algebraic invariants
  
  • Girard Benjamin
  Acta Arithmetica, Instytut Matematyczny PAN, 2008, 135 (3), pp.231-246. In this article, we study the maximal cross number of long zero-sumfree sequences in a finite Abelian group. Regarding this inverse-type problem, we formulate a general conjecture and prove, among other results, that this conjecture holds true for finite cyclic groups, finite Abelian p-groups and for finite Abelian groups of rank two. Also, the results obtained here enable us to improve, via the resolution of a linear integer program, a result of W. Gao and A. Geroldinger concerning the minimal number of elements with maximal order in a long zero-sumfree sequence of a finite Abelian group of rank two. (10.4064/aa135-3-3)
  DOI : 10.4064/aa135-3-3
• Rational invariant tori, phase space tunneling, and spectra for non-selfadjoint operators in dimension 2
  
  • Sjostrand Johannes
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2008, pp.511-571.
• Semi-classical approach for Anosov diffeomorphisms and Ruelle resonances
  
  • Faure Frederic
  • Roy Nicolas
  • Sjöstrand Johannes
  Open math journal, 2008, 1, pp.35--81. In this paper, we show that some spectral properties of Anosov diffeomorphisms can be obtained by semi-classical analysis. In particular the Ruelle resonances which are eigenvalues of the Ruelle transfer operator acting in suitable anisotropic Sobolev spaces and which govern the decay of dynamical correlations, can be treated as the quantum resonances of open quantum systems in the Aguilar-Baslev-Combes theory or the more recent Helffer-Sjostrand phase-space theory.
• Valuations and Plurisubharmonic Singularities
  
  • Boucksom Sébastien
  • Favre Charles
  • Jonsson Mattias
  Publications of the Research Institute for Mathematical Sciences, European Mathematical Society, 2008, 44 (2), pp.449-494. We extend to higher dimensions some of the valuative analysis of singularities of plurisubharmonic (psh) functions developed by the first two authors. Following Kontsevich and Soibelman we describe the geometry of the space V of all normalized valuations on C[x1,. .. , xn] centered at the origin. It is a union of simplices naturally endowed with an affine structure. Using relative positivity properties of divisors living on modifications of C n above the origin, we define formal psh functions on V, designed to be analogues of the usual psh functions. For bounded formal psh functions on V, we define a mixed Monge-Ampère operator which reflects the intersection theory of divisors above the origin of C n. This operator associates to any (n − 1)-tuple of formal psh functions a positive measure of finite mass on V. Next, we show that the collection of Lelong numbers of a given germ u of a psh function at all infinitely near points induces a formal psh functionûfunctionˆfunctionû on V. When ϕ is a psh Hölder weight in the sense of Demailly, the generalized Lelong number νϕ(u) equals the integral ofûofˆofû against the Monge-Ampère measure ofˆϕofˆ ofˆϕ. In particular, any generalized Lelong number is an average of valuations. We also show how to compute the multiplier ideal of u and the relative type of u with respect to ϕ in the sense of Rashkovskii, in terms ofûofˆofû andˆϕandˆ andˆϕ. (10.2977/prims/1210167334)
  DOI : 10.2977/prims/1210167334
• Quantum cohomology of the Grassmannian and alternate Thom-Sebastiani
  
  • Kim Bumsig
  • Sabbah Claude
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2008, pp.221-246. We introduce the notion of alternate product of Frobenius manifolds and we give, after [math.AG/0610265], an interpretation of the Frobenius manifold structure canonically attached to the quantum cohomology of G(r,n+1) in terms of alternate products. We also investigate the relationship with the alternate Thom-Sebastiani product of Laurent polynomials.
• On Distribution of Energy and Vorticity for Solutions of 2D Navier-Stokes Equation with Small Viscosity
  
  • Kuksin Sergei
  Communications in Mathematical Physics, Springer Verlag, 2008, 284 (2), pp.407-424. We study distributions of some functionals of space-periodic solutions for the randomly perturbed 2D Navier-Stokes equation, and of their limits when the viscosity goes to zero. The results obtained give explicit information on distribution of the velocity field of space-periodic turbulent 2D flows. (10.1007/s00220-008-0577-3)
  DOI : 10.1007/s00220-008-0577-3
• Espaces vectoriels de dimension finie et représentations de de Rham
  
  • Colmez Pierre
  Asterisque, Société Mathématique de France, 2008, 319, pp.117-186.
• Représentations triangulines de dimension 2
  
  • Colmez Pierre
  Asterisque, Société Mathématique de France, 2008, 319, pp.213-258.
• Sur le nombre d'éléments exceptionnels d'une base additive
  
  • Plagne Alain
  Journal für die reine und angewandte Mathematik, Walter de Gruyter, 2008, 616, pp.47-65. In a paper recently published in Crelle's Journal, Deschamps and Grekos study asymptotically (when h tends to infinity) the quantity E(h), introduced by Erdős and Graham, and defined as the maximal number of elements which are necessary to the basicity of an additive basis of order h. They show that the maximal order of this function is (h/log h)^(1/2). The aim of this article is to show that the E function does not have oscillations but, to the contrary, does possess a regular asymptotic behaviour, that we determine explicitly. More precisely, we prove that E(h) ~ 2(h/log h)^(1/2).
• Tunnel effect for Kramers-Fokker-Planck type operators.
  
  • Sjostrand Johannes
  Annales Henri Poincaré, Springer Verlag, 2008, pp.209-274.
• Hommage à André Lichnerowicz (1915-1998)
  
  • Kosmann-Schwarzbach Yvette
  Gazette des Mathématiciens, Société Mathématique de France, 2008, 118, pp.52-56.
• Uniform independence in linear groups
  
  • Breuillard Emmanuel
  • Gelander Tsachik
  Inventiones Mathematicae, Springer Verlag, 2008, 173 (2), pp.225-263. We show that for any finitely generated group of matrices that is not virtually solvable, there is an integer m such that, given an arbitrary finite generating set for the group, one may find two elements a and b that are both products of at most m generators, such that a and b are free generators of a free subgroup. This uniformity result improves the original statement of the Tits alternative. (10.1007/s00222-007-0101-y)
  DOI : 10.1007/s00222-007-0101-y