Publications

2010

• Convergence des polygones de Harder-Narasimhan
  
  • Chen Huayi
  , 2010, pp.120. On reformule la théorie des polygones de Harder-Narasimhan par le langage des $\mathbb R$-filtrations. En utlisant une variante du lemme de Fekete et un argument combinatoire des monômes, on établit la convergence uniforme des polygones associés à une algèbre graduée munie des filtrations. Cela conduit à l'existence de plusieur invariants arithmétiques dont un cas très particulier est la capacité sectionnelle. Deux applications de ce résultat dans la géométrie d'Arakelov sont abordées~: le théorème de Hilbert-Samuel arithmétique ainsi que l'existence et l'interprétation géométrique de la pente maximale asymptotique.
• Examples of $\mathcal{C}^r$ interval map with large symbolic extension entropy
  
  • Burguet David
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2010, 26 (3), pp.873-899. For any integer $r\geq2$ and any real $\epsilon>0$, we construct an explicit example of $\mathcal{C}^r$ interval map $f$ with symbolic extension entropy $h_{sex}(f)\geq\frac{r}{r-1}\log\|f'\|_{\infty}-\epsilon$ and $\|f'\|_{\infty}\geq 2$. T.Downarawicz and A.Maass \cite{Dow} proved that for $\mathcal{C}^r$ interval maps with $r>1$, the symbolic extension entropy was bounded above by $\frac{r}{r-1}\log\|f'\|_{\infty}$. So our example prove this bound is sharp. Similar examples had been already built by T.Downarowicz and S.Newhouse for diffeomorphisms in higher dimension by using generic arguments on homoclinic tangencies. (10.3934/dcds.2010.26.873)
  DOI : 10.3934/dcds.2010.26.873
• Wermer examples and currents
  
  • Dujardin Romain
  Geometric And Functional Analysis, Springer Verlag, 2010, 20 (2), pp.398-415. In this paper we give the first examples of positive closed currents in $\mathbb{C}^2$ with continuous potentials, vanishing self-intersection, and which are not laminar. More precisely, they are supported on sets "without analytic structure". The result is mostly interesting when the potential has regularity close to $C^2$, because laminarity is expected to hold in that case. We actually construct examples which are $C^{1,\alpha}$ for all $\alpha<1$. (10.1007/s00039-010-0066-7)
  DOI : 10.1007/s00039-010-0066-7
• Vey theorem in infinite dimensions and its application to KdV
  
  • Kuksin Sergei
  • Perelman Galina
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2010, 27 (1), pp.1-24. We consider an integrable infinite-dimensional Hamiltonian system in a Hilbert space $H=\{u=(u_1^+,u_1^-; u_2^+,u_2^-;....)\}$ with integrals $I_1, I_2,...$ which can be written as $I_j={1/2}|F_j|^2$, where $F_j:H\to \R^2$, $F_j(0)=0$ for $j=1,2,...$ . We assume that the maps $F_j$ define a germ of an analytic diffeomorphism $F=(F_1,F_2,...):H\to H$, such that dF(0)=id$, $(F-id)$ is a $\kappa$-smoothing map ($\kappa\geq 0$) and some other mild restrictions on $F$ hold. Under these assumptions we show that the maps $F_j$ may be modified to maps $F_j^\prime$ such that $F_j-F_j^\prime=O(|u|^2)$ and each $\frac12|F'_j|^2$ still is an integral of motion. Moreover, these maps jointly define a germ of an analytic symplectomorphism $F^\prime: H\to H$, the germ $(F^\prime-id)$ is $\kappa$-smoothing, and each $I_j$ is an analytic function of the vector $(\frac12|F'_j|^2,j\ge1)$. Next we show that the theorem with $\kappa=1$ applies to the KdV equation. It implies that in the vicinity of the origin in a functional space KdV admits the Birkhoff normal form and the integrating transformation has the form 'identity plus a 1-smoothing analytic map'. (10.3934/dcds.2010.27.1)
  DOI : 10.3934/dcds.2010.27.1
• The Optimal Partial Transport Problem
  
  • Figalli Alessio
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2010, 195 (2), pp.533-560. (10.1007/s00205-008-0212-7)
  DOI : 10.1007/s00205-008-0212-7
• Holomorphic self-maps of singular rational surfaces
  
  • Favre Charles
  Publicacions Matemàtiques, Universitat Autònoma de Barcelona, 2010, 54 (2), pp.389-432. We give a new proof of the classification of normal singular surface germs admitting non-invertible holomorphic self-maps and due to J. Wahl. We then draw an analogy between the birational classification of singular holomorphic foliations on surfaces, and the dynamics of holomorphic maps. Following this analogy, we introduce the notion of minimal holomorphic model for holomorphic maps. We give sufficient conditions which ensure the uniqueness of such a model. (10.5565/PUBLMAT_54210_06)
  DOI : 10.5565/PUBLMAT_54210_06
• Projections in several complex variables
  
  • Hsiao Chin-Yu
  Mémoires de la Société Mathématique de France. Nouvelle Série, 2010, 123, pp.131 pp. This work consists two parts. In the first part, we completely study the heat equation method of Menikoff-Sjostrand and apply it to the Kohn Laplacian defined on a compact orientable connected CR manifold. We then get the full asymptotic expansion of the Szego projection for (0,q) forms when the Levi formis nondegenerate. This generalizes a result of Boutet de Monvel and Sjostrand for (0,0) forms. Our main tool is Fourier integral operators with complex valued phase functions of Melin and Sjostrand. 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 de Monvel and Sjostrand 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 of Menikoff and Sjostrand to this operator. We obtain a description of a new Szego projection up to smoothing operators. Finally, by using the Poisson operator, we get our main result.
• Inverse zero-sum problems in finite Abelian p-groups
  
  • Girard Benjamin
  Colloquium Mathematicum, 2010, 120, pp.7-21. In this paper, we study the minimal number of elements of maximal order within a zero-sumfree sequence in a finite Abelian p-group. For this purpose, in the general context of finite Abelian groups, we introduce a new number, for which lower and upper bounds are proved in the case of finite Abelian p-groups. Among other consequences, the method that we use here enables us to show that, if we denote by exp(G) the exponent of the finite Abelian p-group G which is considered, then a zero-sumfree sequence S with maximal possible length in G must contain at least exp(G)-1 elements of maximal order, which improves a previous result of W. Gao and A. Geroldinger. (10.4064/cm120-1-2)
  DOI : 10.4064/cm120-1-2
• Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds
  
  • Bordeaux Montrieux William
  • Sjostrand Johannes
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2010, 19 (3-4), pp.567-587. (10.5802/afst.1257)
  DOI : 10.5802/afst.1257
• Asymptotically flat conformal structures.
  
  • Vassal Guillaume
  Communications in Mathematical Physics, Springer Verlag, 2010, pp.503-529.
• Loi de Weyl presque sûre pour des opérateurs différentiels non-autoadjoints et estimations de résolvante près du bord de l'image du symbole
  
  • Bordeaux Montrieux William
  Séminaire Équations aux Dérivées Partielles. École polytechnique, 2010, 2008-2009 (Exp. no. IV), pp.18 pp.
• Entropy of semiclassical measures in dimension 2
  
  • Riviere Gabriel
  Duke Mathematical Journal, Duke University Press, 2010, 155 (2), pp.271-335. We study the asymptotic properties of eigenfunctions of the Laplacian in the case of a compact Riemannian surface of Anosov type. We show that the Kolmogorov-Sinai entropy of a semiclassical measure for the geodesic flow is bounded from below by half of the Ruelle upper bound (10.1215/00127094-2010-056)
  DOI : 10.1215/00127094-2010-056
• Dirac induction for Harish-Chandra modules
  
  • Renard David
  • Pandžić Pavle
  Journal of Lie Theory, Heldermann Verlag, 2010, 20 (4), pp.617-641.
• Unitary dual of ${\rm GL}(n)$ at Archimedean places and global Jacquet-Langlands correspondence.
  
  • Badulescu Alexandru Ioan
  • Renard David
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2010, 146 (5), pp.1115-1164. In a paper by Badulescu, results on the global Jacquet-Langlands correspondence, (weak and strong) multiplicity-one theorems and the classification of automorphic representations for inner forms of the general linear group over a number field were established, under the assumption that the local inner forms are split at archimedean places. In this paper, we extend the main local results of that article to archimedean places so that the above condition can be removed. Along the way, we collect several results about the unitary dual of general linear groups over R, C or H which are of independent interest.
• A note on Elkin's improvement of Behrend's construction
  
  • Green Ben
  • Wolf Julia
  , 2010, pp.141-144. We provide a short proof of a recent result of Elkin in which large subsets of the integers 1 up to N free of 3-term progressions are constructed.
• Extrema of low eigenvalues of the Dirichlet-Neumann Laplacian on a disk
  
  • Legendre Eveline
  Canadian Journal of Mathematics = Journal Canadien de Mathématiques, University of Toronto Press, 2010, 62 (4), pp.808-826. We study extrema of the first and the second mixed eigenvalues of the Laplacian on the disk among some families of Dirichlet-Neumann boundary conditions. We show that the minimizer of the second eigenvalue among all mixed boundary conditions lies in a compact 1-parameter family for which an explicit description is given. Moreover, we prove that among all partitions of the boundary with bounded number of parts on which Dirichlet and Neumann conditions are imposed alternately, the first eigenvalue is maximized by the uniformly distributed partition. (10.4153/CJM-2010-042-8)
  DOI : 10.4153/CJM-2010-042-8
• Représentations de GL2(Qp) et (φ,Γ)-modules
  
  • Colmez Pierre
  Asterisque, Société Mathématique de France, 2010, 330, pp.281-509.
• The minimum number of monochromatic 4-term progressions in $\mathbb(Z)_p$
  
  • Wolf Julia
  Journal of Combinatorics, International Press, 2010, 1 (1), pp.53-68.
• Groups and Symmetries. From Finite Groups to Lie Groups
  
  • Kosmann-Schwarzbach Yvette
  , 2010, pp.194. (10.1007/978-0-387-78866-1)
  DOI : 10.1007/978-0-387-78866-1
• On the Boltzmann-Grad limit for the two dimensional periodic Lorentz gas
  
  • Caglioti Emanuele
  • Golse François
  Journal of Statistical Physics, Springer Verlag, 2010, 141 (2), pp.264-317. The two-dimensional, periodic Lorentz gas, is the dynamical system corresponding with the free motion of a point particle in a planar system of fixed circular obstacles centered at the vertices of a square lattice in the Euclidian plane. Assuming elastic collisions between the particle and the obstacles, this dynamical system is studied in the Boltzmann-Grad limit, assuming that the obstacle radius r and the reciprocal mean free path are asymptotically equivalent small quantities, and that the particle's distribution function is slowly varying in the space variable. In this limit, the periodic Lorentz gas cannot be described by a linear Boltzmann equation (see [F. Golse, Ann. Fac. Sci. Toulouse 17 (2008), 735--749]), but involves an integro-differential equation conjectured in [E. Caglioti, F. Golse, C.R. Acad. Sci. Ser. I Math. 346 (2008) 477--482] and proved in [J. Marklof, A. Stroembergsson, preprint arXiv:0801.0612], set on a phase-space larger than the usual single-particle phase-space. The main purpose of the present paper is to study the dynamical properties of this integro-differential equation: identifying its equilibrium states, proving a H Theorem and discussing the speed of approach to equilibrium in the long time limit. In the first part of the paper, we derive the explicit formula for a transition probability appearing in that equation following the method sketched in [E. Caglioti, F. Golse, loc. cit.]. (10.1007/s10955-010-0046-1)
  DOI : 10.1007/s10955-010-0046-1
• Equidistribution of Dense Subgroups on Nilpotent Lie Groups
  
  • Breuillard Emmanuel
  Ergodic Theory and Dynamical Systems, Cambridge University Press (CUP), 2010, 30 (1), pp.131-150. Let $\Gamma$ be a dense subgroup of a simply connected nilpotent Lie group $G$ generated by a finite symmetric set $S$. We consider the $n$-ball $S_n$ for the word metric induced by $S$ on $\Gamma$. We show that $S_n$ (with uniform measure) becomes equidistributed on $G$ with respect to the Haar measure as n tends to infinity. We give rates and also prove the analogous result for random walk averages (i.e. the local limit theorem). (10.1017/S0143385709000091)
  DOI : 10.1017/S0143385709000091
• Density of vector bundles periodic under the action of Frobenius
  
  • Ducrohet Laurent
  • Mehta Vikram
  Bulletin des Sciences Mathématiques, Elsevier, 2010, 134 (5), pp.454-460. (10.1016/j.bulsci.2009.11.001)
  DOI : 10.1016/j.bulsci.2009.11.001
• Compatible structures on Lie algebroids and Monge-Ampére operators
  
  • Kosmann-Schwarzbach Yvette
  • Rubtsov Vladimir
  Acta Applicandae Mathematicae, Springer Verlag, 2010, 109 (1), pp.101-135. We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Ampére structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dimensional manifolds, such Monge-Ampére operators give rise to generalized complex structures or generalized product structures on the cotangent bundle of the manifold. (10.1007/s10440-009-9444-2)
  DOI : 10.1007/s10440-009-9444-2
• KAM for the Non-Linear Schrödinger Equation
  
  • Eliasson Hakan
  • Kuksin Sergei
  Annals of Mathematics, Princeton University, Department of Mathematics, 2010, 172 (1), pp.371-435. We consider the $d$-dimensional nonlinear Schrödinger equation under periodic boundary conditions: $-i\dot u=-\Delta u+V(x)*u+\ep \frac{\p F}{\p \bar u}(x,u,\bar u), \quad u=u(t,x), x\in\T^d $ where $V(x)=\sum \hat V(a)e^{i\sc{a,x}}$ is an analytic function with $\hat V$ real, and $F$ is a real analytic function in $\Re u$, $\Im u$ and $x$. (This equation is a popular model for the 'real' NLS equation, where instead of the convolution term $V*u$ we have the potential term $Vu$.) For $\ep=0$ the equation is linear and has time--quasi-periodic solutions $u$, $$ u(t,x)=\sum_{a\in Å}\hat u(a)e^{i(|a|^2+\hat V(a))t}e^{i\sc{a,x}} \quad (|\hat u(a)|>0), $$ where $Å$ is any finite subset of $\Z^d$. We shall treat $\omega_a=|a|^2+\hat V(a)$, $a\inÅ$, as free parameters in some domain $U\subset\R^{Å}$. This is a Hamiltonian system in infinite degrees of freedom, degenerate but with external parameters, and we shall describe a KAM-theory which, under general conditions, will have the following consequence: If $|\ep|$ is sufficiently small, then there is a large subset $U'$ of $U$ such that for all $\omega\in U'$ the solution $u$ persists as a time--quasi-periodic solution which has all Lyapounov exponents equal to zero and whose linearized equation is reducible to constant coefficients. (10.4007/annals.2010.172.371)
  DOI : 10.4007/annals.2010.172.371
• On the vanishing of some non semisimple orbital integrals
  
  • Chenevier Gaëtan
  • Renard David
  Expositiones Math., 2010, 28, pp.276-289. We prove the vanishing of the (possibly twisted) orbital integrals of certain functions on real Lie groups at non semisimple elliptic elements. This applies to Euler-Poincare functions and makes some results of Chenevier and Clozel unconditionnal.