Publications

2012

• Algebraic homotopy classes of rational functions
  
  • Cazanave Christophe
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2012 (45). We compute the set of naive pointed homotopy classes of endomorphisms of the projective line P^1 over the spectrum of a field. Our computation compares well with Fabien Morel's one of the motivic pointed homotopy classes of endomorphisms of P^1: there is an a priori monoid structure on the set of naive homotopy classes such that canonical map from this monoid to group of motivic homotopy classes is a group completion.
• A note on the rank of positive closed currents
  
  • Dujardin Romain
  , 2012. In this short note we prove an estimate on the rank a.e. of the tangent (p,p) vector to a a positive closed current of bidimension (p,p) in CP^k, in terms of the dimension of its trace measure.
• Periodic points of birational maps on projective surfaces
  
  • Xie Junyi
  , 2012. We classify birational maps of projective smooth surfaces whose non-critical periodic points are Zariski dense. In particular, we show that if the first dynamical degree is greater than one, then the periodic points are Zariski dense.
• Eigenvarieties for classical groups and complex conjugations in Galois representations
  
  • Taïbi Olivier
  , 2012. The goal of this paper is to remove the irreducibility hypothesis in a theorem of Richard Taylor describing the image of complex conjugations by $p$-adic Galois representations associated with regular, algebraic, essentially self-dual, cuspidal automorphic representations of $\GL_{2n+1}$ over a totally real number field $F$. We also extend it to the case of representations of $\GL_{2n}/F$ whose multiplicative character is ''odd''. We use a $p$-adic deformation argument, more precisely we prove that on the eigenvarieties for symplectic and even orthogonal groups, there are ''many'' points corresponding to (quasi-)irreducible Galois representations. The recent work of James Arthur describing the automorphic spectrum for these groups is used to define these Galois representations, and also to transfer self-dual automorphic representations of the general linear group to these classical groups.
• Invariant four-forms and symmetric pairs
  
  • Moroianu Andrei
  • Semmelmann Uwe
  , 2012. We give criteria for real, complex and quaternionic representations to define s-representations, focusing on exceptional Lie algebras defined by spin representations. As applications, we obtain the classification of complex representations whose second exterior power is irreducible or has an irreducible summand of co-dimension one, and we give a conceptual computation-free argument for the construction of the exceptional Lie algebras of compact type.
• Weakly complex homogeneous spaces
  
  • Moroianu Andrei
  • Semmelmann Uwe
  , 2012. We complete our recent classification of compact inner symmetric spaces with weakly complex tangent bundle by filling up a case which was left open, and extend this classification to the larger category of compact homogeneous spaces with positive Euler characteristic. We show that a simply connected compact equal rank homogeneous space has weakly complex tangent bundle if and only if it is a product of compact equal rank homogeneous spaces which either carry an invariant almost complex structure (and are classified by Hermann), or have stably trivial tangent bundle (and are classified by Singhof and Wemmer), or belong to an explicit list of weakly complex spaces which have neither stably trivial tangent bundle, nor carry invariant almost complex structures.
• The supports of higher bifurcation currents
  
  • Dujardin Romain
  , 2012. Let (f_\lambda) be a holomorphic family of rational mappings of degree d on the Riemann sphere, with k marked critical points c_1,..., c_k, parameterized by a complex manifold \Lambda. To this data is associated a closed positive current T_1\wedge ... \wedge T_k of bidegree (k,k) on \Lambda, aiming to describe the simultaneous bifurcations of the marked critical points. In this note we show that the support of this current is accumulated by parameters at which c_1,..., c_k eventually fall on repelling cycles. Together with results of Buff, Epstein and Gauthier, this leads to a complete characterization of Supp(T_1\wedge ... \wedge T_k).
• Bifurcation currents and equidistribution on parameter space
  
  • Dujardin Romain
  , 2012. In this paper we review the use of techniques of positive currents for the study of parameter spaces of one-dimensional holomorphic dynamical systems (rational mappings on P^1 or subgroups of the Moebius group PSL(2,C)). The topics covered include: the construction of bifurcation currents and the characterization of their supports, the equidistribution properties of dynamically defined subvarieties on parameter space.
• ON SOME GEOMETRY OF PROPAGATION IN DIFFRACTIVE TIME SCALES
  
  • Cheverry Christophe
  • Paul Thierry
  Discrete and Continuous Dynamical Systems - Series A, American Institute of Mathematical Sciences, 2012, 32 (2), pp.Pages: 499 - 538. In this article, we develop a non linear geometric optics which presents the two main following features. It is valid in diffractive times and it extends the classical approaches to the case of fast variable coefficients. In this context, we can show that the energy is transported along the rays associated with some non usual long-time hamiltonian. Our analysis needs structural assumptions and initial data suitably polrarized to be implemented. All the required conditions are met concerning a current model arising in fluid mechanics and which was the original motivation of our work. As a by product, we get results complementary to the litterature concerning the propagation of the Rossby waves which play a part in the description of large oceanic currents, like Gulf stream or Kuroshio. (10.3934/dcds.2012.32.499)
  DOI : 10.3934/dcds.2012.32.499
• On a twisted de Rham complex, II
  
  • Sabbah Claude
  , 2012. We prove an algebraic formula, conjectured by M. Kontsevich, for computing the monodromy of the vanishing cycles of a regular function on a smooth complex algebraic variety.
• Harmonic maps, second homology classes of smooth manifolds, and bounded cohomology
  
  • Ville Marina
  International Journal of Mathematics, World Scientific Publishing, 2012, 04 (06), pp.997-1006. (10.1142/S0129167X93000467)
  DOI : 10.1142/S0129167X93000467
• Harmonic morphisms from Einstein 4-manifolds to Riemann surfaces
  
  • Ville Marina
  International Journal of Mathematics, World Scientific Publishing, 2012, 14 (03), pp.327-337. If M and N are Riemannian manifolds, a harmonic morphism f : M → N is a map which pulls back local harmonic functions on N to local harmonic functions on M. If M is an Einstein 4-manifold and N is a Riemann surface, John Wood showed that such an f is holomorphic w.r.t. some integrable complex Hermitian structure defined on M away from the singular points of f. In this paper we extend this complex structure to the entire manifold M. It follows that there are no non-constant harmonic morphisms from S4 or CP2 to a Riemann surface. The proof relies heavily on the real analyticity of the whole situation. We conclude by an example of a non-constant harmonic morphism CP2#CP2 from to S2. (10.1142/S0129167X0300179X)
  DOI : 10.1142/S0129167X0300179X
• 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
• A uniform open image theorem for ℓ-adic representations, I
  
  • Cadoret Anna
  • Tamagawa Akio
  Duke Mathematical Journal, Duke University Press, 2012, 161 (13), pp.2605-2634. (10.1215/00127094-1812954)
  DOI : 10.1215/00127094-1812954
• Optimally small sumsets in groups IV. Counting multiplicities and the λ_G functions
  
  • Plagne Alain
  Israel Journal of Mathematics, Springer, 2012, 191 (2), pp.739-754. We continue our investigation on how small a sumset can be in a given abelian group. Here "small" takes into account not only the size of the sumset itself but also the number of elements which are repeated at least twice. A function λ_G (r, s) computing the minimal size (in this sense) of the sum of two sets with respective cardinalities r and s is introduced. (Lower and upper) bounds are obtained, which coincide in most cases. While upper bounds are obtained by constructions, lower bounds follow in particular from the use of a recent theorem by Grynkiewicz.
• 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
• Geometric interpretation of simplicial formulas for the Chern-Simons invariant
  
  • Marche Julien
  Algebraic and Geometric Topology, Mathematical Sciences Publishers, 2012, 12 (2), pp.805-827. We give a direct interpretation of Neumann's combinatorial formula for the Chern-Simons invariant of a 3-manifold with a representation in PSL(2,C) whose restriction to the boundary takes values in upper triangular matrices. Our construction does not involve group homology or Bloch group but is based on the construction of an explicit flat connection for each tetrahedron of a simplicial decomposition of the manifold. (10.2140/agt.2012.12.805)
  DOI : 10.2140/agt.2012.12.805
• 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].
• 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
• 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
• 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
• The role of minimal surfaces in the study of the Allen-Cahn equation.
  
  • Pacard Frank
  , 2012, pp.137-163.
• 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