Publications

2011

• Strong phase-space semiclassical asymptotics
  
  • Paul Thierry
  • Athanassoulis Agissilaos
  SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 2011, 43 (5), pp.2116-2149. Wigner and Husimi transforms have long been used for the phase- space reformulation of SchrÄodinger-type equations, and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are con- cerned with semiclassical limits in the strong topology, i.e. approxima- tion of Wigner functions by solutions of the Liouville equation in L2 and Sobolev norms. The results obtained improve the state of the art, and highlight the role of potential regularity, especially through the regularity of the Wigner equation. It must be mentioned that the strong conver- gence can be shown up to logarithmic time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics. (10.1137/10078712X)
  DOI : 10.1137/10078712X
• The Dirac operator on generalized Taub-NUT spaces
  
  • Moroianu Andrei
  • Moroianu Sergiu
  Comm. Math. Phys., 2011, 305 (3), pp.641-656. We find sufficient conditions for the absence of harmonic $L^2$ spinors on spin manifolds constructed as cone bundles over a compact Kähler base. These conditions are fulfilled for certain perturbations of the Euclidean metric, and also for the generalized Taub-NUT metrics of Iwai-Katayama, thus proving a conjecture of Vi\csinescu and the second author. (10.1007/s00220-011-1263-4)
  DOI : 10.1007/s00220-011-1263-4
• Dynamical compactifications of C^2
  
  • Favre Charles
  • Jonsson Mattias
  Annals of Mathematics, Princeton University, Department of Mathematics, 2011, 173 (1), pp.211-249. We find good dynamical compactifications for arbitrary polynomial mappings of C^2 and use them to show that the degree growth sequence satisfies a linear integral recursion formula. For maps of low topological degree we prove that the Green function is well behaved. For maps of maximum topological degree, we give normal forms. (10.4007/annals.2011.173.1.6)
  DOI : 10.4007/annals.2011.173.1.6
• Distributions propres invariantes sur la paire sym\' etrique (gl(4,R),gl(2,R)*gl(2,R))
  
  • Harinck Pascale
  • Jacquet Nicolas
  Journal of Functional Analysis, Elsevier, 2011, 261 (9), pp.2362-2436. We study orbital integrals and invariant eigendistributions for the symmetric pair (g,h)=(gl(4,R),gl(2,R)*gl(2,R)). Let q=g/h and let N be the set of nilpotents of q. We first obtain an asymptotic behavior of orbital integrals around nonzero semisimple elements of q. We study eigendistributions around such elements and give an explicit basis of eigendistributions on q-N given by a locally integrable function on q-N. (10.1016/j.jfa.2011.06.012)
  DOI : 10.1016/j.jfa.2011.06.012
• Linear forms and quadratic uniformity for functions on $\mathbb{F}_p^n$
  
  • Gowers W. T.
  • Wolf Julia
  Mathematika, University College London, 2011, 57 (2), pp.215-237. We give improved bounds for our theorem in [GW09], which shows that a system of linear forms on $\mathbb{F}_p^n$ with squares that are linearly independent has the expected number of solutions in any linearly uniform subset of $\mathbb{F}_p^n$. While in [GW09] the dependence between the uniformity of the set and the resulting error in the average over the linear system was of tower type, we now obtain a doubly exponential relation between the two parameters. Instead of the structure theorem for bounded functions due to Green and Tao [GrT08], we use the Hahn-Banach theorem to decompose the function into a quadratically structured plus a quadratically uniform part. This new decomposition makes more efficient use of the $U^3$ inverse theorem [GrT08].
• Nijenhuis structures on Courant algebroids
  
  • Kosmann-Schwarzbach Yvette
  Boletim da Sociedade Brasileira de Matemática / Bulletin of the Brazilian Mathematical Society, Springer Verlag, 2011, 42 (4), pp.625-649. We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations. We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skew-symmetric if the square of N is proportional to the identity, and only in this case when the Courant algebroid is irreducible. We derive a necessary and sufficient condition for a skew-symmetric endomorphism to give rise to a deformed Courant structure. In the case of the double of a Lie bialgebroid (A,A*), given an endomorphism n of A that defines a skew-symmetric endomorphism N of the double of A, we prove that the torsion of N is the sum of the torsion of n and that of the transpose of n. (10.1007/s00574-011-0032-5)
  DOI : 10.1007/s00574-011-0032-5
• On a weak variant of the geometric torsion conjecture.
  
  • Cadoret Anna
  • Tamagawa Akio
  Journal of Algebra, Elsevier, 2011, 346 (1), pp.227-247. A consequence of the geometric torsion conjecture for abelian varieties over function fields is the following. Let k be an algebraically closed field of characteristic 0. For any integers d,g⩾0d,g⩾0 there exists an integer N:=N(k,d,g)⩾1N:=N(k,d,g)⩾1 such that for any function field L/kL/k with transcendence degree 1 and genus ⩽g and any d-dimensional abelian variety A→LA→L containing no nontrivial k-isotrivial abelian subvariety, Ators(L)⊂A[N]A(L)tors⊂A[N]. In this paper, we deal with a weak variant of this statement, where A→LA→L runs only over abelian varieties obtained from a fixed (d-dimensional) abelian variety by base change. More precisely, let K/kK/k be a function field with transcendence degree 1 and A→KA→K an abelian variety containing no nontrivial k-isotrivial abelian subvariety. Then we show that if K has genus ⩾1 or if A→KA→K has semistable reduction over all but possibly one place, then, for any integer g⩾0g⩾0, there exists an integer N:=N(A,g)⩾1N:=N(A,g)⩾1 such that for any finite extension L/KL/K with genus ⩽g, Ators(L)⊂A[N]A(L)tors⊂A[N]. Previous works of the authors show that this holds--without any restriction on K--for the ℓ-primary torsion (with ℓ a fixed prime). So, it is enough to prove that there exists an integer N:=N(A,g)⩾1N:=N(A,g)⩾1 such that for any finite extension L/KL/K with genus ⩽g, the prime divisors of |Ators(L)||A(L)tors| are all ⩽N. (10.1016/j.jalgebra.2011.09.002)
  DOI : 10.1016/j.jalgebra.2011.09.002
• Effective dynamics of double solitons for perturbed mKdV
  
  • Perelman Galina
  • Holmer Justin
  • Zworski Maciej
  Communications in Mathematical Physics, Springer Verlag, 2011, 305 (2), pp.363-425. We consider the perturbed mKdV equation ∂_t u=−∂_x (∂^2_x u+2u^3−b(x,t)u) , where the potential b(x,t)=b_0(hx,ht), 0 < h << 1, is slowly varying with a double soliton initial data. On a dynamically interesting time scale the solution is O(h^2) close in H^2 to a double soliton whose position and scale parameters follow an effective dynamics, a simple system of ordinary differential equations. These equations are formally obtained as Hamilton's equations for the restriction of the mKdV Hamiltonian to the submanifold of solitons. The interplay between algebraic aspects of complete integrability of the unperturbed equation and the analytic ideas related to soliton stability is central in the proof. (10.1007/s00220-011-1252-7)
  DOI : 10.1007/s00220-011-1252-7
• Poisson and symplectic functions in Lie algebroid theory
  
  • Kosmann-Schwarzbach Yvette
  , 2011, pp.243-268. Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both a Lie bialgebra action and a Poisson bivector. (10.1007/978-0-8176-4735-3_12)
  DOI : 10.1007/978-0-8176-4735-3_12
• Large energy entire solutions for the Yamabe equation
  
  • Pacard Frank
  • del Pino Manuel
  • Musso Monica
  • Pistoia Angela
  Journal of Differential Equations, Elsevier, 2011, 251 (9), pp.2568-2597. We consider the Yamabe equation Delta u + n(n-2_/4 vertical bar u vertical bar 4/n-2 u = 0 in R(n), n >= 3. Let k >= 1 and xi(k)(j) = (e(2j pi u/k), 0) is an element of R(n) = C x R(n-2). For all large k we find a solution of the form u(k)(x)= u(x) - Sigma(k)(j=1) mu(k) (-n-2/2) U X (mu(-1)(k) (x - xi(j)) +o(1), where U(x) = (2/1+vertical bar x vertical bar(2)) (n-2/2), mu(k) = c(n)/k(2) for n >= 4, mu k = c/k(2)(logk)(2) for n =3 and o(1) -> 0 uniformly as k -> +infinity (10.1016/j.jde.2011.03.008)
  DOI : 10.1016/j.jde.2011.03.008
• Éléments d'analyse et d'algèbre (et de théorie des nombres)
  
  • Colmez Pierre
  , 2011, pp.688.
• Constant curvature foliations in asymptotically hyperbolic spaces.
  
  • Pacard Frank
  • Rafe Mazzeo
  Revista Matemática Iberoamericana, European Mathematical Society, 2011, 27 (1), pp.303-333. Let (M,g) be an asymptotically hyperbolic manifold with a smooth conformal compactification. We establish a general correspondence between semilinear elliptic equations of scalar curvature type on ∂M and Weingarten foliations in some neighbourhood of infinity in M. We focus mostly on foliations where each leaf has constant mean curvature, though our results apply equally well to foliations where the leaves have constant σk-curvature. In particular, we prove the existence of a unique foliation near infinity in any quasi-Fuchsian 3-manifold by surfaces with constant Gauss curvature. There is a subtle interplay between the precise terms in the expansion for g and various properties of the foliation. Unlike other recent works in this area, by Rigger and Neves-Tian, we work in the context of conformally compact spaces, which are more general than perturbations of the AdS-Schwarzschild space, but we do assume a nondegeneracy condition.
• Loi de Weyl presque sûre pour un système différentiel en dimension 1
  
  • Bordeaux Montrieux William
  Annales Henri Poincaré, Springer Verlag, 2011, 12 (1), pp.173-204. (10.1007/s00023-010-0073-0)
  DOI : 10.1007/s00023-010-0073-0
• Examples of non-commutative Hodge structures
  
  • Hertling Claus
  • Sabbah Claude
  Journal of the Institute of Mathematics of Jussieu, Cambridge University Press, 2011, 10 (3), pp.635-674. We show that if the Stokes matrix of a connection with a pole of order two and no ramification gives rise, when added to its adjoint, to a positive semi-definite Hermitian form, then the associated integrable twistor structure (or TERP structure, or non-commutative Hodge structure) is pure and polarized. (10.1017/S147474801100003X)
  DOI : 10.1017/S147474801100003X
• 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
• Quadratic Goldreich-Levin Theorems
  
  • Tulsiani Madhur
  • Wolf Julia
  , 2011, pp.619-628. Decomposition theorems in classical Fourier analysis enable us to express a bounded function in terms of few linear phases with large Fourier coefficients plus a part that is pseudorandom with respect to linear phases. The Goldreich-Levin algorithm can be viewed as an algorithmic analogue of such a decomposition as it gives a way to efficiently find the linear phases associated with large Fourier coefficients. In the study of "quadratic Fourier analysis", higher-degree analogues of such decompositions have been developed in which the pseudorandomness property is stronger but the structured part correspondingly weaker. For example, it has previously been shown that it is possible to express a bounded function as a sum of a few quadratic phases plus a part that is small in the $U^3$ norm, defined by Gowers for the purpose of counting arithmetic progressions of length 4. We give a polynomial time algorithm for computing such a decomposition. A key part of the algorithm is a local self-correction procedure for Reed-Muller codes of order 2 (over $\F_2^n$) for a function at distance $1/2-\epsilon$ from a codeword. Given a function $f:\F_2^n \to \{-1,1\}$ at fractional Hamming distance $1/2-\epsilon$ from a quadratic phase (which is a codeword of Reed-Muller code of order 2), we give an algorithm that runs in time polynomial in $n$ and finds a codeword at distance at most $1/2-\eta$ for $\eta = \eta(\epsilon)$. This is an algorithmic analogue of Samorodnitsky's result, which gave a tester for the above problem. To our knowledge, it represents the first instance of a correction procedure for any class of codes, beyond the list-decoding radius. In the process, we give algorithmic versions of results from additive combinatorics used in Samorodnitsky's proof and a refined version of the inverse theorem for the Gowers $U^3$ norm over $\F_2^n$. (10.1109/FOCS.2011.59)
  DOI : 10.1109/FOCS.2011.59
• Linear forms and higher-degree uniformity for functions on $\mathbb{F}_p^n$
  
  • Gowers W. T.
  • Wolf Julia
  Geometric And Functional Analysis, Springer Verlag, 2011, 21 (1), pp.36-69. In [GW09a] we conjectured that uniformity of degree $k-1$ is sufficient to control an average over a family of linear forms if and only if the $k$th powers of these linear forms are linearly independent. In this paper we prove this conjecture in $\mathbb{F}_p^n$, provided only that $p$ is sufficiently large. This result represents one of the first applications of the recent inverse theorem for the $U^k$ norm over $\mathbb{F}_p^n$ by Bergelson, Tao and Ziegler [BTZ09,TZ08]. We combine this result with some abstract arguments in order to prove that a bounded function can be expressed as a sum of polynomial phases and a part that is small in the appropriate uniformity norm. The precise form of this decomposition theorem is critical to our proof, and the theorem itself may be of independent interest.
• Foliations invariant by rational maps
  
  • Favre Charles
  • Pereira J. Vitorio
  Mathematische Zeitschrift, Springer, 2011, 268 (3-4), pp.753-770. We give a classification of pairs (F, f) where F is a holomorphic foliation on a projective surface and f is a non-invertible dominant rational map preserving F. We prove that both the map and the foliation are integrable in a suitable sense. (10.1007/s00209-010-0693-6)
  DOI : 10.1007/s00209-010-0693-6
• 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
• The Kauffman skein algebra of a surface at $\sqrt{-1}$
  
  • Marché Julien
  Mathematische Annalen, Springer Verlag, 2011, 351, pp.347-364. We study the structure of the Kauffman algebra of a surface with parameter equal to √-1 . We obtain an interpretation of this algebra as an algebra of parallel transport operators acting on sections of a line bundle over the moduli space of flat SU(2)-connections over the surface. We analyse the asymptotics of traces of curve-operators in TQFT in non standard regimes where the root of unity parametrizing the TQFT accumulates to a root of unity. We interpret the case of √-1 in terms of parallel transport operators.
• On the Olson and the Strong Davenport constants
  
  • Ordaz Oscar
  • Philipp Andreas
  • Santos Irene
  • Schmid Wolfgang A.
  Journal de Théorie des Nombres de Bordeaux, Société Arithmétique de Bordeaux, 2011, 23 (3), pp.715-750. A subset $S$ of a finite abelian group, written additively, is called zero-sumfree if the sum of the elements of each non-empty subset of $S$ is non-zero. We investigate the maximal cardinality of zero-sumfree sets, i.e., the (small) Olson constant. We determine the maximal cardinality of such sets for several new types of groups; in particular, $p$-groups with large rank relative to the exponent, including all groups with exponent at most five. These results are derived as consequences of more general results, establishing new lower bounds for the cardinality of zero-sumfree sets for various types of groups. The quality of these bounds is explored via the treatment, which is computer-aided, of selected explicit examples. Moreover, we investigate a closely related notion, namely the maximal cardinality of minimal zero-sum sets, i.e., the Strong Davenport constant. In particular, we determine its value for elementary $p$-groups of rank at most $2$, paralleling and building on recent results on this problem for the Olson constant. (10.5802/jtnb.784)
  DOI : 10.5802/jtnb.784
• 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
• 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
• 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