Publications

2011

• Are systems in entangled states emergent systems?
  
  • Paul Thierry
  • Poinat Sebastien
  , 2011. Several authors argued that systems in entangled states are examples of emergent systems. In this paper, we challenge this point of view. We propose a criterion to distinguish composed systems and noncomposed systems, and show that systems in entangled states don't verify this criterion. Because they are not composed, they cannot be examples of emergent systems.
• Quelques contributions à l'analyse mathématique de l'équation de Gross-Pitaevskii et du modèle de Bogoliubov-Dirac-Fock
  
  • Gravejat Philippe
  , 2011. Ce mémoire présente plusieurs contributions quant à l'analyse mathématique de l'équation de Gross-Pitaevskii et du modèle de Bogoliubov-Dirac-Fock. Au sujet de l'équation de Gross-Pitaevskii, l'analyse commence par la construction variationnelle des ondes progressives minimisantes. La preuve de la stabilité orbitale du soliton noir en dimension un, et la description de la limite transsonique des ondes progressives minimisantes vers les états fondamentaux de l'équation de Kadomtsev-Petviashvili en dimension deux, viennent compléter cette construction. L'analyse s'achève par la dérivation rigoureuse du régime ondes longues vers l'équation de Korteweg-de Vries en dimension un. Quant au modèle de Bogoliubov-Dirac-Fock, il s'agit de construire les états fondamentaux du modèle réduit, puis de préciser le processus de renormalisation de leur charge, lequel autorise le calcul d'un développement asymptotique de la densité de charges du vide polarisé, qui est cohérent avec les développements perturbatifs de l'électrodynamique quantique.
• Equations differentielles p-adiques et modules de Jacquet analytiques
  
  • Dospinescu Gabriel
  , 2011. Using differential techniques, we compute the Jacquet module of the locally analytic vectors of irreducible admissible unitary representations of GL_2(\qp). This gives a direct proof of some results of Colmez, leading to a proof of conjectures by Berger, Breuil and Emerton.
• Le Vierge, le vivace et le bel aujourd'hui : trois mouvements de la structure espace vu des équations différentielles
  
  • Paul Thierry
  , 2011. Nous considérons l'incidence sur la notion d'espace sous-jacent de trois types de dynamiques : la théorie KAM, la dynamique donnée par des potentiels peu réguliers et la limite classique de la dynamique à temps long. Dans les trois cas nous montrons qu'une notion naturelle d'espace est associée, que nous présentons comme la notion d'espace sous-jacente, et qui chacune convoque un des trois labels : le Cantor, le presque et le non commutatif.
• KdV Hamiltonian as function of actions
  
  • Korotyaev Evgeny
  • Kuksin Sergei
  , 2011. We prove that the non-linear part of the Hamiltonian of the KdV equation on the circle, written as a function of the actions, defines a continuous convex function on the $\ell^2$ space and derive for it lower and upper bounds in terms of some functions of the $\ell^2$-norm. The proof is based on a new representation of the Hamiltonian in terms of the quasimomentum and its analysis using the conformal mapping theory.
• Existence and non uniqueness of constant scalar curvature toric Sasaki metrics
  
  • Legendre Eveline
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2011, 147 (05), pp.1613-1634. We study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least 5. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using existence result of [25], we show that a co-oriented compact toric contact 5-manifold whose moment cone has 4 facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on $S^2\times S^3$ admitting two non isometric and non transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.
• Semiclassical approximation and noncommutative geometry
  
  • Paul Thierry
  , 2011. We consider the long time semiclassical evolution for the linear Schrödinger equation. We show that, in the case of chaotic underlying classical dynamics and for times up to $\hbar^{-2+\epsilon},\ \epsilon>0$, the symbol of a propagated observable by the corresponding von Neumann-Heisenberg equation is, in a sense made precise below, precisely obtained by the push-forward of the symbol of the observable at time $t=0$. The corresponding definition of the symbol calls upon a kind of Toeplitz quantization framework, and the symbol itself is an element of the noncommutative algebra of the (strong) unstable foliation of the underlying dynamics.
• From the Boltzmann Equation to the Euler Equations in the Presence of Boundaries
  
  • Golse François
  Computers & Mathematics with Applications, Elsevier, 2013, 65 (6), pp.815-830. The fluid dynamic limit of the Boltzmann equation leading to the Euler equations for an incompressible fluid with constant density in the presence of material boundaries shares some important features with the better known inviscid limit of the Navier-Stokes equations. The present paper slightly extends recent results from [C. Bardos, F. Golse, L. Paillard, Comm. Math. Sci., 10 (2012), 159-190] to the case of boundary conditions for the Boltzmann equation more general than Maxwell's accomodation condition. (10.1016/j.camwa.2012.02.009)
  DOI : 10.1016/j.camwa.2012.02.009
• Recent results in semiclassical approximation with rough potentials
  
  • Paul Thierry
  , 2013. This is an extended abstract for the conference "Microlocal2011 : Microlocal Methods in Mathematical Physics and Global Analysis Universität Tübingen, June 14 - 18, 2011
• Precise Arrhenius law for p-forms: The Witten Laplacian and Morse-Barannikov complex.
  
  • Le Peutrec Dorian
  • Nier Francis
  • Viterbo Claude
  , 2011. Accurate asymptotic expressions are given for the exponentially small eigenvalues of Witten Laplacians acting on p-forms. The key ingredient, which replaces explicit formulas for global quasimodes in the case p = 0, is Barannikov's presentation of Morse theory. (10.1007/s00023-012-0193-9)
  DOI : 10.1007/s00023-012-0193-9
• Generating series and asymptotics of classical spin networks
  
  • Costantino Francesco
  • Marche Julien
  , 2011. We study classical spin networks with group SU(2). In the first part, using gaussian integrals, we compute their generating series in the case where the networks are equipped with holonomies; this generalizes Westbury's formula. In the second part, we use an integral formula for the square of the spin network and perform stationary phase approximation under some non-degeneracy hypothesis. This gives a precise asymptotic behavior when the labels are rescaled by a constant going to infinity.
• Analytic continuation of a parametric polytope and wall-crossing
  
  • Berline Nicole
  • Vergne Michèle
  , 2011. We define a set theoretic "analytic continuation" of a polytope defined by inequalities. For the regular values of the parameter, our construction coincides with the parallel transport of polytopes in a mirage introduced by Varchenko. We determine the set-theoretic variation when crossing a wall in the parameter space, and we relate this variation to Paradan's wall-crossing formulas for integrals and discrete sums. As another application, we refine the theorem of Brion on generating functions of polytopes and their cones at vertices. We describe the relation of this work with the equivariant index of a line bundle over a toric variety and Morelli constructible support function.
• Semaine d'Etude Mathématiques et Entreprises 1 : Géométrie des matrices de covariance pour le traitement de signaux radars
  
  • Bernard Etienne
  • Bosché Aurélien
  • Charon Nicolas
  • El Kolei Salima
  • Lapebie Julie
  • Le Masson Etienne
  • Mirebeau Jean-Marie
  • Richard Thomas
  , 2011. Les radars Doppler permettent de détecter des objets volants petits ou de faible signature radar. Thalès propose ici de réfléchir aux techniques permettant de faire ressortir de la masse des données radar celles qui sont "aberrantes" afin de repérer parmi le bruit de fond dû aux milieux environnants (nuages de pluie,...) la trace d'un objet volant.
• High speed excited multi-solitons in nonlinear Schrödinger equations
  
  • Côte Raphaël
  • Le Coz Stefan
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2011, 96 (2), pp.http://dx.doi.org/10.1016/j.matpur.2011.03.004. We consider the nonlinear Schrödinger equation with a general nonlinearity. In dimension higher than 2, this equation admits travelling wave solutions with a fixed profile which is not the ground state. This kind of profiles are called excited states. In this paper, we construct solutions to NLS behaving like a sum of N excited states which spread up quickly as time grows (which we call multi-solitons). We also show that if the flow around one of these excited states is linearly unstable, then the multi-soliton is not unique, and is unstable. (10.1016/j.matpur.2011.03.004)
  DOI : 10.1016/j.matpur.2011.03.004
• Sur le problème de Kepler
  
  • Guichardet Alain
  , 2011. We give a fairly systemetic exposition in a unified language , firstly of classical results, then of recent works due to mathematicains as well as to physicists , with special attention to the properties of the regularization map from the Kepler phase space to the cotangent bundle of the sphere.
• CONVERGENCE OF A QUANTUM NORMAL FORM AND AN EXACT QUANTIZATION FORMULA
  
  • Graffi Sandro
  • Paul Thierry
  , 2011. Let the quantization of the linear flow of diophantine frequencies $\om$ over the torus $\T^l$, $l>1$, namely the Schrödinger operator $-i\hbar\omega\cdot\nabla$ on $L^2(\T^l)$, be perturbed by the quantization of a function $\V_\om: \R^l\times\T^l\to\R$ of the form \vskip 5pt\noindent $$ \V_\om(\xi,x)=\V(z\circ \L_\om(\xi),x),\quad \L_\om(\xi):= \om_1\xi_1+\ldots+\om_l\xi_l $$ \vskip 4pt\noindent where $z\mapsto \V(z,x): \R\times\T^l \to\R$ is real-holomorphic. We prove that the corresponding quantum normal form converges uniformly with respect to $\hbar\in [0,1]$. Since the quantum normal form reduces to the classical one for $\hbar=0$, this result simultaneously yields an exact quantization formula for the quantum spectrum, as well as a convergence criterion for the Birkhoff normal form, valid for a class of perturbations holomorphic away from the origin. The main technical aspect concerns the quantum homological equation $\ds {[F(-i\hbar\om\cdot\nabla),W]}/{i\hbar}+V=N$, $F:\R\to\R$ being a smooth function $\ep-$close to the identity. Its solution is constructed, and estimated uniformly with respect to $\hbar\in [0,1]$, by solving the equation $\{F(\L_\om),\W\}_M+\V=\N$ for the corresponding symbols. Here $\{\cdot,\cdot\}_M$ stands for the Moyal bracket. As a consequence, the KAM iteration for the symbols of the quantum operators can be implemented, and its convergence proved, uniformly with respect to $(\xi,\hbar,\ep)\in \R^l\times [0,1]\times \{\ep\in\C\,|\;|\ep|<\ep^\ast\}$, where $\ep^\ast>0$ is explicitly estimated in terms only of the diophantine constants. This in turn entails the uniform convergence of the quantum normal form.
• The algebra $U_q(\hat{sl}_\infty)$ and applications
  
  • Hernandez David
  Journal of Algebra, Elsevier, 2011, 329 (1), pp.147-162. In this note we consider the algebra $U_q(\hat{sl}_\infty)$ and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras. In this context, we state a general positivity conjecture for representations of $U_q(\hat{sl}_\infty)$ viewed as representations of quantum toroidal algebras, that we prove for Kirillov-Reshetikhin modules. (10.1016/j.jalgebra.2010.04.002)
  DOI : 10.1016/j.jalgebra.2010.04.002
• Linear forms and quadratic uniformity for functions on $\mathbb{Z}_N$
  
  • Gowers W. T.
  • Wolf Julia
  Journal d'analyse mathématique, Springer, 2011, 115 (1), pp.121-186. A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].
• Strong semiclassical approximation of Wigner functions for the Hartree dynamics.
  
  • Paul Thierry
  • Athanassoulis Agissilaos
  • Pezzotti Federica
  • Pulvirenti Mario
  Rendiconti Lincei. Matematica e Applicazioni, European Mathematical Society, 2011, 22 (4), pp.525-552. We consider the Wigner equation corresponding to a nonlinear Schrodinger evolution of the Hartree type in the semiclassical limit h -> 0. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L-2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L-2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which-as it is well known-is not pointwise positive in general. (10.4171/RLM/613)
  DOI : 10.4171/RLM/613
• Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data
  
  • Ambrosio Luigi
  • Figalli Alessio
  • Friesecke Gero
  • Giannoulis Johannes
  • Paul Thierry
  Communications on Pure and Applied Mathematics, Wiley, 2011, 64 (9), pp.1199-1242. In this paper we study the semiclassical limit of the Schrodinger equation. Under mild regularity assumptions on the potential U, which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness, and stability results for the flow in the space of measures induced by the continuity equation (10.1002/cpa.20371)
  DOI : 10.1002/cpa.20371
• A Height Gap Theorem For Finite Subsets Of GL_d(\bar{Q}) and Non Amenable Subgroups
  
  • Breuillard Emmanuel
  Annals of Mathematics, Princeton University, Department of Mathematics, 2011, 174 (2), pp.1057-1110. We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\bar{\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\hat{h}(F)>\epsilon $ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $\epsilon =\epsilon (n)>0$ is an absolute constant. (10.4007/annals.2011.174.2.7)
  DOI : 10.4007/annals.2011.174.2.7
• Semiclassical Propagation of Coherent States for the Hartree equation
  
  • Athanassoulis Agissilaos
  • Paul Thierry
  • Pezzotti Federica
  • Pulvirenti Mario
  Annales Henri Poincaré, Springer Verlag, 2011, 12 (8), pp.1613-1634. In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion. (10.1007/s00023-011-0115-2)
  DOI : 10.1007/s00023-011-0115-2
• Poisson and symplectic functions in Lie algebroid theory
  
  • Kosmann-Schwarzbach Yvette
  , 2011, pp.243-268. Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector. (10.1007/978-0-8176-4735-3_12)
  DOI : 10.1007/978-0-8176-4735-3_12
• Large energy entire solutions for the Yamabe equation
  
  • Pacard Frank
  • del Pino Manuel
  • Musso Monica
  • Pistoia Angela
  Journal of Differential Equations, Elsevier, 2011, 251 (9), pp.2568-2597. We consider the Yamabe equation Delta u + n(n-2_/4 vertical bar u vertical bar 4/n-2 u = 0 in R(n), n >= 3. Let k >= 1 and xi(k)(j) = (e(2j pi u/k), 0) is an element of R(n) = C x R(n-2). For all large k we find a solution of the form u(k)(x)= u(x) - Sigma(k)(j=1) mu(k) (-n-2/2) U X (mu(-1)(k) (x - xi(j)) +o(1), where U(x) = (2/1+vertical bar x vertical bar(2)) (n-2/2), mu(k) = c(n)/k(2) for n >= 4, mu k = c/k(2)(logk)(2) for n =3 and o(1) -> 0 uniformly as k -> +infinity (10.1016/j.jde.2011.03.008)
  DOI : 10.1016/j.jde.2011.03.008
• Éléments d'analyse et d'algèbre (et de théorie des nombres)
  
  • Colmez Pierre
  , 2011, pp.688.