Showing papers in "International Mathematics Research Notices in 2006"

A characterization of Lp intersection bodies

Abstract: All GL(n) covariant Lp radial valuations on convex polytopes are classified for every p > 0. It is shown that for 0 < p < 1 there is a unique non-trivial such valuation with centrally symmetric images. This establishes a characterization of Lp intersection bodies. 2000 AMS subject classification: 52A20 (52B11, 52B45)

On sharp Strichartz inequalities in low dimensions

Abstract: Recently Foschi gave a proof of a sharp Strichartz inequality in one and two dimensions. In this note, a new representation in terms of an orthogonal projection operator is obtained for the space time norm of solutions of the free Schrödinger equation in dimension one and two. As a consequence, the sharp Strichartz inequality follows from the elementary property that orthogonal projections do not increase the norm.

Some gradient estimates for the heat equation on domains and for an equation by Perelman

Abstract: In the first part, we derive a sharp gradient estimate for the log of Dirichlet heat kernel and Poisson heat kernel on domains, and a sharpened local Li-Yau gradient estimate that matches the global one. In the second part, without explicit curvature assumptions, we prove a global upper bound for the fundamental solution of an equation introduced by G. Perelman, i.e. the heat equation of the conformal Laplacian under backward Ricci flow. Further, under nonnegative Ricci curvature assumption, we prove a qualitatively sharp, global Gaussian upper bound.

Effective computation of Maass cusp forms

Abstract: Author please provide the abstract. Please provide the abstract of this paper that should not exceed 150 words (including spaces) and citation free. 1 Preliminary The aim of this paper is to address theoretical and practical aspects of high-precision computation of Maass forms. Namely, we compute to over 1000 decimal places the Laplacian and Hecke eigenvalues for the first few Maass forms on PSL(2, Z)\H, and certify the Laplacian eigenvalues correct to 100 places. We then use these computations to test certain algebraicity properties of the coefficients. The outline of the paper is as follows. In Section 2, we discuss Hejhal’s algorithm for computation of Maass forms on cofinite Fuchsian groups with cusps, and the details necessary to implement it in high precision. This algorithm is heuristic and does not prove the existence of cusp forms. In Section 3 we turn to the question of rigorously verifying that a proposed eigenvalue, together with a proposed set of Fourier coefficients, indeed correspond to a true Maass cusp form. We will use standard methods to show that the putative eigenfunction has almost all of its spectral support concentrated near the proposed eigenvalue. It is a more subtle point to show that it is close to a cusp form

Integrability of C2-cofinite vertex operator algebras

Abstract: The following integrability theorem for vertex operator algebras V satisfying some finiteness conditions (C2-cofinite and CFT-type) is proved: the vertex operator subalgebra generated by a simple Lie subalgebra g of the weight one subspace V1 is isomorphic to the irreducible highest weight ˆ-module L(k,0) for a positive integer k, and V is an integrable ˆ-module. The case in which g is replaced by an abelian Lie subalgebra is also considered, and several consequences of integrability are discussed. 2000MSC:17B69

Minimality and symplectic sums

Abstract: Let X1, X2 be symplectic 4-manifolds containing symplectic sur- faces F1, F2 of identical positive genus and opposite squares. Let Z denote the symplectic sum of X1 and X2 along the Fi. Using relative Gromov-Witten theory, we determine precisely when the symplectic 4-manifold Z is minimal (i.e., cannot be blown down); in particular, we prove that Z is minimal unless either: one of the Xi contains a ( 1)-sphere disjoint from Fi; or one of the Xi admits a ruling with Fi as a section. As special cases, this proves a conjecture of Stipsicz asserting the minimality of fiber sums of Lefschetz fibrations, and implies that the non-spin examples constructed by Gompf in his study of the geography problem are minimal. Let (X1,!1), (X2,!2) be symplectic 4-manifolds, and let F1 ⊂ X1, F2 ⊂ X2 be two-dimensional symplectic submanifolds with the same genus whose homology classes satisfy (F1) 2 + (F2) 2 = 0, with the !i normalized to give equal area to the surfaces Fi. For i = 1,2, a neighborhood of Fi is symplectically identified by Weinstein's symplectic neighborhood theorem (19) with the disc normal bundlei of Fi in Xi. Choose a smooth isomorphismof the normal bundle to F1 in X1 (which is a complex line bundle) with the dual of the normal bundle to F2 in X2. According to (2) (and independently (11)), the symplectic sum

The Bennequin number, Kauffman polynomial, and ruling invariants of a Legendrian link: the Fuchs conjecture and beyond

Abstract: Author(s): Rutherford, Dan | Abstract: We show that the ungraded ruling invariants of a Legendrian link can be realized as certain coefficients of the Kauffman polynomial which are non-vanishing if and only if the upper bound for the Bennequin number given by the Kauffman polynomial is sharp. This resolves positively a conjecture of Fuchs. Using similar methods a result involving the upper bound given by the HOMFLY polynomial and 2-graded rulings is proved.

Counterexamples of Strichartz Inequalities for Schrodinger Equations with Repulsive Potentials

Abstract: MICHAEL GOLDBERG, LUIS VEGA, AND NICOLA VISCIGLIAAbstract. In each dimension n ≥ 2, we construct a class of nonnegativepotentials that are homogeneous of order −σ, chosen from the range 0 ≤ σ < 2,and for which the perturbed Schro¨dinger equation does not satisfy global intime Strichartz estimates.

On positive solutions to semilinear conformally invariant equations on locally conformally flat manifolds

Abstract: In this paper we study the existence and compactness of positive solutions to a family of conformally invariant equations on closed locally conformally flat manifolds. The family of conformally covariant operators $P_\\alpha$ were introduced via the scattering theory for Poincar\\'{e} metrics associated with a conformal manifold $(M^n, [g])$. We prove that, on a closed and locally conformally flat manifold with Poincar\\'{e} exponent less than $\\frac {n-\\alpha}2$ for some $\\alpha \\in [2, n)$, the set of positive smooth solutions to the equation $$ P_\\alpha u = u^\\frac {n+\\alpha}{n-\\alpha} $$ is compact in the $C^\\infty$ topology. Therefore the existence of positive solutions follows from the existence of Yamabe metrics and a degree theory.

Voronoi formulas on GL(n)

Abstract: In this paper, we give a new, simple, purely analytic proof of the Voronoi formula for Maass forms on GL(3) first derived by Miller and Schmid Our method is based on two lemmas of the first author and Thillainatesan which appear in their recent non-adelic proof of the converse theorem on GL(3) Using a different, even simpler method we derive Voronoi formulas on GL(n) twisted by additive characters of prime conductors We expect that this method will work in general In the final section of the paper Voronoi formulas on GL(n) are obtained, but in this case, the twists are by automorphic forms from lower rank groups

Remarks on the stack of coherent algebras

Abstract: We consider the stack of coherent algebras with proper support, a moduli problem generalizing Alexeev and Knutson's stack of branchvarieties to the case of an Artin stack. The main results are proofs of the existence of Quot and Hom spaces in greater generality than is currently known and several applications to Alexeev and Knutson's original construction: a proof that the stack of branchvarieties is always algebraic, that limits of one-dimensional families always exist, and that the connected components of the stack of branchvarieties are proper over the base under certain hypotheses on the ambient stack.

Counting rational points of quiver moduli

Abstract: It is shown that rational points over finite fields of moduli spaces of stable quiver representations are counted by polynomials with integer coefficients. These polynomials are constructed recursively using an identity in the Hall algebra of a quiver.

On a Gauss-Givental representation of quantum Toda chain wave function

Abstract: We propose group theory interpretation of the integral representation of the quantum open Toda chain wave function due to Givental. In particular we construct the representation of $U((\mathfrak{gl}(N))$ in terms of first order differential operators in Givental variables. The construction of this representation turns out to be closely connected with the integral representation based on the factorized Gauss decomposition. We also reveal the recursive structure of the Givental representation and provide the connection with the Baxter $Q$-operator formalism. Finally the generalization of the integral representation to the infinite and periodic quantum Toda wave functions is discussed.

Rotation equivariant Minkowski valuations

Abstract: The projection body operator Π, which associates with every convex body in Euclidean space Rn its projection body, is a continuous valuation, it is invariant under translations and equivariant under rotations. It is also well known that Π maps the set of polytopes in Rn into itself. We show that Π is the only non-trivial operator with these properties. MSC 2000: 52B45, 52A20

On the uniqueness of generating Hamiltonian for continuous limits of Hamiltonians flows

Abstract: We show that if a sequence of Hamiltonian flows has a C0 limit, and if the generating Hamiltonians of the sequence have a limit, this limit is uniquely determined by the limiting C0 flow. This answers a question by Y.G. Oh in [Oh04].

Quaternionic bundles and Betti numbers of symplectic 4-manifolds with Kodaira dimension zero

Abstract: The Kodaira dimension of a non-minimal manifold is defined to be that of any of its minimal models. It is shown in [12] that, if ω is a Kahler form on a complex surface (M,J), then κ(M,ω) agrees with the usual holomorphic Kodaira dimension of (M,J). It is also shown in [12] that minimal symplectic 4−manifolds with κ = 0 are exactly those with torsion canonical class, thus can be viewed as symplectic Calabi-Yau surfaces. Known examples of symplectic 4−manifolds with torsion canonical class are either Kahler surfaces with (holomorphic) Kodaira dimension zero or T 2−bundles over T 2 ([10], [12]). They all have small Betti numbers and Euler numbers: b+ ≤ 3, b ≤ 19 and b1 ≤ 4; and the Euler number is between 0 and 24. It is speculated in [12] that these are the only ones. In this paper we prove that it is true up to rational homology.

Galleries, Hall-Littlewood polynomials, and structure constants of the spherical Hecke algebra

Abstract: We provide a combinatorial description of the coefficients appearing in the expansion of Hall-Littlewood polynomials in terms of monomial symmetric functions. We also give a Littlewood-Richardson rule for Hall-Littlewood polynomials. For proving this we use galleries to calculate Satake coefficients and structure constants of spherical Hecke algebras with arbitrary parameters.

Schmidt's theorem, Hausdorff measures, and slicing

Abstract: A Hausdorff measure version of W. M. Schmidt's inhomogeneous, linear forms theorem in metric number theory is established. The key ingredient is a "slicing" technique motivated by a standard result in geometric measure theory. In short, "slicing" together with the mass transference principle allows us to transfer Lebesgue measure theoretic statements for lim sup sets associated with linear forms to Hausdorff measure theoretic statements. This extends the approach developed for simultaneous approximation and further demonstrates the surprising fact that the Lebesgue theory for lim sup sets underpins the general Hausdorff theory. Furthermore, we establish a new mass transference principle which incorporates both forms of approximation. As an application we obtain a complete metric theory for a "fully" nonlinear Diophantine problem within the linear forms setup—the first of its kind.

A category of modules for the full toroidal Lie algebra

Abstract: Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody algebras. The theory of affine Lie algebras is rich and beautiful, having connections with diverse areas of mathematics and physics. Toroidal Lie algebras are also proving themselves to be useful for the applications. Frenkel, Jing and Wang [FJW] used representations of toroidal Lie algebras to construct a new form of the McKay correspondence. Inami et al., studied toroidal symmetry in the context of a 4-dimensional conformal field theory [IKUX], [IKU]. There are also applications of toroidal Lie algebras to soliton theory. Using representations of the toroidal algebras one can construct hierarchies of non-linear PDEs [B2], [ISW]. In particular, the toroidal extension of the Korteweg-de Vries hierarchy contains the Bogoyavlensky’s equation, which is not in the classical KdV hierarchy [IT]. One can use the vertex operator realizations to construct n-soliton solutions for the PDEs in these hierarchies. We hope that further development of the representation theory of toroidal Lie algebras will help to find new applications of this interesting class of algebras. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra ġ and considers Fourier polynomial maps from an N + 1-dimensional torus into ġ. Setting tk = e ixk , we may identify the algebra of Fourier polynomials on a torus with the Laurent polynomial algebra R = C[t0 , t ± 1 , . . . , t ± N ], and the Lie algebra of the ġ-valued maps from a torus with the multi-loop algebra C[t±0 , t ± 1 , . . . , t ± N ] ⊗ ġ. When N = 0, this yields the usual loop algebra. Just as for the affine algebras, the next step is to build the universal central extension (R⊗ ġ)⊕K of R⊗ ġ. However unlike the affine case, the center K is infinite-dimensional when N ≥ 1. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of vector fields on a torus, D = Der (R) to (R⊗ ġ)⊕K. The resulting algebra,

Coherent permutations with descent statistic and the boundary problem for the graph of zigzag diagrams

Abstract: The graph of zigzag diagrams is a close relative of Young's lattice. The boundary problem for this graph amounts to describing coherent random permutations with descent-set statistic, and is also related to certain positive characters on the algebra of quasi-symmetric functions. We establish connections to some further relatives of Young's lattice and solve the boundary problem by reducing it to the classification of spreadable total orders on integers, as recently obtained by Jacka and Warren.

Asymptotic correlations for Gaussian and Wishart matrices with external source

Abstract: We consider ensembles of Gaussian (Hermite) and Wishart (Laguerre) $N\times N$ hermitian matrices. We study the effect of finite rank perturbations of these ensembles by a source term. The rank $r$ of the perturbation corresponds to the number of non-null eigenvalues of the source matrix. In the perturbed ensembles, the correlation functions can be written in terms of kernels. We show that for all $N$, the difference between the perturbed and the unperturbed kernels is a degenerate kernel of size $r$ which depends on multiple Hermite or Laguerre functions. We also compute asymptotic formulas for the multiple Laguerre functions kernels in terms multiple Bessel (resp. Airy) functions. This leads to the large $N$ limiting kernels at the hard (resp. soft) edge of the spectrum of the perturbed Laguerre ensemble. Similar results are obtained in the Hermite case.

Crystal structure on rigged configurations

Abstract: Rigged configurations are combinatorial objects originating from the Bethe Ansatz, that label highest weight crystal elements. In this paper a new unrestricted set of rigged configurations is introduced for types ADE by constructing a crystal structure on the set of rigged configurations. In type A an explicit characterization of unrestricted rigged configurations is provided which leads to a new fermionic formula for unrestricted Kostka polynomials or q-supernomial coefficients. The affine crystal structure for type A is obtained as well.

Jacobians and rank 1 perturbations relating to unitary Hessenberg matrices

Abstract: Killip and Nenciu gave random recurrences for the characteristic polynomials of certain unitary and real orthogonal upper Hessenberg matrices. The corresponding eigenvalue probability density functions (p.d.f's) are β-generalizations of the classical groups. Left open was the direct calculation of certain Jacobians. We provide the sought direct calculation. Furthermore, we show how a multiplicative rank 1 perturbation of the unitary Hessenberg matrices provides a joint eigenvalue p.d.f. generalizing the circular β-ensemble, and we show how this joint density is related to known interrelations between circular ensembles. Projecting the joint density onto the real line leads to the derivation of a random three-term recurrence for polynomials with zeros distributed according to the circular Jacobi β-ensemble.