Publications

2009

• The Hermitian Laplace Operator on Nearly Kähler Manifolds
  
  • Moroianu Andrei
  • Semmelmann Uwe
  Communications in Mathematical Physics, Springer Verlag, 2009, 294 (1), pp.251-272. The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6-dimensional homogeneous nearly Kähler manifolds. It turns out that the nearly Kähler structure is rigid except for the flag manifold F(1,2)=SU_3/T^2, which carries an 8-dimensional moduli space of infinitesimal nearly Kähler deformations, modeled on the Lie algebra su_3 of the isometry group. (10.1007/s00220-009-0903-4)
  DOI : 10.1007/s00220-009-0903-4
• Chaotic vibrations and strong scars
  
  • Anantharaman Nalini
  • Nonnenmacher Stéphane
  EMS Newsletter, European Mathematical Society, 2009, 74, pp.19-24. This article aims at popularizing some aspects of "quantum chaos", in particular the study of eigenmodes of classically chaotic systems, in the semiclassical (or high frequency) limit.
• A survey on the Theorem of Chekhanov
  
  • Tonnelier Benoit
  , 2009. The theorem of Chekhanov asserts that a Lagrangian submanifold L has positive displacement energy under natural assumptions on the symplectic topology at infinity. It is greater than or equal to the minimal area of holomorphic disks bounded by L. This estimate was obtained by Y.V. Chekhanov in 1998. Section 1 presents a direct proof based on the use of holomorphic curves and their Hamiltonian perturbations. In section 2, we define a filtered version of the Lagrangian Floer homology, without any assumption on L. This is compared with the Morse homology groups, via the continuation maps (subsubsection 2.3.2) or the PSS maps (subsection 3.2).
• Délocalisation des mesures semi-classiques pour des systèmes dynamiques chaotiques
  
  • Riviere Gabriel
  , 2009. Dans cette thèse, on étudie deux paradigmes du chaos quantique: celui des symplectomorphismes linéaires du tore et celui du flot géodésique sur une variété riemannienne compacte. Dans les deux cas, on étudie le problème d'ergodicité quantique associé. Les résultats obtenus sont de deux sortes. D'une part, on obtient des bornes inférieures sur l'entropie des mesures semi-classiques en dimension 2. D'autre part, on obtient des résultats de type grandes déviations semi-classiques en toute dimension.
• Puzzles of Quasi-Finite Type, Zeta Functions and Symbolic Dynamics for Multi-Dimensional Maps
  
  • Buzzi Jerome
  , 2009. We introduce "puzzles of quasi-finite type" which are the counterparts of our subshifts of quasi-finite type (Invent. Math. 159 (2005)) in the setting of combinatorial puzzles as defined in complex dynamics. We are able to analyze these dynamics defined by entropy conditions rather completely, obtaining a complete classification with respect to large entropy measures and a description of their measures with maximum entropy and periodic orbits. These results can in particular be applied to entropy-expanding maps like (x,y)-->(1.8-x^2+sy,1.9-y^2+sx) for small s. We prove in particular the meromorphy of the Artin-Mazur zeta function on a large disk. This follows from a similar new result about strongly positively recurrent Markov shifts where the radius of meromorphy is lower bounded by an "entropy at infinity" of the graph.
• Transformations of locally conformally Kähler manifolds
  
  • Moroianu Andrei
  • Ornea Liviu
  Manuscripta mathematica, Springer Verlag, 2009, 130 (1), pp.93-100. We consider several transformation groups of a locally conformally Kähler manifold and discuss their inter-relations. Among other results, we prove that all conformal vector fields on a compact Vaisman manifold which is neither locally conformally hyperkähler nor a diagonal Hopf manifold are Killing, holomorphic and that all affine vector fields with respect to the minimal Weyl connection of a locally conformally Kähler manifold which is neither Weyl-reducible nor locally conformally hyperkähler are holomorphic and conformal (10.1007/s00229-009-0278-z)
  DOI : 10.1007/s00229-009-0278-z
• A direct Lebeau-Robbiano strategy for the observability of heat-like semigroups
  
  • Miller Luc
  , 2009. This paper generalizes and simplifies abstract results of Miller and Seidman on the cost of fast control/observation. It deduces final-observability of an evolution semigroup from some stationary observability property on some spaces associated to the generator, e.g. spectral subspaces when the semigroup has an integral representation via spectral measures. Contrary to the original Lebeau-Robbiano strategy, it does not have recourse to null-controllability and it yields the optimal bound of the cost when applied to the heat equation, i.e. c' exp(c/T), or to the heat diffusion in potential wells observed from cones, i.e. c' exp(c/T^beta) with optimal beta. It also yields simple upper-bounds for c.
• Nonlinear regularizing effect for hyperbolic partial differential equations
  
  • Golse François
  , 2010, pp.433-437. The Tartar-DiPerna compensated compactness method, used initially to construct global weak solutions of hyperbolic systems of conservation laws for large data, can be adapted in order to provide some regularity estimates on these solutions. This note treats two examples: (a) the case of scalar conservation laws with convex flux, and (b) the Euler system for a polytropic, compressible fluid, in space dimension one.
• Holonomy of tame Weyl Structures
  
  • Belgun Florin
  • Moroianu Andrei
  , 2009. A Weyl structure on a compact conformal manifold is not complete in general. If, however, the life-time of incomplete geodesics can be controlled on compact subsets of the tangent bundle, the Weyl connection is called tame. We prove that every closed, non-exact, tame Weyl structure on a compact conformal manifold is either flat, or has irreducible holonomy, generalizing a classical statement for Riemannian cone metrics by S. Gallot.
• Approche multiéchelle du comportement mécanique des matériaux composites à renfort tissé
  
  • Couégnat Guillaume
  • Martin Éric .
  • Lamon Jacques
  • Carrère Nicolas
  , 2009. Ce travail présente une approche multiéchelle du comportement mécanique des matériaux composites à renfort tissé. Le modèle développé repose sur une description de l'architecture du renfort tissé, de l'arrangement des constituants, de leurs propriétés et de leurs modes d'endommagement. Des outils originaux de modélisation géométrique et de maillage ont été développés afin de prendre en compte les particularités des composites tissés. Le modèle a été identifié et validé pour un composite tissé à matrice céramique et appliqué à des cas-tests de calcul de structure.
• Summing a polynomial function over integral points of a polygon. User's guide.
  
  • Baldoni Velleda
  • Berline Nicole
  • Vergne Michèle
  , 2009. This document is a companion for the Maple program \textbf{Summing a polynomial function over integral points of a polygon}. It contains two parts. First, we see what this programs does. In the second part, we briefly recall the mathematical background.
• The Incompressible Navier-Stokes Limit of the Boltzmann Equation for Hard Cutoff Potentials
  
  • Golse François
  • Saint-Raymond Laure
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2009, 91 (5), pp.508-552. The present paper proves that all limit points of sequences of renormalized solutions of the Boltzmann equation in the limit of small, asymptotically equivalent Mach and Knudsen numbers are governed by Leray solutions of the Navier-Stokes equations. This convergence result holds for hard cutoff potentials in the sense of H. Grad, and therefore completes earlier results by the same authors [Invent. Math. 155, 81-161 (2004)] for Maxwell molecules. (10.1016/j.matpur.2009.01.013)
  DOI : 10.1016/j.matpur.2009.01.013
• Differentiability of volumes of divisors and a problem of Teissier
  
  • Boucksom Sébastien
  • Favre Charles
  • Jonsson Mattias
  Journal of Algebraic Geometry, American Mathematical Society, 2009, 18 (2), pp.279-308. We give an algebraic construction of the positive intersection products of pseudo-effective classes first introduced in [BDPP], and use them to prove that the volume function on the Néron-Severi space of a projective variety is C 1-differentiable, expressing its differential as a positive intersection product. We also relate the differential to the restricted volumes introduced in [ELMNP3, Ta]. We then apply our dif-ferentiability result to prove an algebro-geometric version of the Diskant inequality in convex geometry, allowing us to characterize the equality case of the Khovanskii-Teissier inequalities for nef and big classes. (10.1090/S1056-3911-08-00490-6)
  DOI : 10.1090/S1056-3911-08-00490-6
• Highest coefficients of weighted Ehrhart quasi-polynomials for a rational polytope
  
  • Baldoni Velleda
  • Berline Nicole
  • Vergne Michèle
  , 2009. We describe a method for computing the highest degree coefficients of a weighted Ehrhart quasi-polynomial for a rational simple polytope.
• Frobenius inverse image of the semi-stable boundary in the moduli space of vector bundles
  
  • Ducrohet Laurent
  , 2009. We study stable rank 2 vector bundles with trivial determinant whose Frobenius pull back is non stable over a general curve of genus g>1. In genus 2, we apply recent results about the theta divisor associated to the bundle B of locally exact differential forms and we derive a scheme-theoretic description of the Frobenius inverse image of the semi-stable boundary of the moduli space.
• Spectral deviations for the damped wave equation
  
  • Anantharaman Nalini
  , 2009. We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this typical behaviour; the exponent that appears naturally is the `entropy' that gives the deviation rate from the Birkhoff ergodic theorem for the geodesic flow. A Weyl-type lower bound is still far from reach; but in the particular case of arithmetic surfaces, and for a strong enough damping, we can use the trace formula to prove a result going in this direction.
• Simulation of Laser Beam Propagation With a Paraxial Model in a Tilted Frame
  
  • Doumic Marie
  • Duboc Frédéric
  • Golse François
  • Sentis Rémi
  Journal of Computational Physics, Elsevier, 2009, 228 (3), pp.861--880. We study the Schrödinger equation which comes from the paraxial approximation of the Helmholtz equation in the case where the direction of propagation is tilted with respect to the boundary of the domain. In a first part, a mathematical analysis is made which leads to an analytical formula of the solution in the simple case where the refraction index and the absorption coefficients are constant. Afterwards, we propose a numerical method for solving the initial problem which uses the previous analytical expression. Numerical results are presented. We also sketch an extension to a time dependant model which is relevant for laser plasma interaction. (10.1016/j.jcp.2008.10.009)
  DOI : 10.1016/j.jcp.2008.10.009
• Generalized solutions for the Euler equations in one and two dimensions
  
  • Bernot Marc
  • Figalli Alessio
  • Santambrogio Filippo
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2009, 91 (2), pp.137-155. In this paper we study generalized solutions (in the Brenier's sense) for the Euler equations. We prove that uniqueness holds in dimension one whenever the pressure field is smooth, while we show that in dimension two uniqueness is far from being true. In the case of the two-dimensional disc we study solutions to Euler equations where particles located at a point $x$ go to $-x$ in a time $\pi$, and we give a quite general description of the (large) set of such solutions. As a byproduct, we can construct a new class of classical solutions to Euler equations in the disc.
• Weyl-parallel Forms and conformal Products
  
  • Belgun Florin
  • Moroianu Andrei
  , 2009. Motivated by the study of Weyl structures on conformal manifolds admitting parallel weightless forms, we define the notion of conformal product of conformal structures and study its basic properties. We obtain a classification of Weyl manifolds carrying parallel forms, and we use it to describe in further detail the holonomy of the adapted Weyl connection on conformal products.
• Corps de nombres peu ramifies et formes automorphes autoduales
  
  • Chenevier Gaëtan
  • Clozel Laurent
  Journal of the American Mathematical Society, American Mathematical Society, 2009, 22 (2), pp.467-519. Let S be a finite set of primes, p in S, and Q_S a maximal algebraic extension of Q unramified outside S and infinity. Assume that |S|>=2. We show that the natural maps Gal(Q_p^bar/Q_p) --> Gal(Q_S/Q) are injective. Much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of GL(2n,A_Q) with prescribed properties at all places, that we study via the twisted trace formula of J. Arthur. The techniques we develop shed also some lights on the orthogonal/symplectic alternative for selfdual representations of GL(2n).
• Vibrations chaotiques et grosses balafres
  
  • Anantharaman Nalini
  • Nonnenmacher Stéphane
  Gazette des Mathématiciens, Société Mathématique de France, 2009, 119, pp.16-32.
• Geodesics in the space of measure-preserving maps and plans
  
  • Ambrosio Luigi
  • Figalli Alessio
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2009, 194 (2), pp.421-462. We study Brenier's variational models for incompressible Euler equations. These models give rise to a relaxation of the Arnold distance in the space of measure-preserving maps and, more generally, measure-preserving plans. We analyze the properties of the relaxed distance, we show a close link between the Lagrangian and the Eulerian model, and we derive necessary and sufficient optimality conditions for minimizers. These conditions take into account a modified Lagrangian induced by the pressure field. Moreover, adapting some ideas of Shnirelman, we show that, even for non-deterministic final conditions, generalized flows can be approximated in energy by flows associated to measure-preserving maps. (10.1007/s00205-008-0189-2)
  DOI : 10.1007/s00205-008-0189-2
• On Reducibility of Schrödinger Equations with Quasiperiodic in Time Potentials
  
  • Eliasson Hakan
  • Kuksin Sergei
  Communications in Mathematical Physics, Springer Verlag, 2009, 286 (1), pp.125-135. We prove that a linear d-dimensional Schrödinger equation with an x-periodic and t-quasiperiodic potential reduces to an autonomous equation for most values of the frequency vector. The reduction is made by means of a non-autonomous linear transformation of the space of x-periodic functions. This transformation is a quasiperiodic function of t. (10.1007/s00220-008-0683-2)
  DOI : 10.1007/s00220-008-0683-2
• Laure Saint-Raymond reçoit un prix de la Société Mathématiques Européenne
  
  • Gallagher Isabelle
  • Golse François
  Gazette des Mathématiciens, Société Mathématique de France, 2009, 120, pp.82-84.
• Perverse t-structure on Artin stacks
  
  • Laszlo Yves
  • Olsson Martin
  Mathematische Zeitschrift, Springer, 2009, 261 (4), pp.737-748. In this paper we develop the theory of perverse sheaves on Artin stacks continuing the study in "The six operations for sheaves on Artin stacks I: Finite Coefficients" and "The six operations for sheaves on Artin stacks II: Adic Coefficients" (math.AG/0512097 and math.AG/0603680)