Publications

2008

• Entropie et complexité locale des systèmes dynamiques différentiables
  
  • Burguet David
  , 2008. Dans ce travail nous nous intéressons aux systèmes dynamiques du point de vue de l'entropie. Nous rappellons tout d'abord le formalisme des structures d'entropie introduit par T.Downarowicz. Dans ce cadre on donne en particulier une preuve élémentaire du principe variationnel pour l'entropie de queue et on généralise certaines structures d'entropie aux endomorphismes. Dans un deuxième temps, nous reprenons l'approche semi-algébrique de Y. Yomdin et M. Gromov pour contrôler la dynamique locale des applications de classe $C^r$. On présente une preuve complète du lemme algébrique de Gromov, qui est un point clé de la théorie de Yomdin. Aussi nous déduisons de nouvelles applications dynamiques de cette théorie : d'une part nous bornons l'entropie de queue mesurée en fonction de l'exposant de Lyapounov ; d'autre part nous généralisons une formule due à J.Buzzi pour l'entropie k-dimensionnelle d'un produit d'applications de classe $C^{\infty}$. On s'intéresse enfin à la théorie des extensions symboliques due à M.Boyle et T.Downarowicz pour les applications $C^r$ et affines par morceaux du plan. On exhibe en particulier des exemples de dynamique $C^r$ de l'intervalle ayant une grande entropie d'extension symbolique. Nous donnerons aussi une borne de l'entropie d'extensions symboliques pour les applications affines par morceaux du plan.
• Unique continuation estimates for sums of semiclassical eigenfunctions and null-controllability from cones
  
  • Miller Luc
  , 2008. For all sums of eigenfunctions of a semiclassical Schrödinger operator below some given energy level, this paper proves that the ratio of the L² norm on R^d over the L² norm on any given open set is bounded by exp(C/h) for some positive C in the semiclassical limit h tends to 0. Corresponding estimates on a compact manifold are also given. They generalize the unique continuation estimate of Lebeau, with Jerison, Robbiano and Zuazua, on sums of classical eigenfunctions of the Laplacian on a compact manifold below an eigenvalue threshold as this threshold tends to infinity. The main tools are semiclassical Carleman estimates following Lebeau, Robbiano and Burq with a new semiclassical propagation of smallness argument. For sums of classical Hermite functions, or for sums of classical eigenfunctions of homogeneous polynomial potential wells, similar unique continuation estimates from cones are deduced. They apply to the null-controllability from a cone of the heat semigroups corresponding to these Schrödinger operators, with a sharp cost estimate of fast control, following a new version of the strategy of Lebeau and Robbiano.
• Uniform growth of groups acting on Cartan-Hadamard spaces.
  
  • Besson Gérard
  • Courtois Gilles
  • Gallot Sylvestre
  , 2008. Let $X$ be an $n$-dimensional simply connected manifold of pinched sectional curvature $-a^2 \leq K \leq -1$. There exist a positive constant $C(n,a)$ such that for any finitely generated discrete group $\Gamma$ acting on $X$, then either $\Gamma$ is virtually nilpotent or the algebraic entropy $Ent (\Gamma) \geq C(n,a)$.
• Sur la p-dimension des corps
  
  • Gabber Ofer
  • Orgogozo Fabrice
  Inventiones Mathematicae, Springer Verlag, 2008, 174 (1), pp.47-80. Soient A un anneau local nœthérien hensélien excellent intègre, k son corps résiduel de caractéristique p>0 et K son corps des fractions. Utilisant une technique d'algébrisation due au premier auteur, ainsi que le cas crucial de la dimension un, dû à Kazuya KATÔ, on démontre la formule suivante : cd_p(K) = dim(A) + p-rang(k), si k est séparablement clos et K de caractéristique nulle. (cd_p(K) est la p-dimension cohomologique du groupe de Galois absolu de K, dim(A) la dimension de Krull de l'anneau et p-rang(k) le p-rang du corps k). Un énoncé semblable est également vrai sans ces restrictions sur k et K. (10.1007/s00222-008-0133-y)
  DOI : 10.1007/s00222-008-0133-y
• Twistor Forms on Riemannian Products
  
  • Moroianu Andrei
  • Semmelmann Uwe
  Journal of Geometry and Physics, Elsevier, 2008, 58 (10), pp.1343-1345. We study twistor forms on products of compact Riemannian manifolds and show that they are defined by Killing forms on the factors. The main result of this note is a necessary step in the classification of compact Riemannian manifolds with non-generic holonomy carrying twistor forms. (10.1016/j.geomphys.2008.05.007)
  DOI : 10.1016/j.geomphys.2008.05.007
• Structures conformes asymptotiquement plates
  
  • Vassal Guillaume
  , 2008. In the first part of this article we revisit the theory of weighted spinors on conformal manifolds. In the second part we introduce the notions of asymptotically flat Weyl structures and of associated mass, and we prove a conformal version of the positive mass theorem on conformal spin manifolds.
• On the slowing down of charged particles in a binary stochastic mixture
  
  • Clouet Jean-François
  • Golse François
  • Sentis Remi
  • Puel Marjolaine
  Kinetic and Related Models, AIMS, 2008, 1 (3), pp.387-404. A kinetic equation is addressed for the straight line slowing-down of charged particles, the geometrical domain consists of randomly distributed spherical grains of dense material imbedded in a light material. The dense material is assumed to be a Boolean medium (the sphere centers are sampled according to a Poisson random field). We focus on the fraction of particles $P$ which stop in the light medium. After setting some properties of the Boolean medium, we perform an asymptotic analysis in two extreme cases corresponding to grain radius very small and very large with respect to the stopping distance of the dense material. A fitted analytic formula is proposed for the quantity P and results of numerical simulations are presented in order to validate the proposed formula. (10.3934/krm.2008.1.387)
  DOI : 10.3934/krm.2008.1.387
• Vector partition functions and index of transversally elliptic operators
  
  • de Concini Corrado
  • Procesi Claudio C.
  • Vergne Michele
  , 2008. Let G be a torus acting linearly on a complex vector space M, and let X be the list of weights of G in M. We determine the equivariant K-theory of the open subset of M consisting of points with finite stabilizers. We identify it to the space DM(X) of functions on the lattice of weights of G, satisfying the cocircuit difference equations associated to X, introduced by Dahmen--Micchelli in the context of the theory of splines in order to study vector partition functions. This allows us to determine the range of the index map from G-transversally elliptic operators on M to generalized functions on G and to prove that the index map is an isomorphism on the image. This is a setting studied by Atiyah-Singer which is in a sense universal for index computations.
• Vector partition function and generalized Dahmen-Micchelli spaces
  
  • de Concini Corrado
  • Procesi Claudio C.
  • Vergne Michèle
  , 2008. This is the first of two papers on partition functions and the index theory of transversally elliptic operators. In this paper we only discuss algebraic and combinatorial issues related to partition functions. The applications to index theory will appear in a subsequent paper.
• Computing volume function on projective bundle over a curve
  
  • Chen Huayi
  , 2008. We establish an explicit link between the volume function on a projective variety fibered on a curve and the asymptotic behaviour of the canonical filtration of direct images. As an application, we calculate explicitly the volume function on projective bundle over a curve.
• Projections in several complex variables
  
  • Hsiao Chin-Yu
  , 2008. This thesis consists two parts. In the first part, we completely study the heat equation method ofMenikoff-Sjöstrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szegö projection for (0,q) forms when the Levi formis nondegenerate. This generalizes a result of Boutet de Monvel and Sjöstrand for (0,0) forms. Our main tool is Fourier integral operators with complex valued phase functions of Melin and Sjöstrand. In the second part, we obtain the full asymptotic expansion of the Bergman projection for (0,q) forms when the Levi form is non-degenerate. This also generalizes a result of Boutet deMonvel and Sjöstrand for (0,0) forms. We introduce a new operator analogous to the Kohn Laplacian defined on the boundary of a domain and we apply the heat equation method ofMenikoff and Sjöstrand to this operator. We obtain a description of a new Szegö projection up to smoothing operators. Finally, by using the Poisson operator, we get our main result.
• Nonlinear Regularizing Effect for Conservation Laws
  
  • Golse François
  , 2009, Vol. 1: Plenary & Invited Talks, pp.73-92. Compactness of families of solutions --- or of approximate solutions --- is a feature that distinguishes certain classes of nonlinear hyperbolic equations from the case of linear hyperbolic equations, in space dimension one. This paper shows that some classical compactness results in the context of hyperbolic conservation laws, such as the Lax compactness theorem for the entropy solution semigroup associated with a nonlinear scalar conservation laws with convex flux, or the Tartar-DiPerna compensated compactness method, can be turned into quantitative compactness estimates --- in terms of epsilon-entropy, for instance --- or even nonlinear regularization estimates. This regularizing effect caused by the nonlinearity is discussed in detail on two examples: a) the case of a scalar conservation law with convex flux, and b) the case of isentropic gas dynamics, in space dimension one.
• The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow
  
  • Desvillettes Laurent
  • Golse François
  • Ricci Valeria
  Journal of Statistical Physics, Springer Verlag, 2008, 131 (5), pp.941-967. We propose a mathematical derivation of Brinkman's force for a cloud of particles immersed in an incompressible fluid. Our starting point is the Stokes or steady Navier-Stokes equations set in a bounded domain with the disjoint union of N balls of radius 1/N removed, and with a no-slip boundary condition for the fluid at the surface of each ball. The large N limit of the fluid velocity field is governed by the same (Navier-)Stokes equations in the whole domain, with an additional term (Brinkman's force) that is (minus) the total drag force exerted by the fluid on the particle system. This can be seen as a generalization of Allaire's result in [Arch. Rational Mech. Analysis 113 (1991), 209-259] who treated the case of motionless, periodically distributed balls. Our proof is based on slightly simpler, though similar homogenization techniques, except that we avoid the periodicity assumption and use instead the phase-space empirical measure for the particle system. Similar equations are used for describing the fluid phase in various models for sprays. (10.1007/s10955-008-9521-3)
  DOI : 10.1007/s10955-008-9521-3
• A Strong Tits Alternative
  
  • Breuillard Emmanuel
  , 2008. We show that for every integer $d$, there is a constant $N(d)$ such that if $K$ is any field and $F$ is a finite subset of $GL_d(K)$, which generates a non amenable subgroup, then $F^{N(d)}$ contains two elements, which freely generate a non abelian free subgroup. This improves the original statement of the Tits alternative. It also implies a growth gap and a co-growth gap for non-amenable linear groups, and has consequences about the girth and uniform expansion of small sets in finite subgroups of $GL_d(\Bbb{F}_q)$ as well as other diophantine properties of non-discrete subgroups of Lie groups.
• The Boltzmann-Grad limit of the periodic Lorentz gas in two space dimensions
  
  • Caglioti Emanuele
  • Golse François
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2008, 346 (7-8), pp.477-482. The periodic Lorentz gas is the dynamical system corresponding to the free motion of a point particle in a periodic system of fixed spherical obstacles of radius $r$ centered at the integer points, assuming all collisions of the particle with the obstacles to be elastic. In this Note, we study this motion on time intervals of order $1/r$ and in the limit as $r\to 0^+$, in the case of two space dimensions. (10.1016/j.crma.2008.01.016)
  DOI : 10.1016/j.crma.2008.01.016
• THE ROSSELAND LIMIT FOR RADIATIVE TRANSFER IN GRAY MATTER
  
  • Golse François
  • Salvarani Francesco
  , 2008. This paper establishes the Rosseland approximation of the radiative transfer equations in a gray atmosphere, i.e. assuming that the opacity is independent of the radiation frequency.
• Positive degree and arithmetic bigness
  
  • Chen Huayi
  , 2008. We establish, for a generically big Hermitian line bundle, the convergence of truncated Harder-Narasimhan polygons and the uniform continuity of the limit. As applications, we prove a conjecture of Moriwaki asserting that the arithmetic volume function is actually a limit instead of a sup-limit, and we show how to compute the asymptotic polygon of a Hermitian line bundle, by using the arithmetic volume function.
• Paradan's wall crossing formula for partition functions and Khovanski-Pukhlikov differential operator.
  
  • Boysal Arzu
  • Vergne Michele
  , 2008. We give an elementary algebraic proof of Paradan's wall crossing formulae for partition functions. We also express such jumps in volume and partition functions by one dimensional residue formulae. Subsequently we reprove the relation between them as given by the application of a generalized Khovanskii-Pukhlikov differential operator.
• Analysis of the boundary layer equation in the kinetic theory of gases
  
  • Golse François
  Bulletin of the Institute of Mathematics, Academia Sinica (New Series), Institute of Mathematics, Academia sinica, 2008, 3 (1), pp.211-242. The present paper completes an earlier result by S. Ukai, T. Yang, and S.-H. Yu [Commun. Math. Phys. 236 (2003), 373--393] on weakly nonlinear half-space problems for the steady Boltzmann equation with hard-sphere potential.
• Conformally Einstein Products and Nearly Kähler Manifolds
  
  • Moroianu Andrei
  • Ornea Liviu
  Annals of Global Analysis and Geometry, Springer Verlag, 2008, 33 (1), pp.11-18. In the first part of this note we study compact Riemannian manifolds (M,g) whose Riemannian product with R is conformally Einstein. We then consider compact 6--dimensional almost Hermitian manifolds of type W_1+W_4 in the Gray--Hervella classification admitting a parallel vector field and show that (under some regularity assumption) they are obtained as mapping tori of isometries of compact Sasaki-Einstein 5-dimensional manifolds. In particular, we obtain examples of inhomogeneous locally (non-globally) conformal nearly Kähler compact manifolds. (10.1007/s10455-007-9071-y)
  DOI : 10.1007/s10455-007-9071-y
• Solution non universelle pour le problème KV-78
  
  • Albert Luc
  • Harinck Pascale
  • Torossian Charles
  , 2008. In 78' M. Kashiwara and Vergne conjectured some property on the Campbell-Hausdorff series in such way a trace formula is satisfied. They proposed an explicit solution in the case of solvable Lie algebras. In this note we prove that this "solvable solution" is not universal. Our method is based on computer calculation. Furthermore our programs prove up to degree 16, Drinfeld's Lie algebra $\mathfrak{grt}_1$ coincides with the Lie algebra $\widehat{kv_2}$ defined in \cite{AT}.
• Deformations of Nearly Kähler Structures
  
  • Moroianu Andrei
  • Nagy Paul-Andi
  • Semmelmann Uwe
  Pacific Journal of Mathematics, Mathematical Sciences Publishers, 2008, 235 (1), pp.57-72. We study the space of nearly Kähler structures on compact 6-dimensional manifolds. In particular, we prove that the space of infinitesimal deformations of a strictly nearly Kähler structure (with scalar curvature scal) modulo the group of diffeomorphisms, is isomorphic to the space of primitive co-closed (1,1)-eigenforms of the Laplace operator for the eigenvalue 2scal/5.
• Degree growth of meromorphic surface maps
  
  • Boucksom Sébastien
  • Favre Charles
  • Jonsson Mattias
  Duke Mathematical Journal, Duke University Press, 2008, 141 (3), pp.519-538. We study the degree growth of iterates of meromorphic self-maps of compact Kähler surfaces. Using cohomology classes on the Riemann-Zariski space we show that the degrees grow similarly to those of mappings that are algebraically stable on some bimeromorphic model. (10.1215/00127094-2007-004)
  DOI : 10.1215/00127094-2007-004
• Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications
  
  • Hérau Frédéric
  • Hitrik Michael
  • Sjöstrand Johannes
  , 2008. In the first part of this work, we consider second order supersymmetric differential operators in the semiclassical limit, including the Kramers-Fokker-Planck operator, such that the exponent of the associated Maxwellian $\phi$ is a Morse function with two local minima and one saddle point. Under suitable additional assumptions of dynamical nature, we establish the long time convergence to the equilibrium for the associated heat semigroup, with the rate given by the first non-vanishing, exponentially small, eigenvalue. In the second part of the paper, we consider the case when the function $\phi$ has precisely one local minimum and one saddle point. We also discuss further examples of supersymmetric operators, including the Witten Laplacian and the infinitesimal generator for the time evolution of a chain of classical anharmonic oscillators.
• On the non-randomness of modular arthmetic progressions : a solution to a problem by V.I. Arnold
  
  • Cesaratto Eda
  • Plagne Alain
  • Vallée Brigitte
  , 2008. We solve a problem by V. I. Arnold dealing with “how random” modular arithmetic progressions can be. After making precise how Arnold proposes to measure the randomness of a modular sequence, we show that this measure of randomness takes a simplified form in the case of arithmetic progressions. This simplified expression is then estimated using the methodology of dynamical analysis, which operates with tools coming from dynamical systems theory.