Publications

2013

• Classification of one-dimensional superattracting germs in positive characteristic
  
  • Ruggiero Matteo
  , 2013. We give a classification of superattracting germs in dimension one over a complete normed algebraically closed field of positive characteristic up to conjugacy. In particular we show that formal and analytic classifications coincide for these germs. We also give a higher dimensional version of some of these results.
• Higher rank homogeneous Clifford structures
  
  • Moroianu Andrei
  • Pilca Mihaela
  , 2013. We give an upper bound for the rank $r$ of homogeneous (even) Clifford structures on compact manifolds of non-vanishing Euler characteristic. More precisely, we show that if $r=2^a\cdot b$ with $b$ odd, then $r\le 9$ for $a=0$, $r\le 10$ for $a=1$, $r\le 12$ for $a=2$ and $r\le 16$ for $a\ge 3$. Moreover, we describe the four limiting cases and show that there is exactly one solution in each case.
• On the size of attractors in $\mathbb{P}^k$
  
  • Daurat Sandrine
  Mathematische Zeitschrift, Springer, 2014, 277 (3-4), pp.629-650. Let $f$ be a holomorphic endomorphism of $\mathbb{P}^k(\C)$ having an attracting set $\A$. In this paper, we address the question of the ''size" of $\A$ in a pluripolar sense. We introduce a conceptually simple framework to have non-algebraic attracting sets. We prove that adding a dimensional condition, these sets support a closed positive current with bounded quasi-potential (which answers a question from T.C. Dinh). Therefore, they are not pluripolar. Moreover, the examples are abundant on $\mathbb{P}^2$. (10.1007/s00209-013-1269-z)
  DOI : 10.1007/s00209-013-1269-z
• Torus action on S^n and sign changing solutions for conformally invariant equations
  
  • del Pino Manuel
  • Musso Monica
  • Pacard Frank
  • Pistoia Angela
  Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, Scuola Normale Superiore, 2013, 12 (1), pp.209-237. (10.2422/2036-2145.201010_011)
  DOI : 10.2422/2036-2145.201010_011
• Decaying Turbulence in Generalised Burgers Equation
  
  • Boritchev Alexandre
  , 2013. We consider the generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, where f is strongly convex and \nu is small and positive. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for small-scale quantities which characterise the Burgers turbulence, i.e. the dissipation length scale, the structure functions and the energy spectrum. Our proof uses a quantitative version of arguments by Aurell, Frisch, Lutsko and Vergassola \cite{AFLV92}. Our estimates remain valid in the inviscid limit.
• Dynamical Mordell-Lang conjecture for birational polynomial morphisms on $\mathbb{A}^2$
  
  • Xie Junyi
  Mathematische Annalen, Springer Verlag, 2014, 360 (1-2), pp.457-480. We prove the dynamical Mordell-Lang conjecture for birational polynomial morphisms on $\mathbb{A}^2$. (10.1007/s00208-014-1039-1)
  DOI : 10.1007/s00208-014-1039-1
• Pseudo-effective classes and pushforwards
  
  • Debarre O.
  • Jiang Z.
  • Voisin C.
  , 2013. Given a morphism between complex projective varieties, we make several conjectures on the relations between the set of pseudo-effective (co)homology classes which are annihilated by pushforward and the set of classes of varieties contracted by the morphism. We prove these conjectures for classes of curves or divisors. We also prove that one of these conjectures implies Grothendieck's generalized Hodge conjecture for varieties with Hodge coniveau at least 1.
• Ambitoric geometry I: Einstein metrics and extremal ambikaehler structures
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  , 2013. We present a local classification of conformally equivalent but oppositely oriented 4-dimensional Kaehler metrics which are toric with respect to a common 2-torus action. In the generic case, these "ambitoric" structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable. We use this description to classify Einstein 4-metrics which are hermitian with respect to both orientations, as well a class of solutions to the Einstein-Maxwell equations including riemannian analogues of the Plebanski-Demianski metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4-manifolds, obtained independently by R. Bryant and the first and third authors. These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kaehler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kaehler metrics. For both the Einstein-Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kaehler metrics on toric 4-orbifolds with second betti number b2=2.
• Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds
  
  • Apostolov Vestislav
  • Calderbank David M. J.
  • Gauduchon Paul
  , 2013. We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(M)=2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals. Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.
• Rossby waves trapped by quantum mechanics
  
  • Paul Thierry
  , 2013. Extended abstract for the workshop "Geophysical Fluid Dynamics " held in Oberwolfach, 17 February - 23 February 2013, to be published by the Mathematisches Forschungsinstitut Oberwolfach in the "Oberwolfach Reports" series.
• Growth of attraction rates for iterates of a superattracting germ in dimension two
  
  • Gignac William
  • Ruggiero Matteo
  , 2013. We study the sequence of attraction rates of iterates of a dominant superattracting holomorphic fixed point germ f:(C^2,0)->(C^2,0). By using valuative techniques similar to those developed by Favre-Jonsson, we show that this sequence eventually satisfies an integral linear recursion relation, which, up to replacing f by an iterate, can be taken to have order at most two. In addition, when the germ f is finite, we show the existence of a bimeromorphic model of (C^2,0) where f satisfies a weak local algebraic stability condition.
• The interior regularity of the Calabi flow on a toric surface
  
  • Chen Xiuxiong
  • Huang Hongnian
  • Sheng Li
  , 2013. Let X be a toric surface with Delzant polygon P and u(t) be a solution of the Calabi flow equation on P. Suppose the Calabi flow exists in [0, T). By studying local estimates of the Riemann curvature and the geodesic distance under the Calabi flow, we prove a uniform interior estimate of u(t) for t < T.
• Deformation of extremal metrics, complex manifolds and the relative Futaki invariant
  
  • Rollin Yann
  • Simanca Santiago
  • Tipler Carl
  Mathematische Zeitschrift, Springer, 2013, 273 (1-2), pp.547-568. (10.1007/s00209-012-1019-7)
  DOI : 10.1007/s00209-012-1019-7
• An Averaging Theorem for Perturbed KdV Equation
  
  • Huang Guan
  , 2013. We consider a perturbed KdV equation: [\dot{u}+u_{xxx} - 6uu_x = \epsilon f(x,u(\cdot)), \quad x\in \mathbb{T}, \quad\int_\mathbb{T} u dx=0.] For any periodic function $u(x)$, let $I(u)=(I_1(u),I_2(u),...)\in\mathbb{R}_+^{\infty}$ be the vector, formed by the KdV integrals of motion, calculated for the potential $u(x)$. Assuming that the perturbation $\epsilon f(x,u(\cdot))$ is a smoothing mapping (e.g. it is a smooth function $\epsilon f(x)$, independent from $u$), and that solutions of the perturbed equation satisfy some mild a-priori assumptions, we prove that for solutions $u(t,x)$ with typical initial data and for $0\leqslant t\lesssim \epsilon^{-1}$, the vector $I(u(t))$ may be well approximated by a solution of the averaged equation.
• A splitting theorem on toric manifolds
  
  • Huang Hongnian
  Mathematical Research Letters, International Press, 2013, 20 (2), pp.273-278. Using the Calabi flow, we prove that any extremal Kähler metric ωE on a product toric variety X1×X2 is a product extremal Kähler metric. (10.4310/MRL.2013.v20.n2.a5)
  DOI : 10.4310/MRL.2013.v20.n2.a5
• On the hyperbolicity of minimizers for 1D random Lagrangian systems
  
  • Boritchev Alexandre
  • Khanin Konstantin
  Nonlinearity, IOP Publishing, 2013, 26 (1), pp.65 - 80. We prove hyperbolicity of global minimizers for random Lagrangian systems in dimension 1. The proof considerably simplifies a related result in [2]. The conditions for hyperbolicity are almost optimal: they are essentially the same as conditions for uniqueness of a global minimizer in [3]. (10.1088/0951-7715/26/1/65)
  DOI : 10.1088/0951-7715/26/1/65
• Les crochets de Poisson, de la mécanique céleste à la mécanique quantique.
  
  • Kosmann-Schwarzbach Yvette
  , 2013, pp.369-401.
• Éléments de distorsion de ${\rm Diff}^\infty_0(M)$
  
  • Militon Emmanuel
  Bull. Soc. Math. France, 2013, pp.35-46. Dans cet article, on montre que, dans le groupe Di ff_0^oo(M) des diff éomorphismes isotopes à l'identité d'une variété compacte M, tout élément récurrent est de distorsion. Pour ce faire, on généralise une méthode de démonstration utilisée par Avila pour le cas de Diff_0^oo(S^ 1). La méthode nous permet de retrouver un résultat de Calegari et Freedman selon lequel tout homéomorphisme de la sphère isotope à l'identité est un élément de distorsion.
• Courant Algebroids. A Short History
  
  • Kosmann-Schwarzbach Yvette
  Symmetry, Integrability and Geometry : Methods and Applications, National Academy of Science of Ukraine, 2013, 9, pp.014. The search for a geometric interpretation of the constrained brackets of Dirac led to the definition of the Courant bracket. The search for the right notion of a "double" for Lie bialgebroids led to the definition of Courant algebroids. We recount the emergence of these concepts. (10.3842/SIGMA.2013.014)
  DOI : 10.3842/SIGMA.2013.014
• Intermediate Sums on Polyhedra: Computation and Real Ehrhart Theory
  
  • Baldoni Velleda
  • Berline Nicole
  • Köppe Matthias
  • Vergne Michèle
  Mathematika, University College London, 2013, 59 (1), pp.1-22. We study intermediate sums, interpolating between integrals and discrete sums, which were introduced by A. Barvinok [Computing the Ehrhart quasi-polynomial of a rational simplex, Math. Comp. 75 (2006), 1449--1466]. For a given semi-rational polytope P and a rational subspace L, we integrate a given polynomial function h over all lattice slices of the polytope P parallel to the subspace L and sum up the integrals. We first develop an algorithmic theory of parametric intermediate generating functions. Then we study the Ehrhart theory of these intermediate sums, that is, the dependence of the result as a function of a dilation of the polytope. We provide an algorithm to compute the resulting Ehrhart quasi-polynomials in the form of explicit step polynomials. These formulas are naturally valid for real (not just integer) dilations and thus provide a direct approach to real Ehrhart theory. (10.1112/S0025579312000101)
  DOI : 10.1112/S0025579312000101
• Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation
  
  • Boritchev Alexandre
  Geometric And Functional Analysis, Springer Verlag, 2013, 23, pp.1730 - 1771. We consider the non-homogeneous generalised Burgers equation$$∂u/∂t + f (u) ∂u/∂x − \nu ∂^2u/∂x^2 = η, t ≥ 0, x ∈ S^1 .$$Here $f$ is strongly convex and satisfies a growth condition, $\nu$ is small and positive, while $\eta$ is a random forcing term, smooth in space and white in time. For any solution $u$ of this equation we consider the quasi-stationary regime, corresponding to $t ≥ T_1$ , where $T_1$ depends only on $f$ and on the distribution of $η$. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on $t$ for$ t ≥ T_1$ , and hold uniformly in $\nu$. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their article. (10.1007/s00039-013-0245-4)
  DOI : 10.1007/s00039-013-0245-4
• On the selection of the classical limit for potentials with BV derivatives
  
  • Athanassoulis Agissilaos
  • Paul Thierry
  Journal of Dynamics and Differential Equations, Springer Verlag, 2013, 25 (1), pp.33-47. We consider the classical limit of the quantum evolution, with some rough potential, of wave packets concentrated near singular trajectories of the underlying dynamics. We prove that under appropriate conditions, even in the case of BV vector fields, the correct classical limit can be selected.
• Normal surface singularities admitting contracting automorphisms
  
  • Favre Charles
  • Ruggiero Matteo
  Annales de la Faculté des Sciences de Toulouse. Mathématiques., Université Paul Sabatier _ Cellule Mathdoc, 2013. We show that a complex normal surface singularity admitting a contracting automorphism is necessarily quasihomogeneous. We also describe the geometry of a compact complex surface arising as the orbit space of such a contracting automorphism. (10.5802/afst.1425)
  DOI : 10.5802/afst.1425
• Wild Hodge theory and Hitchin-Kobayashi correspondence [after T. Mochizuki]
  
  • Sabbah Claude
  Asterisque, Société Mathématique de France, 2013, 352, pp.Exposé n°2050. T. Mochizuki constructs a theory of variations of wild Hodge structure for which the underlying flat connection can have irregular singularities at infinity. He extends in this way the correspondence of Corlette and Simpson between irreducible flat bundles and stables Higgs bundles, taking into account objects with irregular singularities. As an application, he proves a conjecture of Kashiwara concerning a generalization of the Hard Lefschetz theorem when the coefficients are the de Rham complex of a simple holonomic D-module on smooth complex projective variety.
• On the convergence to equilibrium for degenerate transport problems
  
  • Bernard Etienne
  • Salvarani Francesco
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2013, 208 (3), pp.977-984. We give a counterexample which shows that the asymptotic rate of convergence to the equilibrium state for the transport equation, with a degenerate cross section and in the periodic setting, cannot be better than $t^{-1/2}$ in the general case. We suggest moreover that the geometrical properties of the cross section are the key feature of the problem and impose, through the distribution of the forward exit time, the speed of convergence to the stationary state. (10.1007/s00205-012-0608-2)
  DOI : 10.1007/s00205-012-0608-2