Publications

2011

• How to Integrate a Polynomial over a Simplex
  
  • Baldoni Velleda
  • Berline Nicole
  • de Loera Jesús A.
  • Köppe Matthias
  • Vergne Michèle
  Mathematics of Computation, American Mathematical Society, 2011, 80 (273), pp.297-325. This paper settles the computational complexity of the problem of integrating a polynomial function f over a rational simplex. We prove that the problem is NP-hard for arbitrary polynomials via a generalization of a theorem of Motzkin and Straus. On the other hand, if the polynomial depends only on a fixed number of variables, while its degree and the dimension of the simplex are allowed to vary, we prove that integration can be done in polynomial time. As a consequence, for polynomials of fixed total degree, there is a polynomial time algorithm as well. We conclude the article with extensions to other polytopes and discussion of other available methods. (10.1090/S0025-5718-2010-02378-6)
  DOI : 10.1090/S0025-5718-2010-02378-6
• From the Boltzmann equation to hydrodynamic equations in thin layers
  
  • Golse François
  Bollettino dell'Unione Matematica Italiana, Springer Verlag, 2011, 4, pp.163-186. The present paper discusses an asymptotic theory for the Boltzmann equation leading to either the Prandtl incompressible boundary layer equations, or the incompressible hydrostatic equations. These results are formal, and based on the same moment method used in [C. Bardos, F. Golse, D. Levermore, J. Stat. Phys 63 (1991), pp. 323--344] to derive the incompressible Euler and Navier-Stokes equations from the Boltzmann equation.
• 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
• 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
• 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
• 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
• 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
• 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
• Semiclassical Propagation of Coherent States for the Hartree equation
  
  • Athanassoulis Agissilaos
  • Paul Thierry
  • Pezzotti Federica
  • Pulvirenti Mario
  Annales Henri Poincaré, Springer Verlag, 2011, 12 (8), pp.1613-1634. In this paper we consider the nonlinear Hartree equation in presence of a given external potential, for an initial coherent state. Under suitable smoothness assumptions, we approximate the solution in terms of a time dependent coherent state, whose phase and amplitude can be determined by a classical flow. The error can be estimated in $L^2$ by $C \sqrt {\var}$, $\var$ being the Planck constant. Finally we present a full formal asymptotic expansion. (10.1007/s00023-011-0115-2)
  DOI : 10.1007/s00023-011-0115-2
• Strong semiclassical approximation of Wigner functions for the Hartree dynamics.
  
  • Paul Thierry
  • Athanassoulis Agissilaos
  • Pezzotti Federica
  • Pulvirenti Mario
  Rendiconti Lincei. Matematica e Applicazioni, European Mathematical Society, 2011, 22 (4), pp.525-552. We consider the Wigner equation corresponding to a nonlinear Schrodinger evolution of the Hartree type in the semiclassical limit h -> 0. Under appropriate assumptions on the initial data and the interaction potential, we show that the Wigner function is close in L-2 to its weak limit, the solution of the corresponding Vlasov equation. The strong approximation allows the construction of semiclassical operator-valued observables, approximating their quantum counterparts in Hilbert-Schmidt topology. The proof makes use of a pointwise-positivity manipulation, which seems necessary in working with the L-2 norm and the precise form of the nonlinearity. We employ the Husimi function as a pivot between the classical probability density and the Wigner function, which-as it is well known-is not pointwise positive in general. (10.4171/RLM/613)
  DOI : 10.4171/RLM/613
• A Height Gap Theorem For Finite Subsets Of GL_d(\bar{Q}) and Non Amenable Subgroups
  
  • Breuillard Emmanuel
  Annals of Mathematics, Princeton University, Department of Mathematics, 2011, 174 (2), pp.1057-1110. We show a global adelic analog of the classical Margulis Lemma from hyperbolic geometry. We introduce a conjugation invariant normalized height $\hat{h}(F)$ of a finite set of matrices $F$ in $GL_{n}(\bar{\Bbb{Q}})$ which is the adelic analog of the minimal displacement on a symmetric space. We then show, making use of theorems of Bilu and Zhang on the equidistribution of Galois orbits of small points, that $\hat{h}(F)>\epsilon $ as soon as $F$ generates a non-virtually solvable subgroup of $SL_{n}(\bar{\Bbb{Q}}),$ where $\epsilon =\epsilon (n)>0$ is an absolute constant. (10.4007/annals.2011.174.2.7)
  DOI : 10.4007/annals.2011.174.2.7
• Linear forms and quadratic uniformity for functions on $\mathbb{Z}_N$
  
  • Gowers W. T.
  • Wolf Julia
  Journal d'analyse mathématique, Springer, 2011, 115 (1), pp.121-186. A very useful fact in additive combinatorics is that analytic expressions that can be used to count the number of structures of various kinds in subsets of Abelian groups are robust under quasirandom perturbations, and moreover that quasirandomness can often be measured by means of certain easily described norms, known as uniformity norms. However, determining which uniformity norms work for which structures turns out to be a surprisingly hard question. In [GW09a] and [GW09b, GW09c] we gave a complete answer to this question for groups of the form $G=\mathbb{F}_p^n$, provided $p$ is not too small. In $\mathbb{Z}_N$, substantial extra difficulties arise, of which the most important is that an "inverse theorem" even for the uniformity norm $\|.\|_{U^3}$ requires a more sophisticated (local) formulation. When $N$ is prime, $\mathbb{Z}_N$ is not rich in subgroups, so one must use regular Bohr neighbourhoods instead. In this paper, we prove the first non-trivial case of the main conjecture from [GW09a].
• Semiclassical limit of quantum dynamics with rough potentials and well-posedness of transport equations with measure initial data
  
  • Ambrosio Luigi
  • Figalli Alessio
  • Friesecke Gero
  • Giannoulis Johannes
  • Paul Thierry
  Communications on Pure and Applied Mathematics, Wiley, 2011, 64 (9), pp.1199-1242. In this paper we study the semiclassical limit of the Schrodinger equation. Under mild regularity assumptions on the potential U, which include Born-Oppenheimer potential energy surfaces in molecular dynamics, we establish asymptotic validity of classical dynamics globally in space and time for "almost all" initial data, with respect to an appropriate reference measure on the space of initial data. In order to achieve this goal we prove existence, uniqueness, and stability results for the flow in the space of measures induced by the continuity equation (10.1002/cpa.20371)
  DOI : 10.1002/cpa.20371
• The algebra $U_q(\hat{sl}_\infty)$ and applications
  
  • Hernandez David
  Journal of Algebra, Elsevier, 2011, 329 (1), pp.147-162. In this note we consider the algebra $U_q(\hat{sl}_\infty)$ and we study the category O of its integrable representations. The main motivations are applications to quantum toroidal algebras. In this context, we state a general positivity conjecture for representations of $U_q(\hat{sl}_\infty)$ viewed as representations of quantum toroidal algebras, that we prove for Kirillov-Reshetikhin modules. (10.1016/j.jalgebra.2010.04.002)
  DOI : 10.1016/j.jalgebra.2010.04.002
• 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
• 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.
• 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
• 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.
• 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
• 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
• 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
• 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.
• 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
• 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
• 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