Publications

2018

• The quantum N -body problem in the mean-field and semiclassical regime
  
  • Golse François
  Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Royal Society, The, 2018, 376 (2118). (10.1098/rsta.2017.0229)
  DOI : 10.1098/rsta.2017.0229
• Une remarque sur l'article "Un th\'eor\`eme \`a la "Thom-Sebastiani" pour les int\'egrales-fibres" de D. Barlet
  
  • Sabbah Claude
  , 2018. Dans cette note, nous donnons une autre d\'emonstration du r\'esultat d\'emontr\'e par D. Barlet dans http://aif.cedram.org/aif-bin/fitem?id=AIF_2010__60_1_319_0 (arXiv:0809.4981), en nous appuyant sur la transformation de Mellin et des propri\'et\'es classiques des fonctions de Bessel (au lieu de la convolution). Le th\'eor\'eme principal de cette note est un peu plus pr\'ecis que l'\'enonc\'e donn\'e dans loc. cit. [English: In this article, we give another proof of the result shown by D. Barlet in http://aif.cedram.org/aif-bin/fitem?id=AIF_2010__60_1_319_0 (arXiv:0809.4981), relying on the Mellin transform and on classical properties of Bessel functions (instead of convolution). The main theorem of this note is somewhat more precise than the statement given in loc. cit.]
• Control from an Interior Hypersurface
  
  • Galkowski Jeffrey
  • Léautaud Matthieu
  , 2018. We consider a compact Riemannian manifold M (possibly with boundary) and Σ ⊂ M \ ∂M an interior hypersurface (possibly with boundary). We study observation and control from Σ for both the wave and heat equations. For the wave equation, we prove controllability from Σ in time T under the assumption (T GCC) that all generalized bicharacteristics intersect Σ transversally in the time interval (0, T). For the heat equation we prove unconditional controllability from Σ. As a result, we obtain uniform lower bounds for the Cauchy data of Laplace eigenfunctions on Σ under T GCC and unconditional exponential lower bounds on such Cauchy data.
• Continuity of the Green function in meromorphic families of polynomials
  
  • Favre Charles
  • Gauthier Thomas
  , 2018. We prove that along any marked point the Green function of a meromorphic family of polynomials parameterized by the punctured unit disk explodes exponentially fast near the origin with a continuous error term.
• Sur les paquets d'Arthur de $\mathbf{Sp}(2n,\mathbb{R})$ contenant des modules unitaires de plus haut poids, scalaires
  
  • Renard David
  • Moeglin Colette
  , 2018. Soit $\pi$ un module de plus haut poids unitaire du groupe $G=\mathbf{Sp}(2n,\mathbb{R})$. On s'int\'eresse au paquets d'Arthur contenant $\pi$. Lorsque le plus haut poids est scalaire, on d\'etermine les param\`etres de ces paquets. Let $\pi$ be an irreducible unitary highest weight module for $G=\mathbf{Sp}(2n,\mathbb{R})$. We would like to determine the Arthur packets containing $\pi$. We give a complete answer when the highest weight is scalar.
• UNIFORM K-STABILITY AND ASYMPTOTICS OF ENERGY FUNCTIONALS IN KÄHLER GEOMETRY
  
  • Boucksom Sébastien
  • Hisamoto Tomoyuki
  • Jonsson Mattias
  , 2018. Consider a polarized complex manifold (X, L) and a ray of positive metrics on L defined by a positive metric on a test configuration for (X, L). For many common functionals in Kähler geometry, we prove that the slope at infinity along the ray is given by evaluating the non-Archimedean version of the functional (as defined in our earlier paper [BHJ15]) at the non-Archimedean metric on L defined by the test configuration. Using this asymptotic result, we show that coercivity of the Mabuchi functional implies uniform K-stability, as defined in [Der15, BHJ15]. As a partial converse, we show that uniform K-stability implies coercivity of the Mabuchi functional when restricted to Bergman metrics.
• SINGULAR SEMIPOSITIVE METRICS ON LINE BUNDLES ON VARIETIES OVER TRIVIALLY VALUED FIELDS
  
  • Boucksom Sébastien
  • Jonsson Mattias
  , 2018. Let X be a smooth projective Berkovich space over a trivially or discretely valued field k of residue characteristic zero, and let L be an ample line bundle on X. We develop a theory of plurisubharmonic (or semipositive) metrics on L. In particular we show that the (non-Archimedean) Monge-Ampère operator induces a bijection between plurisubharmonic metrics and Radon probability measures of finite energy. In the discretely valued case, these results refine earlier work obtained in collaboration with C. Favre. In the trivially valued case, the results are new and will in subsequent work be shown to have ramifications for the study of K-stability.
• A VARIATIONAL APPROACH TO THE YAU-TIAN-DONALDSON CONJECTURE
  
  • Berman Robert
  • Boucksom Sébastien
  • Jonsson Mattias
  , 2018. We give a new proof of a uniform version of the Yau-Tian-Donaldson conjecture for Fano manifolds with finite automorphism group, and of the semistable case of the conjecture. Our approach does not involve the continuity method or Cheeger-Colding-Tian theory. Instead, the proof is variational and uses pluripotential theory and certain non-Archimedean considerations.
• Inelasticity of soliton collisions for the 5D energy critical wave equation
  
  • Martel Yvan
  • Merle Frank
  , 2018. For the focusing energy critical wave equation in 5D, we construct a solution showing the inelastic nature of the collision of any two solitons, except the special case of two solitons of same scaling and opposite signs. Beyond its own interest as one of the first rigorous studies of the collision of solitons for a non-integrable model, the case of the quartic gKdV equation being partially treated by the authors in previous works, this result can be seen as part of a wider program aiming at establishing the soliton resolution conjecture for the critical wave equation. This conjecture has already been established in the 3D radial case and in the general case in 3, 4 and 5D along a sequence of times by Duyckaerts, Kenig and Merle. The study of the nature of the collision requires a refined approximate solution of the two-soliton problem and a precise determination of its space asymptotics. To prove inelasticity, these asymptotics are combined with the method of channels of energy.
• Sur les paquets d'Arthur des groupes classiques r\'eels
  
  • Renard David
  • Moeglin Colette
  , 2018. This article is part of a project which consists of investigating Arthur packets for real classical groups. Our goal is to give an explicit description of these packets and to establish the multiplicity one property (which is known to hold for $p$-adic and complex groups). The main result in this paper is a construction of packets from unipotent packets on $c$-Levi factors using cohomological induction. An important tool used in the argument is a statement of commutativity between cohomological induction and spectral endoscopic transfer.
• Paquets d'Arthur des groupes classiques complexes
  
  • Renard David
  • Moeglin Colette
  , 2018. Nous d\'ecrivons explicitement les paquets d'Arthur des groupes classiques complexes, ainsi que leur param\'etrisation interne par les caract\`eres du groupe des composantes connexes du centralisateur de leur param\`etre. Nous montrons d'abord qu'ils sont obtenus par induction parabolique pr\'eservant l'irr\'eductibilit\'e \`a partir des paquets unipotents de "bonne parit\'e". Pour ceux-ci, nous montrons qu'ils co\"incident avec les paquets d\'efinis par Barbasch-Vogan. Nous utilisons des r\'esultats profonds de Barbasch entrant dans sa classification du dual unitaire de ces groupes. We describe explicitly Arthur packets for complex classical groups, as well as their internal parametrization by the group of characters of the component group of the stabilizer of their parameter. We first show that they are obtained by parabolic induction preserving irreducibility from unipotent packets of "good parity". For these, we show that they coincide with the packets defined by Barbasch and Vogan. We use deep results of Barbasch entering his classification of the unitary dual of these groups.
• Sur les paquets d'Arthur aux places r\'eelles, translation
  
  • Renard David
  • Moeglin Colette
  , 2018. This article is part of a project which aims to describe as explicitly as possible the Arthur packets of classical real groups and to prove a multiplicity one result for them. Let $G$ be a symplectic or special orthogonal real group, and $\psi: W_{\mathbb R}\times \mathbf{SL}_2(\mathbb C)\rightarrow {}^LG$ be an Arthur parameter for $G$. Let $A(\psi)$ the component group of the centralizer of $\psi$ in $\hat G$. Attached to $\psi$ is a finite length unitary representation $\pi^A(\psi)$ of $G\times A(\psi)$, which is characterized by the endoscopic identities (ordinary and twisted) it satisfies. In [arXiv:1703.07226] we gave a description of the irreducible components of $\pi^A(\psi)$ when the parameter $\psi$ is "very regular, with good parity". In the present paper, we use translation of infinitesimal character to describe $\pi^A(\psi)$ in the general good parity case from the representation $\pi^A(\psi_+)$ attached to a very regular, with good parity, parameter $\psi_+$ obtained from $\psi$ by a simple shift.
• Paquets d'Arthur des groupes classiques et unitaires
  
  • Renard David
  • Arancibia Nicolás
  • Moeglin Colette
  , 2018. Let $G=\mathbf{G}(\mathbb{R})$ be the group of real points of a quasi-split connected reductive algebraic group defined over $\mathbb{R}$. Assume furthermore that $G$ is a classical group (symplectic, special orthogonal or unitary). We show that the packets of irreducible unitary cohomological representations defined by Adams and Johnson in 1987 coincide with the ones defined recently by J. Arthur in his work on the classification of the discrete automorphic spectrum of classical groups (C.-P. Mok for unitary groups). For this, we compute the endoscopic transfer of the stable distributions on $G$ supported by these packets to twisted $\mathbf{GL}_N$ in terms of standard modules and show that it coincides with the twisted trace prescribed by Arthur.
• Classification of Special Curves in the Space of Cubic Polynomials
  
  • Favre Charles
  • Gauthier Thomas
  International Mathematics Research Notices, Oxford University Press (OUP), 2018. We describe all special curves in the parameter space of complex cubic polynomials, that is all algebraic irreducible curves containing infinitely many post-critically finite polynomials. This solves in a strong form a conjecture by Baker and DeMarco for cubic polynomials. We also prove that an irreducible component of the algebraic curve consisting of those cubic polynomials that admit an orbit of any given period and multiplier is special if and only if the multiplier is 0. (10.1093/imrn/rnw245)
  DOI : 10.1093/imrn/rnw245
• Null-controllability of hypoelliptic quadratic differential equations
  
  • Beauchard Karine
  • Pravda-Starov Karel
  Journal de l'École polytechnique — Mathématiques, École polytechnique, 2018, 5, pp.1-43. We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space. (10.5802/jep.62)
  DOI : 10.5802/jep.62
• The Vlasov-Navier-Stokes system in a 2D pipe: existence and stability of regular equilibria
  
  • Moussa Ayman
  • Glass Olivier
  • Han-Kwan Daniel
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2018, 230 (2), pp.593–639. In this paper, we study the Vlasov-Navier-Stokes system in a 2D pipe with partially absorbing boundary conditions. We show the existence of stationary states for this system near small Poiseuille flows for the fluid phase, for which the kinetic phase is not trivial. We prove the asymptotic stability of these states with respect to appropriately compactly supported perturbations. The analysis relies on geometric control conditions which help to avoid any concentration phenomenon for the kinetic phase. (10.1007/s00205-018-1253-1)
  DOI : 10.1007/s00205-018-1253-1
• Some properties and applications of Brieskorn lattices
  
  • Sabbah Claude
  Journal of Singularities, Worldwide Center of Mathematics, LLC, 2018. (10.5427/jsing.2018.18k)
  DOI : 10.5427/jsing.2018.18k
• A NUMERICAL NOTE ON UPPER BOUNDS FOR B 2 [g] SETS
  
  • Habsieger Laurent
  • Plagne Alain
  Experimental Mathematics, Taylor & Francis, 2018, 27 (2), pp.208-214. Sidon sets are those sets such that the sums of two of its elements never coincide. They go back to the 30s when Sidon asked for the maximal size of a subset of consecutive integers with that property. This question is now answered in a satisfactory way. Their natural generalization, called B 2 [g] sets and defined by the fact that there are at most g ways (up to reordering the summands) to represent a given integer as a sum of two elements of the set, are much more difficult to handle and not as well understood. In this article, using a numerical approach, we improve the best upper estimates on the size of a B 2 [g] set in an interval of integers in the cases g = 2, 3, 4 and 5.
• Strongly interacting blow up bubbles for the mass critical NLS
  
  • Martel Yvan
  • Raphaël Pierre
  Annales Scientifiques de l'École Normale Supérieure, Gauthier-Villars ; Société mathématique de France, 2018. We construct a new class of multi-solitary wave solutions for the mass critical two dimensional nonlinear Schrodinger equation (NLS). Given any integer K>1, there exists a global (for positive time) solution of (NLS) that decomposes asymptotically into a sum of solitary waves centered at the vertices of a K-sided regular polygon and concentrating at a logarithmic rate in large time. This solution blows up in infinite time with logarithmic rate. Using the pseudo-conformal transform, this yields the first example of solution blowing up in finite time with a rate strictly above the pseudo-conformal one. Such solution concentrates K bubbles at a point. These special behaviors are due to strong interactions between the waves, in contrast with previous works on multi-solitary waves of (NLS) where interactions do not affect the blow up rate.
• Non-existence of torically maximal hypersurfaces
  
  • Brugallé Erwan
  • Mikhalkin Grigory
  • Risler Jean-Jacques
  • Shaw Kristin
  Algebra i Analiz, Nauka, Leningradskoe otdelenie, 2018, 30 (1), pp.20-31. Torically maximal curves (known also as simple Harnack curves) are real algebraic curves in the projective plane such that their logarithmic Gau{\ss} map is totally real. In this paper we show that hyperplanes in projective spaces are the only torically maximal hypersurfaces of higher dimensions.
• Rayleigh-Schrödinger series and Birkhoff decomposition
  
  • Novelli Jean-Christophe
  • Paul Thierry
  • Sauzin David
  • Thibon Jean-Yves
  Letters in Mathematical Physics, Springer Verlag, 2018, 108 (7), pp.1583-1600. We derive new expressions for the Rayleigh-Schr\"odinger series describing the perturbation of eigenvalues of quantum Hamiltonians. The method, somehow close to the so-called dimensional renormalization in quantum field theory, involves the Birkhoff decomposition of some Laurent series built up out of explicit fully non-resonant terms present in the usual expression of the Rayleigh-Schr\"odinger series. Our results provide new combinational formulae and a new way \ff{of deriving} perturbation series in Quantum Mechanics. More generally we prove that such a decomposition provides solutions of general normal form problems in Lie algebras. (10.1007/s11005-017-1040-1)
  DOI : 10.1007/s11005-017-1040-1
• Duality, refined partial Hasse invariants and the canonical filtration
  
  • Bijakowski Stéphane
  Mathematical Research Letters, International Press, 2018, 25 (4), pp.1109-1142. Let G be a p-divisible group over the ring of integers of C-p, and assume that it is endowed with an action of the ring of integers of a finite unramified extension F of Q(p). Let us fix the type mu of this action on the sheaf of differentials omega(G). V. Hernandez, following a construction of Goldring and Nicole, defined partial Hasse invariants for G. The product of these invariants is the mu-ordinary Hasse invariant, and it is non-zero if and only if the p-divisible group is mu-ordinary (i.e. the Newton polygon is minimal given the type of the action). We show that if the valuation of the mu-ordinary Hasse invariant is small enough, then each of these partial Hasse invariants is a product of other sections, the refined partial Hasse invariants. We also give a condition for the construction of these invariants over an arbitrary scheme of characteristic p. We then give a simple, natural and elegant proof of the compatibility with duality for the classical Hasse invariant, and show how to adapt it to the case of the refined partial Hasse invariants. Finally, we show how these invariants allow us to compute the partial degrees of the canonical filtration (if it exists). (10.4310/MRL.2018.v25.n4.a3)
  DOI : 10.4310/MRL.2018.v25.n4.a3
• On the size of chaos in the mean field dynamics
  
  • Paul Thierry
  • Pulvirenti Mario
  • Simonella Sergio
  Archive for Rational Mechanics and Analysis, Springer Verlag, 2018, 231, pp.285–317. We consider the error arising from the approximation of an N-particle dynamics with its description in terms of a one-particle kinetic equation. We estimate the distance between the j-marginal of the system and the factorized state, obtained in a mean field limit as N → ∞. Our analysis relies on the evolution equation for the " correlation error " rather than on the usual BBGKY hierarchy. The rate of convergence is shown to be O(j 2 N) in any bounded interval of time (size of chaos), as expected from heuristic arguments. Our formalism applies to an abstract hierarchical mean field model with bounded collision operator and a large class of initial data, covering (a) stochastic jump processes converging to the homogeneous Boltzmann and the Povzner equation and (b) quantum systems giving rise to the Hartree equation.
• Einstein Equations Under Polarized ${\mathbb{U}}$ (1) Symmetry in an Elliptic Gauge
  
  • Huneau Cécile
  • Luk Jonathan
  Communications in Mathematical Physics, Springer Verlag, 2018, 361 (3), pp.873-949. We prove local existence of solutions to the Einstein–null dust system under polarized ${\mathbb{U}}$ (1) symmetry in an elliptic gauge. Using in particular the previous work of the first author on the constraint equations, we show that one can identify freely prescribable data, solve the constraints equations, and construct a unique local in time solution in an elliptic gauge. Our main motivation for this work, in addition to merely constructing solutions in an elliptic gauge, is to provide a setup for our companion paper in which we study high frequency backreaction for the Einstein equations. In that work, the elliptic gauge we consider here plays a crucial role to handle high frequency terms in the equations. The main technical difficulty in the present paper, in view of the application in our companion paper, is that we need to build a framework consistent with the solution being high frequency, and therefore having large higher order norms. This difficulty is handled by exploiting a reductive structure in the system of equations. (10.1007/s00220-018-3167-z)
  DOI : 10.1007/s00220-018-3167-z
• Construction of multi-bubble solutions for the critical gKdV equation
  
  • Combet Vianney
  • Martel Yvan
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2018, 50 (4), pp.3715-3790. We prove the existence of solutions of the mass critical generalized Korteweg-de Vries equation $\partial_t u + \partial_x(\partial_{xx} u + u^5) = 0$ containing an arbitrary number $K\geq 2$ of blow up bubbles, for any choice of sign and scaling parameters: for any $\ell_1>\ell_2>\cdots>\ell_K>0$ and $\epsilon_1,\ldots,\epsilon_K\in\{\pm1\}$, there exists an $H^1$ solution $u$ of the equation such that \[ u(t) - \sum_{k=1}^K \frac {\epsilon_k}{\lambda_k^\frac12(t)} Q\left( \frac {\cdot - x_k(t)}{\lambda_k(t)} \right) \longrightarrow 0 \quad\mbox{ in }\ H^1 \mbox{ as }\ t\downarrow 0, \] with $\lambda_k(t)\sim \ell_k t$ and $x_k(t)\sim -\ell_k^{-2}t^{-1}$ as $t\downarrow 0$. The construction uses and extends techniques developed mainly by Martel, Merle and Rapha\"el. Due to strong interactions between the bubbles, it also relies decisively on the sharp properties of the minimal mass blow up solution (single bubble case) proved by the authors in arXiv:1602.03519. (10.1137/17M1140595)
  DOI : 10.1137/17M1140595