Publications

2014

• The KdV equation under periodic boundary conditions and its perturbations
  
  • Huang Guan
  • Kuksin Sergei
  Nonlinearity, IOP Publishing, 2014, 27 (9), pp.61-88. In this paper we discuss properties of the Korteweg–de Vries (KdV) equation under periodic boundary conditions, especially those which are important for studying perturbations of the equation. We then review what is known about the long-time behaviour of solutions for perturbed KdV equations. (10.1088/0951-7715/27/9/R61)
  DOI : 10.1088/0951-7715/27/9/R61
• On the universal CH0 group of cubic hypersurfaces
  
  • Voisin Claire
  , 2014.
• Dynamique algébrique des applications rationnelles de surfaces
  
  • Xie Junyi
  , 2014. Cette thèse se se compose de trois parties. La première partie est consacrée à l'étude des points périodiques des applications birationnelles des surfaces projectives. Nous montrons que toute application birationnelle de surface dont la croissance des degrés est exponentielle admet un ensemble de points périodiques Zariski dense. Dans la seconde partie, nous démontrons la conjecture de Mordell-Lang dynamique pour toute application polynomiale birationnelle du plan affine définie sur un corps de caractéristique nulle. Notre approche donne une nouvelle démonstration de cette conjecture pour les automorphismes polynomiaux du plan. Enfin la troisième partie porte sur un problème de géométrie affine inspiré par la généralisation au cas de toutes les applications polynomiales du plan affine de la conjecture de Mordell-Lang dynamique. Etant donné un ensemble fini S de valuations sur l'anneau de polynomes k[x,y] sur un corps algébriquement clos k triviales sur k, nous donnons une condition nécessaire et suffisante pour que le corps des fractions de l'intersection des anneaux de valuations de S avec k[x,y] soit de degré de transcendance 2 sur k.
• Laure Saint-Raymond: des molécules aux fluides
  
  • Golse François
  Gazette des Mathématiciens, Société Mathématique de France, 2014, 141, pp.100-106.
• On maximally inflected hyperbolic curves
  
  • Arroyo Aubin
  • Brugallé Erwan
  • López de Medrano Lucia
  Discrete and Computational Geometry, Springer Verlag, 2014, 52 (1), pp.140-152. (10.1007/s00454-014-9603-8)
  DOI : 10.1007/s00454-014-9603-8
• On Welschinger invariants of symplectic 4-manifolds
  
  • Brugallé Erwan
  • Puignau Nicolas
  , 2014.
• Inertial-sensor bias estimation from brightness/depth images and based on SO(3)-invariant integro/partial-differential equations on the unit sphere
  
  • Zarrouati-Vissiere Nadege
  • Beauchard Karine
  • Rouchon Pierre
  SIAM Journal on Control and Optimization, Society for Industrial and Applied Mathematics, 2014, 52 (6), pp.3463–3495. Constant biases associated to measured linear and angular velocities of a moving ob- ject can be estimated from measurements of a static scene by embedded brightness and depth sensors. We propose here a Lyapunov-based observer taking advantage of the SO(3)-invariance of the partial differential equations satisfied by the measured brightness and depth fields. The resulting asymptotic observer is governed by a non-linear integro/partial differential system where the two independent scalar variables indexing the pixels live on S2. The observer design and analysis are strongly simplified by coordinate-free differential calculus on S2 equipped with its natural Riemannian structure. The observer convergence is investigated under C1 regularity assumptions on the object motion and its scene. It relies on Ascoli-Arzela theorem and pre-compactness of the observer trajectories. It is proved that the estimated biases converge towards the true ones, if and only if, the scene admits no cylindrical symmetry. The observer design can be adapted to realistic sensors where brightness and depth data are only available on a subset of S2. Preliminary simulations with synthetic brightness and depth images (corrupted by noise around 10%) indicate that such Lyapunov-based observers should be robust and convergent for much weaker regularity assumptions.
• Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula
  
  • Taïbi Olivier
  , 2014. We consider the problem of explicitly computing dimensions of spaces of automorphic or modular forms in level one, for a split classical group $\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series. Our main contribution is an algorithm calculating orbital integrals for the characteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of $\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side in Arthur's specialisation of his invariant trace formula involving stable discrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore we explicitly compute the Euler-Poincaré characteristic of the level one discrete automorphic spectrum of $\mathbf{G}$ with respect to a finite-dimensional representation of $\mathbf{G}(\mathbb{R})$. For such a group $\mathbf{G}$, Arthur's endoscopic classification of the discrete spectrum allows to analyse precisely this Euler-Poincaré characteristic. For example one can deduce the number of everywhere unramified automorphic representations $\pi$ of $\mathbf{G}$ such that $\pi_{\infty}$ is isomorphic to a given discrete series representation of $\mathbf{G}(\mathbb{R})$. Dimension formulae for the spaces of vector-valued Siegel modular forms are easily derived.
• An averaging theory for nonlinear partial differential equations
  
  • Huang Guan
  , 2014. This Ph.D thesis focuses on studying the long-time behavior of solutions for non-linear PDEs that are close to a linear or an integrable Hamiltonian PDE. An averaging theory for nonlinear PDEs is presented. The model equations are the perturbed Korteweg-de Vries (KdV) equations and some weakly nonlinear partial differential equations.
• Unirational threefolds with no universal codimension $2$ cycle
  
  • Voisin Claire
  , 2014. We prove that the general quartic double solid with $k\leq 7$ nodes does not admit a Chow theoretic decomposition of the diagonal, or equivalently has a nontrivial universal ${\rm CH}_0$ group. The same holds if we replace in this statement "Chow theoretic" by "cohomological". In particular, it is not stably rational. We also prove that the general quartic double solid with seven nodes does not admit a universal codimension $2$ cycle parameterized by its intermediate Jacobian, and even does not admit a parametrization with rationally connected fibres of its Jacobian by a family of $1$-cycles. This implies that its third unramified cohomology group is not universally trivial.
• Local controllability of 1D Schrödinger equations with bilinear control and minimal time
  
  • Beauchard Karine
  • Morancey Morgan
  Mathematical Control and Related Fields, AIMS, 2014, 4 (2), pp.125-160. We consider a linear Schrödinger equation, on a bounded interval, with bilinear control. Beauchard and Laurent proved that, under an appropriate non degeneracy assumption, this system is controllable, locally around the ground state, in arbitrary time. Coron proved that a positive minimal time is required for this controllability, on a particular degenerate example. In this article, we propose a general context for the local controllability to hold in large time, but not in small time. The existence of a positive minimal time is closely related to the behaviour of the second order term, in the power series expansion of the solution. (10.3934/mcrf.2014.4.125)
  DOI : 10.3934/mcrf.2014.4.125
• Some new results on modified diagonals
  
  • Voisin Claire
  , 2014. O'Grady studied recently $m$-th modified diagonals for a smooth projective variety, generalizing the Gross-Schoen modified small diagonal. These cycles $\Gamma^m(X,a)$ depend on a choice of reference point $a\in X$ (or more generally a degree $1$ zero-cycle). We prove that for any $X,a$, the cycle $\Gamma^m(X,a)$ vanishes for large $m$. We also prove the following conjecture of O'Grady: if $X$ is a double cover of $Y$ and $\Gamma^m(Y,a)$ vanishes (where $a$ belongs to the branch locus), then $\Gamma^{2m-1}(X,a)$ vanishes, and we provide a generalization to higher degree finite covers. We finally prove the vanishing $\Gamma^{n+1}(X,o_X)=0$ when $X=S^{[m]}$, $S$ a $K3$ surface, and $n=2m$, which was conjectured by O'Grady and proved by him for $m=2,3$.
• Solution to a non-Archimedean Monge-Ampére equation
  
  • Boucksom S.
  • Favre Charles
  • Jonsson M.
  Journal of the American Mathematical Society, American Mathematical Society, 2014. Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, and assume that X is defined over a function field admitting K as a completion. Let further m be a positive measure on X and L be an ample line bundle such that the mass of m is equal to the degree of L. Then we show the existence a continuous semipositive metric whose associated measure is equal to m in the sense of Zhang and Chambert-Loir. This we do under a technical assumption on the support of m, which is, for instance, fulfilled if the support is a finite set of divisorial points. Our method draws on analogues of the variational approach developed to solve complex Monge-Ampére equations on compact Kähler manifolds by Berman, Guedj, Zeriahi and the first named author, and of Ko{\l}odziej's continuity estimates. It relies in a crucial way on the compactness properties of singular semipositive metrics, as defined and studied in a companion article. (10.1090/S0894-0347-2014-00806-7)
  DOI : 10.1090/S0894-0347-2014-00806-7
• Orbital measures on SU(2)/SO(2)
  
  • Anchouche Boudjemaa
  • Gupta Sanjiv Kumar
  • Plagne Alain
  , 2014.
• The Davenport constant of a box
  
  • Plagne Alain
  • Tringali Salvatore
  , 2014.
• Conjugacy class of homeomorphisms and distortion elements in groups of homeomorphisms
  
  • Militon Emmanuel
  , 2014. Let S be a compact connected surface and let f be an element of the group Homeo_0(S) of homeomorphisms of S isotopic to the identity. Denote by \tilde{f} a lift of f to the universal cover of S. Fix a fundamental domain D of this universal cover. The homeomorphism f is said to be non-spreading if the sequence (d_{n}/n) converges to 0, where d_{n} is the diameter of \tilde{f}^{n}(D). Let us suppose now that the surface S is orientable with a nonempty boundary. We prove that, if S is different from the annulus and from the disc, a homeomorphism is non-spreading if and only if it has conjugates in Homeo_{0}(S) arbitrarily close to the identity. In the case where the surface S is the annulus, we prove that a homeomorphism is non-spreading if and only if it has conjugates in Homeo_{0}(S) arbitrarily close to a rotation (this was already known in most cases by a theorem by Béguin, Crovisier, Le Roux and Patou). We deduce that, for such surfaces S, an element of Homeo_{0}(S) is distorted if and only if it is non-spreading.
• Global exact controllability of 1D Schrödinger equations with a polarizability term
  
  • Morancey Morgan
  • Nersesyan Vahagn
  Comptes rendus de l'Académie des sciences. Série I, Mathématique, Elsevier, 2014, 352 (5), pp.http://dx.doi.org/10.1016/j.crma.2014.03.013. We consider a quantum particle in a 1D interval submitted to a potential. The evolution of this particle is controlled using an external electric field. Taking into account the so-called polarizability term in the model (quadratic with respect to the control), we prove global exact controllability in a suitable space for arbitrary potential and arbitrary dipole moment. This term is relevant both from the mathematical and physical points of view. The proof uses tools from the bilinear setting and a perturbation argument. (10.1016/j.crma.2014.03.013)
  DOI : 10.1016/j.crma.2014.03.013
• Simultaneous local exact controllability of 1D bilinear Schrödinger equations
  
  • Morancey Morgan
  Annales de l'Institut Henri Poincaré (C), Analyse non linéaire (Nonlinear Analysis), EMS, 2014, pp.501–529. We consider N independent quantum particles, in an infinite square potential well coupled to an external laser field. These particles are modelled by a system of linear Schrödinger equations on a bounded interval. This is a bilinear control system in which the state is the N-tuple of wave functions. The control is the real amplitude of the laser field. For N=1, Beauchard and Laurent proved local exact controllability around the ground state in arbitrary time. We prove, under an extra generic assumption, that their result does not hold in small time if N is greater or equal than 2. Still, for N=2, we prove using Coron's return method that local controllability holds either in arbitrary time up to a global phase or exactly up to a global delay. We also prove that for N greater or equal than 3, local controllability does not hold in small time even up to a global phase. Finally, for N=3, we prove that local controllability holds up to a global phase and a global delay. (10.1016/j.anihpc.2013.05.001)
  DOI : 10.1016/j.anihpc.2013.05.001
• Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces
  
  • Brugallé Erwan
  , 2014.
• Lifting harmonic morphisms II: tropical curves and metrized complexes
  
  • Amini Omid
  • Baker Matthew
  • Brugallé Erwan
  • Rabinoff Joseph
  , 2014.
• The generalized Hodge and Bloch conjectures are equivalent for general complete intersections, II
  
  • Voisin Claire
  , 2014. We prove an unconditional (but slightly weakened) version of the main result of our earlier paper with the same title, which was, starting from dimension $4$, conditional to the Lefschetz standard conjecture. Let $X$ be a variety with trivial Chow groups, (i.e. the cycle class map to cohomology is injective on $CH(X)_\mathbb{Q}$). We prove that if the cohomology of a general very ample hypersurface $Y$ in $X$ is ''parameterized by cycles of dimension $c$'', then the Chow groups $CH_{i}(Y)_\mathbb{Q}$ are trivial for $i\leq c-1$.
• Sous-groupes de GL2 et arbres
  
  • Bellaïche Joël
  • Chenevier Gaëtan
  Journal of Algebra, Elsevier, 2014, 410, pp.501-525. (10.1016/j.jalgebra.2014.02.005)
  DOI : 10.1016/j.jalgebra.2014.02.005
• Sums of dilates in ordered groups
  
  • Plagne Alain
  • Tringali Salvatore
  , 2014.
• A thin-film limit in the Landau–Lifshitz–Gilbert equation relevant for the formation of Néel walls
  
  • Côte Raphaël
  • Ignat Radu
  • Miot Evelyne
  Journal of Fixed Point Theory and Applications, Springer Verlag, 2014, 15 (1), pp.241-272. (10.1007/s11784-014-0183-2)
  DOI : 10.1007/s11784-014-0183-2
• Null controllability of Kolmogorov-type equations
  
  • Beauchard Karine
  Mathematics of Control, Signals, and Systems, Springer Verlag, 2014, 26 (1), pp.145-176. We study the null controllability of Kolmogorov-type equations in a rectangle, under an additive control supported in an open subset of the rectangle. These equations couple a diffusion in variable v with a transport in variable x at speed v^a. For a=1, with periodic-type boundary conditions, we prove that null controllability holds in any positive time, with any control support.This improves the previous result [5], in which the control support was a horizontal strip. With Dirichlet boundary conditions and a horizontal strip as control support, we prove that null controllability holds in any positive time if a = 1, or if a= 2 and the control support contains the segment {v = 0}, and only in large time if a = 2 and the control support does not contain the segment {v = 0}. Our approach, inspired from [7, 31], is based on 2 key ingredients: the observability of the Fourier components of the solution of the adjoint system, uniformly with respect to the frequency, and the explicit exponential decay rate of these Fourier components. (10.1007/s00498-013-0110-x)
  DOI : 10.1007/s00498-013-0110-x