Publications

2011

• Dynamics of meromorphic mappings with small topological degree II : energy and invariant measure
  
  • Diller Jeffrey
  • Dujardin Romain
  • Guedj Vincent
  Commentarii Mathematici Helvetici, European Mathematical Society, 2011, 86 (2), pp.277-316. (10.4171/CMH/224)
  DOI : 10.4171/CMH/224
• Local semiconvexity of Kantorovich potentials on non-compact manifolds
  
  • Figalli Alessio
  • Gigli Nicola
  ESAIM: Control, Optimisation and Calculus of Variations, EDP Sciences, 2011, 17 (3), pp.648-653. (10.1051/cocv/2010011)
  DOI : 10.1051/cocv/2010011
• The sign of Galois representations attached to automorphic forms for unitary groups
  
  • Bellaïche Joël
  • Chenevier Gaëtan
  Compositio Mathematica, Foundation Compositio Mathematica / Cambridge University Press, Cambridge, 2011, 147 (5), pp.1337-1352. We determine the sign of the polarization of any polarized irreducible factor of a Galois representation attached to a polarized cohomological cuspidal automorphic form of GL_n of a CM field: it is always +1, as was conjectured by Gross. In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GL_n(A_F) when F is a totally real number field. (10.1112/S0010437X11005264)
  DOI : 10.1112/S0010437X11005264
• Construction of multi-soliton solutions for the L2-supercritical gKdV and NLS equations
  
  • Côte Raphaël
  • Martel Yvan
  • Merle Frank
  Revista Matemática Iberoamericana, European Mathematical Society, 2011, 27 (1), pp.273-302. Multi-soliton solutions, i.e. solutions behaving as the sum of N given solitons as $t\to +\infty$, were constructed in previous works for the L2 critical and subcritical (NLS) and (gKdV) equations. In this paper, we extend the construction of multi-soliton solutions to the L2 supercritical case both for (gKdV) and (NLS) equations, using a topological argument to control the direction of instability. (10.4171/RMI/636)
  DOI : 10.4171/RMI/636
• Rigidity of Rank-One Factors of Compact Symmetric Spaces
  
  • Clarke Andrew
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2011, 61 (2), pp.491-509. We consider the decomposition of a compact-type symmetric space into a product of factors and show that the rank-one factors, when considered as totally geodesic submanifolds of the space, are isolated from inequivalent minimal submanifolds. (10.5802/aif.2621)
  DOI : 10.5802/aif.2621
• Non-commutative Hodge structures
  
  • Sabbah Claude
  Annales de l'Institut Fourier, Association des Annales de l'Institut Fourier, 2011, 61 (7), pp.2681-2717. This article gives a survey of recent results on a generalization of the notion of a Hodge structure. The main example is related to the Fourier-Laplace transform of a variation of polarizable Hodge structure on the punctured affine line, like the Gauss-Manin systems of a proper or tame algebraic function on a smooth quasi-projective variety. Variations of non-commutative Hodge structures often occur on the tangent bundle of Frobenius manifolds, giving rise to a tt* geometry. (10.5802/aif.2790)
  DOI : 10.5802/aif.2790
• Yahya Ould Hamidoune: the Mauritanian mathematician, 1948-11 March 2011.
  
  • Plagne Alain
  Combinatorics, Probability and Computing, Cambridge University Press (CUP), 2011, 20 (5), pp.641-645. Yahya ould Hamidoune passed away in Paris on 11 March 2011 after a brief illness, leaving insufficient time for his friends and colleagues to express their indebtedness to him for his kindness and generosity, both in mathematics and in everyday life. Yahya was a discreet individual, always looking for the essential rather than the superficial, and certainly did not receive the recognition he deserved. May this modest testimony render justice to this singular man. (10.1017/S0963548311000332)
  DOI : 10.1017/S0963548311000332
• Extremal metrics on blowups
  
  • Pacard Frank
  • Arezzo Claudio
  • Singer Michael
  Duke Mathematical Journal, Duke University Press, 2011, 157 (1), pp.1-51. In this paper we provide conditions that are sufficient to guarantee the existence of extremal metrics on blowups at finitely many points of Kähler manifolds which already carry an extremal metric. As a particular case, we construct extremal metrics on $\mathbb {P}^2$ blown-up k points in general position, with $k \lt m+2$.
• Anticanonical divisors and curve classes on Fano manifolds
  
  • Hoering Andreas
  • Voisin Claire
  Pure and Applied Mathematics Quarterly, International Press, 2011.
• Extremal Kähler metrics on projective bundles over a curve
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  • Tønnesen-Friedman Christina W.
  Advances in Mathematics, Elsevier, 2011, 227 (6), pp.2385-2424. Let $M=P(E)$ be the complex manifold underlying the total space of the projectivization of a holomorphic vector bundle $E \to \Sigma$ over a compact complex curve $\Sigma$ of genus $\ge 2$. Building on ideas of Fujiki, we prove that $M$ admits a Kähler metric of constant scalar curvature if and only if $E$ is polystable. We also address the more general existence problem of extremal Kähler metrics on such bundles and prove that the splitting of $E$ as a direct sum of stable subbundles is necessary and sufficient condition for the existence of extremal Kähler metrics in sufficiently small Kähler classes. The methods used to prove the above results apply to a wider class of manifolds, called {\it rigid toric bundles over a semisimple base}, which are fibrations associated to a principal torus bundle over a product of constant scalar curvature Kähler manifolds with fibres isomorphic to a given toric Kähler variety. We discuss various ramifications of our approach to this class of manifolds. (10.1016/j.aim.2011.05.006)
  DOI : 10.1016/j.aim.2011.05.006
• On an overdetermined elliptic problem
  
  • Pacard Frank
  • Hélein Frédéric
  • Hauswirth Laurent
  Pacific Journal of Mathematics, Mathematical Sciences Publishers, 2011, 250 (2), pp.319-334. A smooth flat Riemannian manifold is called an exceptional domain if it admits positive harmonic functions having vanishing Dirichlet boundary data and constant (nonzero) Neumann boundary data. In analogy with minimal surfaces, a representation formula is derived and applied to the classification of exceptional domains. Some interesting open problems are proposed along the way. (10.2140/pjm.2011.250.319)
  DOI : 10.2140/pjm.2011.250.319
• An application of coding theory to estimating Davenport constants
  
  • Plagne Alain
  • Schmid Wolfgang A.
  Designs, Codes and Cryptography, Springer Verlag, 2011, 61 (1), pp.105-118. We investigate a certain well-established generalization of the Davenport constant. For j a positive integer (the case j = 1, is the classical one) and a finite Abelian group (G,+,0), the invariant D_j(G) is defined as the smallest l such that each sequence over G of length at least l has j disjoint non-empty zero-sum subsequences. We investigate these quantities for elementary 2-groups of large rank (relative to j). Using tools from coding theory, we give fairly precise estimates for these quantities. We use our results to give improved bounds for the classical Davenport constant of certain groups. (10.1007/s10623-010-9441-5)
  DOI : 10.1007/s10623-010-9441-5
• Sums of dilates in groups of prime order
  
  • Plagne Alain
  Combinatorics, Probability and Computing, Cambridge University Press (CUP), 2011, 20 (6), pp.867-873. We obtain a first non-trivial estimate for the sum of dilates problem in the case of groups of prime order, by showing that if t is an integer different from 0, 1 or -1 and if A < ℤ/pℤ is not too large (with respect to p), then | A + t A| is significantly larger than 2|A| (unless |t| = 3). In the important case |t| = 2, we obtain for instance | A+ tA| ≥ 2.08 |A|−2. (10.1017/S0963548311000447)
  DOI : 10.1017/S0963548311000447
• Endoscopy for real reductive groups
  
  • Renard David
  , 2011, pp.95-141.
• Toric geometry of convex quadrilaterals
  
  • Legendre Eveline
  Journal of Symplectic Geometry, International Press, 2011, 9, pp.343-385. We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler-Einstein and toric Sasaki-Einstein metrics constructed in [6,22,14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler-Einstein ones, and show that for a toric orbi-surface with 4 fixed points of the torus action, the vanishing of the Futaki invariant is a necessary and sufficient condition for the existence of Kähler metric with constant scalar curvature. Our results also provide explicit examples of relative K-unstable toric orbi-surfaces that do not admit extremal metrics.
• Entropy of quantum limits for symplectic linear maps of the multidimensional torus
  
  • Riviere Gabriel
  International Mathematics Research Notices, Oxford University Press (OUP), 2011, 2011 (11), pp.2396-2443. In the case of a linear symplectic map A of the 2d-torus, semiclassical measures are A-invariant probability measures associated to sequences of high energy quantum states. Our main result is an explicit lower bound on the entropy of any semiclassical measure of a given quantizable matrix A in Sp(2d,Z). In particular, our result implies that if A has an eigenvalue outside the unit circle, then a semiclassical measure cannot be carried by a closed orbit of A. (10.1093/imrn/rnq155)
  DOI : 10.1093/imrn/rnq155