Publications

2012

• Uniform boundedness of p-primary torsion of abelian schemes
  
  • Cadoret Anna
  • Tamagawa Akio
  Inventiones Mathematicae, Springer Verlag, 2012, 188 (1), pp.83-125. (10.1007/s00222-011-0343-6)
  DOI : 10.1007/s00222-011-0343-6
• DIFFERENTIABLE RIGIDITY UNDER RICCI CURVATURE LOWER BOUND
  
  • Bessières Laurent
  • Besson Gérard
  • Courtois Gilles
  • Gallot Sylvestre
  Duke Mathematical Journal, Duke University Press, 2012, 161 (1), pp.29-67. One proves the following gap theorem, involving the volume and the Ricci curvature : For any integer $n \ge 3$ and $d > 0$, there exists $\epsilon(n, d) > 0 such that the following holds. Let $(X, g_0 )$ be a $n$-dimensional hyperbolic compact manifold with diameter $\le d$ and let $Y$ be a compact manifold which admits a continuous map $f : Y \rightarrow X$ of degree one. Then Y has a metric $g$ such that $Ric_g \geq -(n - 1)g$ and $vol_g (Y ) \leq (1 + \epsilon) vol_{g_0} (X )$ if and only if $f$ is homotopic to a diffeomorphism. (10.1215/00127094-1507272)
  DOI : 10.1215/00127094-1507272
• The role of minimal surfaces in the study of the Allen-Cahn equation.
  
  • Pacard Frank
  , 2012, pp.137-163.
• The space of 4-ended solutions to the Allen-Cahn equation in the plane.
  
  • Kowalczyk Michal
  • Liu Yong
  • Pacard Frank
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2012, 29 (5), pp.761-781. We are interested in entire solutions of the Allen-Cahn equation Delta u - F'(u) = 0 which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4-ended solutions. The main result of our paper states that, for any theta is an element of (0, pi/2), there exists a 4-ended solution of the Allen-Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles theta, pi - theta, pi + theta and 2 pi - theta with the x-axis. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k >= 2. (10.1016/j.anihpc.2012.04.003)
  DOI : 10.1016/j.anihpc.2012.04.003
• Solutions of the Allen-Cahn equation which are invariant under screw-motion
  
  • Pacard Frank
  • del Pino Manuel
  • Musso Monica
  Manuscripta Math., 2012, 138 (3-4), pp.273-286. We study entire solutions of the Allen-Cahn equation which are defined in the 3-dimensional Euclidean space and which are invariant under screw-motion. In particular, we discuss the existence and non existence of nontrivial solutions whose nodal set is a helicoid of R^3 .
• A non-archimedean Montel's theorem
  
  • Favre Charles
  • Kiwi Jan
  • Trucco Eugenio
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2012, 148 (3), pp.966-990. We prove a version of Montel's theorem for analytic functions over a non-archimedean complete valued field. We propose a definition of normal family in this context, and give applications of our results to the dynamics of non-archimedean entire functions. (10.1112/S0010437X11007470)
  DOI : 10.1112/S0010437X11007470
• Fatou directions along the Julia set for endomorphisms of CP^k
  
  • Dujardin Romain
  Journal de Mathématiques Pures et Appliquées, Elsevier, 2012, 98 (6), pp.591-615. Not much is known about the dynamics outside the support of the maximal entropy measure $\mu$ for holomorphic endomorphisms of $\mathbb{CP}^k$. In this article we study the structure of the dynamics on the Julia set, which is typically larger than $Supp(\mu)$. The Julia set is the support of the so-called Green current $T$, so it admits a natural filtration by the supports of the exterior powers of $T$. For $1\leq q \leq k$, let $J_q= Supp(T^q)$. We show that for a generic point of $J_q\setminus J_{q+1}$ there are at least $(k-q)$ "Fatou directions" in the tangent space. We also give estimates for the rate of expansion in directions transverse to the Fatou directions. (10.1016/j.matpur.2012.05.004)
  DOI : 10.1016/j.matpur.2012.05.004
• Convergence of a quantum normal form and an exact quantization formula
  
  • Paul Thierry
  • Graffi Sandro
  Journal of Functional Analysis, Elsevier, 2012, 262 (7), pp.3340-3393. The operator - i (h) over bar omega.Delta on L-2(T-1), quantizing the linear flow of diophantine frequencies omega = (omega(1),...,omega(l)) over T-l, l > 1, is perturbed by the operator quantizing a function V-omega(xi, x) = V(omega . xi, x) : R-1 x T-l > R, z bar right arrow V(z, x) : R x T-l -> R real-holomorphic. The corresponding quantum normal form (QNF) is proved to converge uniformly in (h)over bar> is an element of [0,]. This yields non-trivial examples of quantum integrable systems, an exact quantization formula for the spectrum, and a convergence criterion for the Birkhoff normal form, valid for perturbations holomorphic away from the origin. The main technical aspect concerns the solution of the quantum homological equation, which is constructed and estimated by solving the Moyal equation for the operator symbols. The KAM iteration can thus be implemented on the symbols, and its convergence proved. This entails the convergence of the QNF, with radius estimated in terms only of the diophantine constants of omega (10.1016/j.jfa.2012.01.010)
  DOI : 10.1016/j.jfa.2012.01.010
• Flat meromorphic connections of Frobenius manifolds with tt*-structure
  
  • Lin Jiezhu
  • Sabbah Claude
  Journal of Geometry and Physics, Elsevier, 2012, 62, pp.37-46. The base space of a semi-universal unfolding of a hypersurface singularity carries a rich geometric structure, which was axiomatized as a CDV-structure by C. Hertling. For any CDV-structure on a Frobenius manifold M, the pull-back of the (1,0)-tangent bundle of M to the product of M by the complex line carries two natural holomorphic structures equipped with flat meromorphic connections. We show that, for any semi-simple CDV-structure, there is a formal isomorphism between these two bundles compatible with connections. Moreover, if we assume that the super-symmetric index Q vanishes, we give a necessary and sufficient condition for such a formal isomorphism to be convergent, and we make it explicit for dim M = 2. (10.1016/j.geomphys.2011.09.006)
  DOI : 10.1016/j.geomphys.2011.09.006
• Homogenization and kinetic models in extended phase space
  
  • Golse François
  Riv. Math. Univ. Parma, 2012, 3 (1), pp.71-89. This paper reviews recent results obtained in collaboration with E. Bernard and E. Caglioti on the homogenization problem for the linear Boltzmann equation for a monokinetic population of particles set in a periodically perforated domain, assuming that particles are absorbed by the holes. We distinguish a critical scale for the hole radius in terms of the distance between neighboring holes, derive the homogenized equation under this scaling assumption, and study the asymptotic mass loss rate in the long time limit. The homogenized equation so obtained is set on an extended phase space as it involves an extra time variable, which is the time since the last jump in the stochastic process driving the linear Boltzmann equation. The present paper proposes a new proof of exponential decay for the mass which is based on a priori estimates on the homogenized equation instead of the renewal theorem used in [Bernard-Caglioti-Golse, SIAM J. Math. Anal. 42 (2010), 2082--2113].
• Random walks, Kleinian groups, and bifurcation currents
  
  • Deroin Bertrand
  • Dujardin Romain
  Inventiones Mathematicae, Springer Verlag, 2012, 190 (1), pp.57-118. Let (\rho_\lambda)_{\lambda\in \Lambda} be a holomorphic family of representations of a finitely generated group G into PSL(2,C), parameterized by a complex manifold \Lambda . We define a notion of bifurcation current in this context, that is, a positive closed current on \Lambda describing the bifurcations of this family of representations in a quantitative sense. It is the analogue of the bifurcation current introduced by DeMarco for holomorphic families of rational mappings on the Riemann sphere. Our definition relies on the theory of random products of matrices, so it depends on the choice of a probability measure \mu on G. We show that under natural assumptions on \mu, the support of the bifurcation current coincides with the bifurcation locus of the family. We also prove that the bifurcation current describes the asymptotic distribution of several codimension 1 phenomena in parameter space, like accidental parabolics or new relations, or accidental collisions between fixed points. (10.1007/s00222-012-0376-5)
  DOI : 10.1007/s00222-012-0376-5
• Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schrödinger equation.
  
  • Pacard Frank
  • Musso Monica
  • Wei Juncheng
  Journal of the European Mathematical Society, European Mathematical Society, 2012, 14 (6), pp.1923-1953. We address the problem of the existence of finite energy solitary waves for nonlinear Klein-Gordon or Schrödinger type equations Δu−u+f(u)=0 in RN, u∈H1(RN), where N≥2. Under natural conditions on the nonlinearity f, we prove the existence of infinitely many nonradial solutions in any dimension N≥2. Our result complements earlier works of Bartsch and Willem (N=4 or N≥6) and Lorca-Ubilla (N=5) where solutions invariant under the action of O(2)×O(N−2) are constructed. In contrast, the solutions we construct are invariant under the action of Dk×O(N−2) where Dk⊂O(2) denotes the dihedral group of rotations and reflexions leaving a regular planar polygon with k sides invariant, for some integer k≥7, but they are not invariant under the action of O(2)×O(N−2). (10.4171/JEMS/351)
  DOI : 10.4171/JEMS/351
• On the de Rham complex of mixed twistor D-modules
  
  • Fernandes Teresa Monteiro
  • Sabbah Claude
  International Mathematics Research Notices, Oxford University Press (OUP), 2012, pp.1-18. Given a complex manifold S, we introduce for each complex manifold X a t-structure on the bounded derived category of C-constructible complexes of O_S-modules on X x S. We prove that the de Rham complex of a holonomic D_{XxS/S}-module which is O_S-flat as well as its dual object is perverse relatively to this t-structure. This result applies to mixed twistor D-modules. (10.1093/imrn/rns197)
  DOI : 10.1093/imrn/rns197
• Actions infinitésimales dans la correspondance de Langlands locale p-adique
  
  • Dospinescu Gabriel
  Mathematische Annalen, Springer Verlag, 2012, 354 (2), pp.627-657. Let V be a two-dimensional absolutely irreducible p-adic Galois representation and let Pi be the p-adic Banach space representation associated to V via Colmez's p-adic Langlands correspondence. We establish a link between the infinitesimal action of GL_2(Q_p) on the locally analytic vectors of Pi, the differential equation associated to V via the theory of Fontaine and Berger, and the Sen polynomial of V. This answers a question of Harris and gives a new proof of a theorem of Colmez: Pi has nonzero locally algebraic vectors if and only if V is potentially semi-stable with distinct Hodge-Tate weights. (10.1007/s00208-011-0736-2)
  DOI : 10.1007/s00208-011-0736-2
• Recent Results on the Periodic Lorentz Gas
  
  • Golse François
  , 2012, pp.39-99. The Drude-Lorentz model for the motion of electrons in a solid is a classical model in statistical mechanics, where electrons are represented as point particles bouncing on a fixed system of obstacles (the atoms in the solid). Under some appropriate scaling assumption --- known as the Boltzmann-Grad scaling by analogy with the kinetic theory of rarefied gases --- this system can be described in some limit by a linear Boltzmann equation, assuming that the configuration of obstacles is random [G. Gallavotti, [Phys. Rev. (2) {\bf 185} (1969), 308]). The case of a periodic configuration of obstacles (like atoms in a crystal) leads to a completely different limiting dynamics. These lecture notes review several results on this problem obtained in the past decade as joint work with J. Bourgain, E. Caglioti and B. Wennberg. (10.1007/978-3-0348-0191-1)
  DOI : 10.1007/978-3-0348-0191-1
• The Incompressible Euler Limit of the Boltzmann Equation with Accommodation Boundary Condition
  
  • Bardos Claude
  • Golse François
  • Paillard Lionel
  Communications in Mathematical Sciences, International Press, 2012, 10 (1), pp.159-190. The convergence of solutions of the Navier-Stokes equations set in a domain with boundary to solutions of the Euler equations in the large Reynolds number limit is a challenging open problem both in 2 and 3 space dimensions. In particular it is distinct from the question of existence in the large of a smooth solution of the initial-boundary value problem for the Euler equations. The present paper proposes three results in that direction. First, if the solutions of the Navier-Stokes equations satisfy a slip boundary condition with vanishing slip coefficient in the large Reynolds number limit, we show by an energy method that they converge to the classical solution of the Euler equations on its time interval of existence. Next we show that the incompressible Navier-Stokes limit of the Boltzmann equation with Maxwell's accommodation condition at the boundary is governed by the Navier-Stokes equations with slip boundary condition, and we express the slip coefficient at the fluid level in terms of the accommodation parameter at the kinetic level. This second result is formal, in the style of [Bardos-Golse-Levermore, J. Stat. Phys. 63 (1991), 323--344]. Finally, we establish the incompressible Euler limit of the Boltzmann equation set in a domain with boundary with Maxwell's accommodation condition assuming that the accommodation parameter is small enough in terms of the Knudsen number. Our proof uses the relative entropy method following closely [L. Saint-Raymond, Arch. Ration. Mech. Anal. 166 (2003), 47--80] in the case of the 3-torus, except for the boundary terms, which require special treatment.
• Towards classification of multiple-end solutions to the Allen-Cahn equation in $\Bbb R^2$.
  
  • Pacard Frank
  • Kowalczyk Michal
  • Yong Liu
  Netw. Heterog. Media, 2012, 7 (4), pp.837-855. An entire solution of the Allen-Cahn equation \Delta u=f(u), where f is an odd function and has exactly three zeros at \pm 1 and 0, e.g. f(u)=u(u^2-1), is called a 2k-ended solution if its nodal set is asymptotic to 2k half lines, and if along each of these half lines the function u looks (up to a multiplication by -1) like the one dimensional, odd, heteroclinic solution H, of H''=f(H). In this paper we present some recent advances in the theory of the multiple-end solutions. We begin with the description of the moduli space of such solutions. Next we move on to study a special class of these solutions with just four ends. A special example is the saddle solutions U whose nodal lines are precisely the straight lines y=\pm x. We describe the connected components of the moduli space of 4-ended solutions. Finally we establish a uniqueness result which gives a complete classification of these solutions. It says that all 4-ended solutions are continuous deformations of the saddle solution.
• The Mean-Field Limit for a Regularized Vlasov-Maxwell Dynamics
  
  • Golse François
  Communications in Mathematical Physics, Springer Verlag, 2012, 310 (no. 3), pp.789-816. The present work establishes the mean-field limit of a N-particle system towards a regularized variant of the relativistic Vlasov-Maxwell system, following the work of Braun-Hepp [Comm. in Math. Phys. 56 (1977), 101--113] and Dobrushin [Func. Anal. Appl. 13 (1979), 115--123] for the Vlasov-Poisson system. The main ingredients in the analysis of this system are (a) a kinetic formulation of the Maxwell equations in terms of a distribution of electromagnetic potential in the momentum variable, (b) a regularization procedure for which an analogue of the total energy --- i.e. the kinetic energy of the particles plus the energy of the electromagnetic field --- is conserved and (c) an analogue of Dobrushin's stability estimate for the Monge-Kantorovich-Rubinstein distance between two solutions of the regularized Vlasov-Poisson dynamics adapted to retarded potentials. (10.1007/s00220-011-1377-8)
  DOI : 10.1007/s00220-011-1377-8
• Semiclassical limit for mixed states with singular and rough potentials
  
  • Figalli Alessio
  • Ligabo Marilena
  • Paul Thierry
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2012, 61 (1), pp.193-222. We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to $BV$; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data to ensure that the quantum dynamics converges to the classical one. More precisely, we consider the Husimi functions of the solution of the Heisenberg-von Neumann equation, and under suitable assumptions on the initial data we prove that they converge, as $\e \to 0$, to the unique bounded solution of the Liouville equation (locally uniformly in time).
• Stabilization for an ensemble of half-spin systems
  
  • Beauchard Karine
  • Pereira da Silva Paulo Sergio
  • Rouchon Pierre
  Automatica, Elsevier, 2012, 48 (1), pp.68-76. Feedback stabilization of an ensemble of non interacting half spins described by the Bloch equations is considered. This system may be seen as an interesting example for infinite dimensional systems with continuous spectra. We propose an explicit feedback law that stabilizes asymptotically the system around a uniform state of spin 1/2 or -1/2. The proof of the convergence is done locally around the equilibrium in the H1 topology. This local convergence is shown to be a weak asymptotic convergence for the H1 topology and thus a strong convergence for the C0 topology. The proof relies on an adaptation of the LaSalle invariance principle to infinite dimensional systems. Numerical simulations illustrate the efficiency of these feedback laws, even for initial conditions far from the equilibrium. (10.1016/j.automatica.2011.09.050)
  DOI : 10.1016/j.automatica.2011.09.050
• Poincaré face au négatif : une méthodologie ?
  
  • Paul Thierry
  Matapli, Société de Mathématiques Appliquées et Industrielles (SMAI), 2012, 98, pp.36-51. Poincaré a eu au début de sa carrière lors de l'erreur du prix du Roi de Suède, et à la fin de sa vie dans un passionnant article sur les quanta, l'occasion de se confronter à des résultats négatifs. Il a d'ailleurs lourdement insisté sur cette (prétendue) négativité dans plusieurs écrits. Nous essayons dans ce court article de cibler ce qui suscite l'intérêt du négatif chez Poincaré. Nous évoquons comment il se pourrait qu'il y ait chez lui une véritable méthodologie de la dynamique du négatif, plutôt qu'une vision classique de la dualité positif/négatif.
• STRONG AND WEAK SEMICLASSICAL LIMIT FOR SOME ROUGH HAMILTONIANS
  
  • Athanassoulis Agissilaos
  • Paul Thierry
  Mathematical Models and Methods in Applied Sciences, World Scientific Publishing, 2012, 22 (12), pp.1250038. We present several results concerning the semiclassical limit of the time dependent Schrödinger equation with potentials whose regularity doesn't guarantee the uniqueness of the underlying classical flow. Different topologies for the limit are considered and the situation where two bicharateristics can be obtained out of the same initial point is emphasized. (10.1142/S0218202512500388)
  DOI : 10.1142/S0218202512500388
• Dirac pairs
  
  • Kosmann-Schwarzbach Yvette
  Journal of Geometric Mechanics, 2012, 4 (2), pp.165-180. We extend the definition of the Nijenhuis torsion of an endomorphism of a Lie algebroid to that of a relation, and we prove that the torsion of the relation defined by a bi-Hamiltonian structure vanishes. Following Gelfand and Dorfman, we then define Dirac pairs, and we analyze the relationship of this general notion with the various kinds of compatible structures on manifolds, more generally on Lie algebroids. (10.3934/jgm.2012.4.165)
  DOI : 10.3934/jgm.2012.4.165
• Arithmetic and polynomial progressions in the primes [after Gowers, Green, Tao and Ziegler]
  
  • Wolf Julia
  , 2013, 352 (Exp. No. 1054), pp.389-427.
• Computation of the highest coefficients of weighted Ehrhart quasi-polynomials of rational polyhedra
  
  • Baldoni Velleda
  • Berline Nicole
  • de Loera Jesús A.
  • Köppe Matthias
  • Vergne Michèle
  Foundations of Computational Mathematics, Springer Verlag, 2012, 12 (4), pp.435-469. This article concerns the computational problem of counting the lattice points inside convex polytopes, when each point must be counted with a weight associated to it. We describe an efficient algorithm for computing the highest degree coefficients of the weighted Ehrhart quasi-polynomial for a rational simple polytope in varying dimension, when the weights of the lattice points are given by a polynomial function h. Our technique is based on a refinement of an algorithm of A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), pp. 1449--1466] in the unweighted case (i.e., h = 1). In contrast to Barvinok's method, our method is local, obtains an approximation on the level of generating functions, handles the general weighted case, and provides the coefficients in closed form as step polynomials of the dilation. To demonstrate the practicality of our approach we report on computational experiments which show even our simple implementation can compete with state of the art software. (10.1007/s10208-011-9106-4)
  DOI : 10.1007/s10208-011-9106-4