Recent publications (since 2009)

Here are some recent publications (since 2009). Preprints not yet published (or accepted) are here, papers published before 2009 are there.
There are some other texts (surveys, etc) on another page.
Please send me an e-mail (or a s-mail) if you wish to receive a preprint or a reprint.
If you have a Mathscinet access you can check the Math Reviews of those papers and of some older stuff.

  • Symplectic Wick rotations between moduli spaces of 3-manifolds. Carlos Scarinci, Jean-Marc Schlenker. arXiv:1411.4772. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, to appear.
    Given a closed hyperbolic surface S, let $\cQF$ denote the space of quasifuchsian hyperbolic metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$, called "Wick rotations", defined in terms of special surfaces (e.g. minimal/maximal surfaces, CMC surfaces, pleated surfaces) and prove that these maps are at least C1 smooth and symplectic with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described. These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure c and a holomorphic quadratic differential q on S to the pair of hyperbolic metrics (mL,mR) such that the harmonic maps isotopic to the identity from (S,c) to (S,mL) and to (S,mR) have, respectively, Hopf differentials equal to iq and −iq, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric m and a measured lamination l on S to the pair (EL(m,l),ER(m,l)), where EL and ER denote the left and right earthquakes. We describe how such 2-dimensional double maps are related to 3-dimensional Wick rotations and prove that they are also C1 smooth and symplectic.

  • The renormalized volume and uniformisation of conformal structures. Colin Guillarmou, Sergiu Moroianu, Jean-Marc Schlenker. arXiv:1211.6507. J. Inst. Math. Jussieu, to appear.
    We study the renormalized volume of asymptotically hyperbolic Einstein (AHE in short) manifolds $(M,g)$ when the conformal boundary $\pl M$ has dimension $n$ even. Its definition depends on the choice of metric $h_0$ on $\partial M$ in the conformal class at infinity determined by $g$, we denote it by ${\rm Vol}_R(M,g;h_0)$. We show that ${\rm Vol}_R(M,g;\cdot)$ is a functional admitting a "Polyakov type" formula in the conformal class $[h_0]$ and we describe the critical points as solutions of some non-linear equation $v_n(h_0)={\rm const}$, satisfied in particular by Einstein metrics. In dimension $n=2$, choosing extremizers in the conformal class amounts to uniformizing the surface, while in dimension $n=4$ this amounts to solving the $\sigma_2$-Yamabe problem. Next, we consider the variation of ${\rm Vol}_R(M,\cdot;\cdot)$ along a curve of AHE metrics $g^t$ with boundary metric $h_0^t$ and we use this to show that, provided conformal classes can be (locally) parametrized by metrics $h$ solving $v_n(h)=\int_{\pl M}v_n(h){\rm dvol}_{h}$, the set of ends of AHE manifolds (up to diffeomorphisms isotopic to Identity) can be viewed as a Lagrangian submanifold in the cotangent space to the space $\mc{T}(\pl M)$ of conformal structures on $\pl M$. We obtain as a consequence a higher-dimensional version of McMullen's quasifuchsian reciprocity. We finally show that conformal classes admitting negatively curved Einstein metrics are local minima for the renormalized volume for a warped product type filling.

  • Small circulant complex Hadamard matrices of Butson type. Gaurush Hiranandani, Jean-Marc Schlenker. arXiv:1311.5390. Online appendix. European Journal of Combinatorics 51, January 2016, 306--314.
    We study the circulant complex Hadamard matrices of order n whose entries are l-th roots of unity. For n=l prime we prove that the only such matrix, up to equivalence, is the Fourier matrix, while for n=p+q,l=pq with p,q distinct primes there is no such matrix. We then provide a list of equivalence classes of such matrices, for small values of n,l.

  • A cyclic extension of the earthquake flow II. Francesco Bonsante, Gabriele Mondello, Jean-Marc Schlenker. arXiv:1208.1738. Annales Scientifiques de l'ENS 48(2015):4, 811-859.
    The landslide flow, introduced in [5], is a smoother analog of the earthquake flow on Teichmüller space which shares some of its key properties. We show here that further properties of earthquakes apply to landslides. The landslide flow is the Hamiltonian flow of a convex function. The smooth grafting map $sgr$ taking values in Teichmüller space, which is to landslides as grafting is to earthquakes, is proper and surjective with respect to either of its variables. The smooth grafting map $SGr$ taking values in the space of complex projective structures is symplectic (up to a multiplicative constant). The composition of two landslides has a fixed point on Teichmüller space. As a consequence we obtain new results on constant Gauss curvature surfaces in 3-dimensional hyperbolic or AdS manifolds. We also show that the landslide flow has a satisfactory extension to the boundary of Teichmüller space.

  • Submatrices of Hadamard matrices: complementation results. Teodor Banica, Ion Nechita, Jean-Marc Schlenker. arXiv:1311.0764. Electronic Journal of Linear Algebra 27 (2014), 197-212.
    Two submatrices $A,D$ of a Hadamard matrix $H$ are called complementary if, up to a permutation of rows and columns, $H=[^A_C{\ }^B_D]$. We find here an explicit formula for the polar decomposition of $D$. As an application, we show that under suitable smallness assumptions on the size of $A$, the complementary matrix $D$ is an almost Hadamard sign pattern, i.e. its rescaled polar part is an almost Hadamard matrix.

  • Analytic aspects of the circulant Hadamard conjecture. Teodor Banica, Ion Nechita, Jean-Marc Schlenker. arXiv:1212.3589. Annales mathématiques Blaise Pascal 21(2014), 25-59.
    We investigate the problem of counting the real or complex Hadamard matrices which are circulant, by using analytic methods. Our main observation is the fact that for $|q_0|=...=|q_{N-1}|=1$ the quantity $\Phi=\sum_{i+k=j+l}\frac{q_iq_k}{q_jq_l}$ satisfies $\Phi\geq N^2$, with equality if and only if $q=(q_i)$ is the eigenvalue vector of a rescaled circulant complex Hadamard matrix. This suggests three analytic problems, namely: (1) the brute-force minimization of $\Phi$, (2) the study of the critical points of $\Phi$, and (3) the computation of the moments of $\Phi$. We explore here these questions, with some results and conjectures.

  • The convex core of quasifuchsian manifolds with particles. Cyril Lecuire, Jean-Marc Schlenker. arXiv:0909.4182. Geometry and topology 18-4 (2014), 2309--2373.
    We consider quasifuchsian manifolds with ``particles'', i.e., cone singularities of fixed angle less than $\pi$ going from one connected component of the boundary at infinity to the other. Each connected component of the boundary at infinity is then endowed with a conformal structure marked by the endpoints of the particles. We prove that this defines a homeomorphism from the space of quasifuchsian metrics with $n$ particles (of fixed angle) and the product of two copies of the Teichm\"uller space of a surface with $n$ marked points. This is analoguous to the Bers theorem in the non-singular case. Quasifuchsian manifolds with particles also have a convex core. Its boundary has a hyperbolic induced metric, with cone singularities at the intersection with the particles, and is pleated along a measured geodesic lamination. We prove that any two hyperbolic metrics with cone singularities (of prescribed angle) can be obtained, and also that any two measured bending laminations, satisfying some obviously necessary conditions, can be obtained, as in [BO] in the non-singular case.

  • Recovering the geometry of a flat spacetime from a background radiation. Francesco Bonsante, Catherine Meusburger, Jean-Marc Schlenker. arXiv:1302.6966. Annales Henri Poincaré 15:9 (2014), 1733--1799.
    We consider globally hyperbolic flat spacetimes in 2+1 and 3+1 dimensions, where a uniform light signal is emitted on the $r$-level surface of the cosmological time for $r\to 0$. We show that the intensity of this signal, as perceived by a fixed observer, is a well-defined, bounded function which is generally not continuous. This defines a purely classical model with anisotropic background radiation that contains information about initial singularity of the spacetime. In dimension 2+1, we show that this observed intensity function is stable under suitable perturbations of the spacetime, and that, under certain conditions, it contains sufficient information to recover its geometry and topology. We compute an approximation of this intensity function in a few simple examples.

  • Productivity and Mobility in Academic Research: Evidence from Mathematicians. Pierre Dubois, Jean-Charles Rochet, Jean-Marc Schlenker. Working paper TSE 10-160 and IDEI 606 , may 2010. Scientometrics 98(2014):3, 1669-1701.
    Using an exhaustive database on academic publications in mathematics, we study the patterns of productivity by world mathematicians over the period 1984-2006. We uncover some surprising facts, such as the absence of age related decline in productivity and the relative symmetry of international movements, rejecting the presumption of a massive "brain drain" towards the U.S. Looking at the U.S. academic market in mathematics, we analyze the determinants of success by top departments. In conformity with recent studies in other fields, we find that selection effects are much stronger than local interaction effects: the best departments are most successful in hiring the most promising mathematicians, but not necessarily at stimulating positive externalities among them. Finally we analyze the impact of career choices by mathematicians: mobility almost always pays, but early specialization does not.

  • The renormalized volume and the volume of the convex core of quasifuchsian manifolds. Jean-Marc Schlenker. arXiv:1109.6663. Mathematical Research Letters 20(2013):4, 773-786.
    We show that the renormalized volume of a quasifuchsian hyperbolic 3-manifold is equal, up to an additive constant, to the volume of its convex core. We also provide a precise upper bound on the renormalized volume in terms of the Weil-Petersson distance between the conformal structures at infinity. As a consequence we show that holomorphic disks in Teichm\"uller space which are large enough must have ``enough'' negative curvature.

  • Collisions of particles in locally AdS spacetimes II. Moduli of globally hyperbolic spaces Thierry Barbot, Francesco Bonsante, Jean-Marc Schlenker. arXiv:1202.5753. Comm. Math. Phys. 327(2014):3, 691-735.
    We investigate 3-dimensional globally hyperbolic AdS manifolds containing ``particles'', i.e., cone singularities of angles less than $2\pi$ along a time-like graph $\Gamma$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partial extension of a result of Mess for non-singular globally hyperbolic AdS manifolds.

  • A cyclic extension of the earthquake flow I. Francesco Bonsante, Gabriele Mondello, Jean-Marc Schlenker. arXiv:1106.0525. Geometry and Topology, 17 (2013), no. 1, 157--234.
    Let $\cT$ be Teichm\"uller space of a closed surface of genus at least 2. For any point $c\in \cT$, we describe an action of the circle on $\cT\times \cT$, which limits to the earthquake flow when one of the parameters goes to a measured lamination in the Thurston boundary of $\cT$. This circle action shares some of the main properties of the earthquake flow, for instance it satisfies an extension of Thurston's Earthquake Theorem and it has a complex extension which is analogous and limits to complex earthquakes. Moreover, a related circle action on $\cT\times \cT$ extends to the product of two copies of the universal Teichm\"uller space.

  • Non-rigidity of spherical inversive distance circle packings. Jiming Ma, Jean-Marc Schlenker. arXiv:1105.1469. Discrete and Computational Geometry 47:3 (2012), 610--617.
    We give a counterexample of Bowers-Stephenson's conjecture in the spherical case: spherical inversive distance circle packings are not determined by their inversive distances.

  • Flippable tilings of constant curvature surfaces. François Fillastre, Jean-Marc Schlenker. arXiv:1012.1594. Illinois J. Math. 56(2012):4, 1213-1256.
    We call ``flippable tilings'' of a constant curvature surface a tiling by ``black'' and ``white'' faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to ``flip'' the tiling by pushing all black faces forward on the left side and backward on the right side. Among those tilings we distinguish the ``symmetric'' ones, for which the metric on the surface does not change under the flip. We provide some existence statements, and explain how to parameterize the space of those tilings (with a fixed number of black faces) in different ways. For instance one can glue the white faces only, and obtain a metric with cone singularities which, in the hyperbolic and spherical case, uniquely determines a symmetric tiling. The proofs are based on the geometry of polyhedral surfaces in 3-dimensional spaces modeled either on the sphere or on the anti-de Sitter space.

  • Fixed points of compositions of earthquakes. Francesco Bonsante, Jean-Marc Schlenker. arXiv:0812.3471. Duke Mathematical Journal 161(2012):6, 1011--1054.
    Let S be a closed surface of genus at least 2, and consider two measured geodesic laminations that fill S. Right earthquakes along these laminations are diffeomorphisms of the Teichmüller space of S. We prove that the composition of these earthquakes has a fixed point in the Teichmüller space. Another way to state this result is that it is possible to prescribe any two measured laminations that fill a surface as the upper and lower measured bending laminations of the convex core of a globally hyperbolic AdS manifold. The proof uses some estimates from the geometry of those AdS manifolds.

  • Collisions of particles in locally AdS spacetimes I. Local description and global examples. Thierry Barbot, Francesco Bonsante and Jean-Marc Schlenker. arXiv:1010.3602. Communications in Mathematical Physics 308(2011):1, 147-200.
    We investigate 3-dimensional globally hyperbolic AdS manifolds (or more generally constant curvature Lorentz manifolds) containing ``particles'', i.e., cone singularities along a graph $\Gamma$. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than $2\pi$ on time-like singular segments). We construct examples of such manifolds, describe the cone singularities that can arise and the way they can interact (the local geometry near the vertices of $\Gamma$). We then adapt to this setting some notions like global hyperbolicity which are natural for Lorentz manifolds, and construct some examples of globally hyperbolic AdS manifolds with interacting particles.

  • Combinatorial aspects of orthogonal group integrals. Teodor Banica and Jean-Marc Schlenker. arXiv:1011.2454. Intern. J. Math. 22:11 (2011), 1611-1646.
    We study the integrals of type $I(a)=\int_{O_n}\prod u_{ij}^{a_{ij}}\,du$, depending on a matrix of exponents $a\in M_{p\times q}(\mathbb N)$, whose exact computation is an open problem. Our results are as follows: (1) an extension of the ``elementary expansion'' formula from the case $a\in M_{2\times q}(2\mathbb N)$ to the general case $a\in M_{p\times q}(\mathbb N)$, (2) the construction of a ``best algebraic normalization'' of $I(a)$, in the case $a\in M_{2\times q}(\mathbb N)$, (3) an explicit formula for $I(a)$, for diagonal matrices $a\in M_{3\times 3}(\mathbb N)$, (4) a modelling result in the case $a\in M_{1\times 2}(\mathbb N)$, in relation with the Euler-Rodrigues formula. Most proofs use various combinatorial techniques.

  • Volume maximization and the extended hyperbolic space. Feng Luo, Jean-Marc Schlenker. arXiv:0908.2023. Proc. Amer. Math. Soc. 140:3 (2012) 1053--1068.
    We consider a volume maximization program to construct hyperbolic structures on triangulated 3-manifolds, for which previous progress has lead to consider angle assignments which do not correspond to a hyperbolic metric on each simplex. We show that critical points of the generalized volume are associated to geometric structures modeled on the extended hyperbolic space -- the natural extension of hyperbolic space by the de Sitter space -- except for the degenerate case where all simplices are Euclidean in a generalized sense.
    Those extended hyperbolic structures can realize geometrically a decomposition of the manifold as connected sum of manifolds admitting a complete hyperbolic metric, along embedded spheres (or projective planes) which are totally geodesic, space-like surfaces in the de Sitter part of the extended hyperbolic structure.

  • On polynomial integrals over the orthogonal group. Teodor Banica, Benoit Collins, Jean-Marc Schlenker. arXiv:0910.1258. Journal of Combinatorial Theory A 118:3 (2011), 778-795.
    We consider integrals of type $\int_{O_n}u_{11}^{a_1}... u_{1n}^{a_n}u_{21}^{b_1}... u_{2n}^{b_n} du$, with respect to the Haar measure on the orthogonal group. We establish several remarkable invariance properties satisfied by such integrals, by using combinatorial methods. We present as well a general formula for such integrals, as a sum of products of factorials.

  • Maximal surfaces and the universal Teichmüller space. Francesco Bonsante, Jean-Marc Schlenker. Inventiones Mathematicae 182(2010):279-333. arXiv:0911.4124.
    We show that any element of the universal Teichmüller space is realized by a unique minimal Lagrangian diffeomorphism from the hyperbolic plane to itself. The proof uses maximal surfaces in the 3-dimensional anti-de Sitter space. We show that, in $AdS^{n+1}$, any subset $E$ of the boundary at infinity which is the boundary at infinity of a space-like hypersurface bounds a maximal space-like hypersurface. In $AdS^3$, if $E$ is the graph of a quasi-symmetric homeomorphism, then this maximal surface is unique, and it has negative sectional curvature. As a by-product, we find a simple characterization of quasi-symmetric homeomorphisms of the circle in terms of 3-dimensional projective geometry.

  • On orthogonal matrices maximizing the 1-norm. Teodor Banica, Benoit Collins, Jean-Marc Schlenker. arXiv:0901.2923. Indiana Univ. Math. J. 59(2010):3, 839-856.
    For $U\in O(N)$ we have $||U||_1\leq N\sqrt{N}$, with equality if and only if $U=H/\sqrt{N}$, with $H$ Hadamard matrix. Motivated by this remark, we discuss in this paper the algebraic and analytic aspects of the computation of the maximum of the 1-norm on O(N). The main problem is to compute the $k$-th moment of the 1-norm, with $k\to\infty$, and we present a number of general comments in this direction.

  • Multi Black Holes and Earthquakes on Riemann surfaces with boundaries. Francesco Bonsante, Kirill Krasnov, Jean-Marc Schlenker. Int. Math. Res. Not. 2011 (2011):3, 487-552. math.GT/0610429. Abstract or pdf.
    We prove an ``Earthquake Theorem'' for hyperbolic metrics with geodesic boundary on a compact surfaces $S$ with boundary: the action of earthquakes on the enhanced Teichmüller space of $S$ is simply transitive. The proof rests on the geometry of ``multi-black holes'', which are 3-dimensional anti-de Sitter manifolds, topologically the product of a surface with boundary by an interval.

  • The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds. Kirill Krasnov and Jean-Marc Schlenker. arXiv:0907.2590. To appear, Handbook of Teichmueller theory, vol III.
    We survey the renormalized volume of hyperbolic 3-manifolds, as a tool for Teichm\"uller theory, using simple differential geometry arguments to recover results sometimes first achieved by other means. One such application is McMullen's quasifuchsian (or more generally Kleinian) reciprocity, for which different arguments are proposed. Another is the fact that the renormalized volume of quasifuchsian (or more generally geometrically finite) hyperbolic 3-manifolds provides a K\"ahler potential for the Weil-Petersson metric on Teichm\"uller space. Yet another is the fact that the grafting map is symplectic, which is proved using a variant of the renormalized volume defined for hyperbolic ends.

  • On the infinitesimal rigidity of polyhedra with vertices in convex position. Ivan Izmestiev, Jean-Marc Schlenker. arXiv:0711.1981. Pacific J. Math. 248(2010):1, 171-190.
    Let $P \subset \R^3$ be a polyhedron. It was conjectured that if $P$ is weakly convex (i. e. its vertices lie on the boundary of a strictly convex domain) and decomposable (i. e. $P$ can be triangulated without adding new vertices), then it is infinitesimally rigid. We prove this conjecture under a weak additional assumption of codecomposability.
    The proof relies on a result of independent interest concerning the Hilbert-Einstein function of a triangulated convex polyhedron. We determine the signature of the Hessian of that function with respect to deformations of the interior edges. In particular, if there are no interior vertices, then the Hessian is negative definite.

  • Profiles of inflated surfaces. Igor Pak, Jean-Marc Schlenker. arXiv:0907.5057. Journal of Nonlinear Mathematical Physics 17:2 (2010) 145-157.
    We study the shape of inflated surfaces introduced in~\cite{B1} and~\cite{P1}. More precisely, we analyze profiles of surfaces obtained by inflating a convex polyhedron, or more generally an almost everywhere flat surface, with a symmetry plane. We show that such profiles are in a one-parameter family of curves which we describe explicitly as the solutions of a certain differential equation.

  • A symplectic map between hyperbolic and complex Teichmüller theory. Kirill Krasnov, Jean-Marc Schlenker. arXiv:0806.0010. Duke Mathematical Journal 150(2009):2, 331-356.
    Let $S$ be a closed, orientable surface of genus at least $2$. The cotangent bundle of the ``hyperbolic'' Teichmüller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations, while the cotangent bundle of the ``complex'' Teichmüller space can be identified with $\CP$ through the Schwarzian derivative. We prove that the resulting map between the two cotangent spaces, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends.

  • Representations of quantum permutation algebras. Teodor Banica, Julien Bichon, Jean-Marc Schlenker. arXiv:0901.2331. J. Funct. Anal. 257 (2009), 2864-2910.
    We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type $\pi:A_s(n)\to B(H)$. We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to $n=6$.

  • On weakly convex star-shaped polyhedra. Jean-Marc Schlenker. arXiv:0704.2901. Discrete Mathematics 309(2009):20, 6139-6149..
    Weakly convex polyhedra which are star-shaped with respect to one of their vertices are infinitesimally rigid. This is a partial answer to the question whether every decomposable weakly convex polyhedron is infinitesimally rigid. The proof uses a recent result of Izmestiev on the geometry of convex caps.

  • Quasi-Fuchsian manifolds with particles. Sergiu Moroianu and Jean-Marc Schlenker. math.DG/0603441. J. Differential Geom. 83:1 (2009), 75-129.
    We consider 3-dimensional hyperbolic cone-manifolds, singular along infinite lines, which are ``convex co-compact'' in a natural sense. We prove an infinitesimal rigidity statement when the angle around the singular lines is less than $\pi$: any first-order deformation changes either one of those angles or the conformal structure at infinity, with marked points corresponding to the endpoints of the singular lines. Moreover, any small variation of the conformal structure at infinity and of the singular angles can be achieved by a unique small deformation of the cone-manifold structure.

  • On the infinitesimal rigidity of weakly convex polyhedra. Robert Connelly and Jean-Marc Schlenker. math.DG/0606681. European Journal of Combinatorics, 31(2010):4, 1080-1090. Special issue, Rigidity and related topics in Geometry.
    The main motivation here is a question: whether any polyhedron which can be subdivided into convex pieces without adding a vertex, and which has the same vertices as a convex polyhedron, is infinitesimally rigid. We prove that it is indeed the case for two classes of polyhedra: those obtained from a convex polyhedron by ``denting'' at most two edges at a common vertex, and suspensions with a natural subdivision.

  • AdS manifolds with particles and earthquakes on singular surfaces. Francesco Bonsante, Jean-Marc Schlenker. math.GT/0609116. Geom. Funct. Anal. 19:1 (2009), 41-82.
    We prove an ``Earthquake Theorem'' for closed hyperbolic surfaces with cone singularities where the total angle is less than $\pi$: any two points in the Teichmüller space are connected by a unique left earthquakes. This is strongly related to another result: the space of ``globally hyperbolic'' AdS manifolds with cone singularities (of given angle) along time-like geodesics is parametrized by the product of two copies of the Teichmüller space with some marked points (corresponding to the cone singularities).

  • Color code:
    3-d hyperbolic geometry
    Teichmueller theory
    AdS or Lorentz geometry
    manifolds with boundary
    manifolds with "particles"
    circle patterns, polyhedral geometry, discrete geometry
    Hadamard matrices, integration over O(n), etc,
    Outside mathematics

    Older papers

    FSCT -- University of Luxembourg