Publications

2012

• 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
• 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
• 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
• 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.
• 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
• 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
• 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
• Arithmetic and polynomial progressions in the primes [after Gowers, Green, Tao and Ziegler]
  
  • Wolf Julia
  , 2013, 352 (Exp. No. 1054), pp.389-427.
• 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
• 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.
• 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
• 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.
• 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).
• 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
• Geometry of representation spaces in SU(2)
  
  • Marche Julien
  IRMA - Lectures in Mathematics and Theoretical Physics, EMS, 2012, 12, pp.333-370. These notes of a course given at IRMA in April 2009 cover some aspects of the representation theory of fundamental groups of manifolds of dimension at most 3 in compact Lie groups, mainly $\su$. We give detailed examples, develop the techniques of twisted cohomology and gauge theory. We review Chern-Simons theory and describe an integrable system for the representation space of a surface. Finally, we explain some basic ideas on geometric quantization. We apply them to the case of representation spaces by computing Bohr-Sommerfeld orbits with metaplectic correction.
• Semiclassical and spectral analysis of oceanic waves
  
  • Cheverry Christophe
  • Gallagher Isabelle
  • Paul Thierry
  • Saint-Raymond Laure
  Duke Mathematical Journal, Duke University Press, 2012, 161 (5), pp.845-892. In this work we prove that the shallow water flow, subject to strong wind forcing and linearized around an adequate stationary profile, develops for large times closed trajectories due to the propagation of Rossby waves, while Poincaré waves are shown to disperse. The methods used in this paper involve semi-classical analysis and dynamical systems for the study of Rossby waves, while some refined spectral analysis is required for the study of Poincaré waves, due to the large time scale involved which is of diffractive type.
• Multicurves and regular functions on the representation variety of a surface in SU(2)
  
  • Charles Laurent
  • Marche Julien
  Commentarii Mathematici Helvetici, European Mathematical Society, 2012, 87 (2), pp.409-431. We consider the representation space of a compact surface, that is the space of morphisms from the fundamental group to SU(2) up to conjugation. We show that the trace functions associated to multicurves on the surface are linearly independent as functions on the representation space. The proof relies on the Fourier decomposition of the trace functions with respect to some torus action provided by a pants decomposition. Consequently the space of trace functions is isomorphic to the skein algebra at A=-1 of the thickened surface. (10.4171/CMH/258)
  DOI : 10.4171/CMH/258
• Configurations of flags and representations of surface groups in complex hyperbolic geometry
  
  • Marché Julien
  • Will Pierre
  Geom. Dedicata, 2012, 156, pp.49-70. In this work, we describe a set of coordinates on the PU(2,1)-representation variety of the fundamental group of an oriented punctured surface Σ with negative Euler characteristic. The main technical tool we use is a set of geometric invariants of a triple of flags in the complex hyperbolic plane H2C . We establish a bijection between a set of decorations of an ideal triangulation of Σ and a subset of the PU(2,1)-representation variety of π 1(Σ).
• Note on the gonality of abstract modular curves
  
  • Cadoret Anna
  , 2012, pp.89-106. (10.1007/978-3-642-23905-2_4)
  DOI : 10.1007/978-3-642-23905-2_4
• Degree growth of monomial maps and McMullen's polytope algebra
  
  • Favre Charles
  • Wulcan Elizabeth
  Indiana University Mathematics Journal, Indiana University Mathematics Journal, 2012, 61 (2), pp.493-524. We compute all dynamical degrees of monomial maps by interpreting them as mixed volumes of polytopes. By exploiting further the isomorphism between the polytope algebra of P. McMullen and the universal cohomology of complete toric varieties, we construct invariant positive cohomology classes when the dynamical degrees have no resonance. (10.1512/iumj.2012.61.4555)
  DOI : 10.1512/iumj.2012.61.4555
• The volume of an isolated singularity
  
  • Boucksom S.
  • de Fernex Tommaso
  • Favre Charles
  Duke Mathematical Journal, Duke University Press, 2012, 161 (8), pp.1455-1520.. We introduce a notion of volume of a normal isolated singularity that generalizes Wahl's characteristic number of surface singularities to arbitrary dimensions. We prove a basic monotonicity property of this volume under finite morphisms. We draw several consequences regarding the existence of non-invertible finite endomorphisms fixing an isolated singularity. Using a cone construction, we deduce that the anticanonical divisor of any smooth projective variety carrying a non-invertible polarized endomorphism is pseudoeffective. Our techniques build on Shokurov's b-divisors. We define the notion of nef Weil b-divisors, and of nef envelopes of b-divisors. We relate the latter to the pull-back of Weil divisors introduced by de Fernex and Hacon. Using the subadditivity theorem for multiplier ideals with respect to pairs recently obtained by Takagi, we carry over to the isolated singularity case the intersection theory of nef Weil b-divisors formerly developed by Boucksom, Favre, and Jonsson in the smooth case. (10.1215/00127094-1593317)
  DOI : 10.1215/00127094-1593317
• The global existence and convergence of the Calabi flow on $\mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$
  
  • Feng Renjie
  • Huang Hongnian
  Journal of Functional Analysis, Elsevier, 2012, 263 (4), pp.1129-1146. In this note, we study the long time existence of the Calabi flow on $X = \mathbb{C}^n/\mathbb{Z}^n + i\mathbb{Z}^n$. Assuming the uniform bound of the total energy, we establish the non-collapsing property of the Calabi flow by using Donaldson's estimates and Streets' regularity theorem. Next we show that the curvature is uniformly bounded along the Calabi flow on $X$ when the dimension is 2, partially confirming Chen's conjecture. Moreover, we show that the Calabi flow exponentially converges to the flat Kähler metric for arbitrary dimension if the curvature is uniformly bounded, partially confirming Donaldson's conjecture. (10.1016/j.jfa.2012.05.017)
  DOI : 10.1016/j.jfa.2012.05.017
• En hommage à Jean-Marie Souriau, quelques souvenirs
  
  • Kosmann-Schwarzbach Yvette
  Gazette des Mathématiciens, Société Mathématique de France, 2012, 133, pp.105-106.
• Geometric properties of maximal psh functions
  
  • Dujardin Romain
  • Guedj Vincent
  , 2012, pp.33-52. (10.1007/978-3-642-23669-3_3)
  DOI : 10.1007/978-3-642-23669-3_3