International Mathematics Research Notices - recent issues

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Fri, 20 Nov 2009 01:19:37 PST

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Fri, 20 Nov 2009 01:19:37 PST

Frame Stabilizers for Framed Vertex Operator Algebras Associated to Lattices Having 4-Frames

Lam, C. H., Shimakura, H. — Fri, 20 Nov 2009 01:19:37 PST

In this paper, we study certain Virasoro frames for lattice vertex operator algebras (VOAs) and their -orbifolds using linear codes over . We also compute the corresponding frame stabilizer from the viewpoint of binary codes and -codes. As an application, we determine the frame stabilizers of several Virasoro frames of the VOA and the moonshine vertex operator algebra V.

Symmetric Waves Are Traveling Waves

Ehrnstrom, M., Holden, H., Raynaud, X. — Fri, 20 Nov 2009 01:19:37 PST

We show that horizontally symmetric water waves are traveling waves. The result is valid for the Euler equations, and is based on a general principle that applies to a large class of nonlinear partial differential equations, including some of the most famous model equations for water waves. A detailed analysis is given for weak solutions of the Camassa–Holm equation. In addition, we establish the existence of nonsymmetric linear rotational waves for the Euler equations.

Moduli of Bundles over Rational Surfaces and Elliptic Curves II: Nonsimply Laced Cases

Leung, N. C., Zhang, J. — Fri, 20 Nov 2009 01:19:37 PST

For any nonsimply laced Lie group G and elliptic curve , we show that the moduli space of flat G bundles over can be identified with the moduli space of rational surfaces with G-configurations which contain as an anticanonical curve. We also construct Lie(G)-bundles over these surfaces. The corresponding results for simply laced groups were obtained by the authors in another paper. Thus, we have established a natural identification for these two kinds of moduli spaces for any Lie group G.

Rational Curves on Smooth Cubic Hypersurfaces

Coskun, I., Starr, J. — Fri, 20 Nov 2009 01:19:37 PST

We prove that the space of rational curves of a fixed degree on any smooth cubic hypersurface of dimension at least four is irreducible and of the expected dimension. Our methods also show that the space of rational curves of a fixed degree on a general hypersurface in of degree 2d ≤ min (n + 4, 2n – 2) and dimension at least three is irreducible and of the expected dimension.

Stable String Operations Are Trivial

Tamanoi, H. — Fri, 20 Nov 2009 01:19:37 PST

We show that in closed string topology and in open-closed string topology with one D-brane, higher genus stable string operations are trivial. This is a consequence of Harer's stability theorem and related stability results on the homology of mapping class groups of surfaces with boundaries. In fact, this vanishing result is a special case of a general result that applies to all homological conformal field theories with the property that in the associated topological quantum field theories, the string operations associated to genus 1 cobordisms with one or two boundaries vanish. In closed string topology, the base manifold can be either finite-dimensional, or infinite-dimensional with finite-dimensional cohomology for its based loop space. The above vanishing result is based on the triviality of string operations associated to the homology classes of mapping class groups that are in the image of stabilizing maps.

Holomorphic Line Bundles on Projective Toric Manifolds from Lagrangian Sections of their Mirrors by SYZ Transformations

Chan, K. — Fri, 20 Nov 2009 01:19:37 PST

The mirror of a projective toric manifold is given by a Landau–Ginzburg model (Y, W). We introduce a class of Lagrangian submanifolds in (Y, W) and show that, under the SYZ mirror transformation, they can be transformed to torus-invariant Hermitian metrics on holomorphic line bundles over . Through this geometric correspondence, we also identify the mirrors of Hermitian–Einstein metrics, which are given by distinguished Lagrangian sections whose potentials satisfy certain Laplace-type equations.

Integral Chow Rings of Toric Stacks

Iwanari, I. — Fri, 20 Nov 2009 01:19:37 PST

The goal of this paper is to compute integral Chow rings of toric stacks and prove that they are naturally isomorphic to Stanley–Reisner rings.

The Fate of the Landau Levels under Perturbations of Constant Sign

Klopp, F., Raikov, G. — Fri, 20 Nov 2009 01:19:37 PST

We show that the Landau levels cease to be eigenvalues if we perturb the 2D Schrödinger operator with a constant magnetic field, by bounded electric potentials of fixed sign. We also show that, if the perturbation is not of fixed sign, then any Landau level may be an eigenvalue of the perturbed problem.

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Wed, 11 Nov 2009 00:03:17 PST

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Wed, 11 Nov 2009 00:03:17 PST

Wed, 11 Nov 2009 00:03:18 PST

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Bourgade, P., Nikeghbali, A., Rouault, A. — Wed, 11 Nov 2009 00:03:18 PST

Using the spectral theory of unitary operators and the theory of orthogonal polynomials on the unit circle, we propose a simple matrix model for the following circular analog of the Jacobi ensemble: with . If e is a cyclic vector for a unitary n x n matrix U, the spectral measure of the pair (U, e) is well parameterized by its Verblunsky coefficients (₀, ..., _n–1). We introduce here a deformation (₀, ..., _n–1) of these coefficients so that the associated Hessenberg matrix (called GGT) can be decomposed into a product r(₀)··· r(_n–1) of elementary reflections parameterized by these coefficients. If ₀, ..., _n–1 are independent random variables with some remarkable distributions, then the eigenvalues of the GGT matrix follow the circular Jacobi distribution above.

These deformed Verblunsky coefficients also allow us to prove that, in the regime = (n) with (n)/ n -> β d/2, the spectral measure and the empirical spectral distribution weakly converge to an explicit nontrivial probability measure supported by an arc of the unit circle. We also prove the large deviations for the empirical spectral distribution.

Action Selectors and Maslov Class Rigidity

Kerman, E. — Wed, 11 Nov 2009 00:03:18 PST

In this paper, we detect new restrictions on the Maslov class of displaceable Lagrangian submanifolds of symplectic manifolds which are symplectically aspherical. These restrictions are established using action selectors for Hamiltonian flows. In particular, we construct and utilize a new action selector for the flows of a special class of Hamiltonian functions which arises naturally in the study of Hamiltonian paths which minimize the Hofer length functional.

Moduli Spaces of Semistable Sheaves on Singular Genus 1 Curves

Ruiperez, D. H., Martin, A. C. L., Gomez, D. S., Prieto, C. T. — Wed, 11 Nov 2009 00:03:18 PST

We find some equivalences of the derived category of coherent sheaves on a Gorenstein genus one curve that preserve the (semi)-stability of pure-dimensional sheaves. Using them we establish new identifications between certain Simpson moduli spaces of semistable sheaves on the curve. For rank zero, the moduli spaces are symmetric powers of the curve whilst for a fixed positive rank there are only a finite number of nonisomorphic spaces. We prove similar results for the relative semistable moduli spaces on an arbitrary genus one fibration with no conditions either on the base or on the total space. For a cycle E_N of projective lines, we show that the unique degree 0 stable sheaves are the line bundles having degree 0 on every irreducible component and the sheaves supported on one irreducible component. We also prove that the connected component of the moduli space that contains vector bundles of rank r is isomorphic to the rth symmetric product of the rational curve with one node.

Desingularization of Orbifolds Obtained from Symplectic Reduction at Generic Coadjoint Orbits

Niederkruger, K., Pasquotto, F. — Wed, 11 Nov 2009 00:03:18 PST

Symplectic reduction is a technique that can be used to decrease the dimension of Hamiltonian manifolds. Unfortunately, this only works under strong assumptions on the group action, and in general, even for a compact Lie group, the reduction at a coadjoint orbit that is transverse to the moment map will only yield a symplectic orbifold.

In this article, we show how to construct resolutions of symplectic orbifolds obtained as quotients of presymplectic manifolds with a torus action. As a corollary, this allows us to desingularize generic symplectic quotients for compact Lie group actions. More precisely, if a point in the Lie coalgebra is regular, that is, its stabilizer is a maximal torus, then we may apply our desingularization result. Regular elements of the Lie coalgebra are generic in the sense that the singular strata have codimension at least three.

Additionally, we show that even though the result of a symplectic cut is an orbifold, it can be modified in an arbitrarily small neighborhood of the cut hypersurface to obtain a smooth symplectic manifold.

Local Structure of the Moduli Space of K3 Surfaces in Positive Characteristic

Yu, J.-D. — Wed, 11 Nov 2009 00:03:18 PST

Let k be a perfect field of characteristic p ≥ 5. Let X be a nonsupersingular K3 surface over k and the enlarged formal Brauer group associated to X. In this paper, we show that the local moduli space of X, with trivial associated deformation of , admits a natural p-divisible formal group structure.

Foliations on Hypersurfaces in Holomorphic Symplectic Manifolds

Sawon, J. — Wed, 11 Nov 2009 00:03:18 PST

Let Y be a hypersurface in a 2n-dimensional holomorphic symplectic manifold X. The restriction |_Y of the holomorphic symplectic form induces a rank one foliation on Y. We investigate situations where this foliation has compact leaves; in such cases, we obtain a space of leaves Y/ F which has dimension 2n – 2 and admits a holomorphic symplectic form.

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Tue, 27 Oct 2009 09:33:21 PDT

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Tue, 27 Oct 2009 09:33:21 PDT

Rigidity, Locally Symmetric Varieties, and the Grothendieck-Katz Conjecture

Farb, B., Kisin, M. — Tue, 27 Oct 2009 09:33:21 PDT

Using Margulis's results on lattices in semisimple Lie groups, we prove the Grothendieck–Katz p-curvature conjecture for many locally symmetric varieties, including Hilbert–Blumenthal modular varieties and the moduli space of abelian varieties when g > 1.

Quantum Alpha-Determinants and q-Deformed Hypergeometric Polynomials

Kimoto, K. — Tue, 27 Oct 2009 09:33:21 PDT

The quantum -determinant is defined as a parametric deformation of the quantum determinant. We investigate the cyclic -submodules of the quantum matrix algebra generated by the powers of the quantum -determinant. For such a cyclic module, there exists a collection of polynomials, which describe the irreducible decomposition of it in the following manner: (i) each polynomial corresponds to a certain irreducible -module, (ii) the cyclic module contains an irreducible submodule if the parameter is not a root of the corresponding polynomial. These polynomials are given as a q-deformation of the hypergeometric polynomials. This is a quantum analog of the result obtained in our previous work [Kimoto, K., S. Matsumoto, and M. Wakayama. "Alpha-determinant cyclic modules and Jacobi polynomials." Transactions of the American Mathematical Society (forthcoming)].

Fusion Subcategories of Representation Categories of Twisted Quantum Doubles of Finite Groups

Naidu, D., Nikshych, D., Witherspoon, S. — Tue, 27 Oct 2009 09:33:21 PDT

We describe all fusion subcategories of the representation category of a twisted quantum double , where G is a finite group and is a 3-cocycle on G. In view of the fact that every group-theoretical braided fusion category can be embedded into some , this gives a complete description of all group-theoretical braided fusion categories. We describe the lattice and give formulas for some invariants of the fusion subcategories of . We also give a characterization of group-theoretical braided fusion categories as equivariantizations of pointed categories.

L2-Betti Numbers and Non-Unitarizable Groups without Free Subgroups

Osin, D. V. — Tue, 27 Oct 2009 09:33:21 PDT

We show that there exist non-unitarizable groups without nonabelian free subgroups. Both torsion and torsion free examples are constructed. As a by-product, we show that there exist finitely generated torsion groups with nonvanishing first L²-Betti numbers. We also relate the well-known problem of whether every hyperbolic group is residually finite to an open question about approximation of L²-Betti numbers.

Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets

Ferrari, F., Valdinoci, E. — Tue, 27 Oct 2009 09:33:21 PDT

A geometric Sobolev–Poincaré inequality for stable solutions of semilinear partial differential equations (PDEs) in the Grushin plane will be obtained. Such inequality will bound the weighted L²-norm of a test function by a weighted L²-norm of its gradient, and the weights will be interesting geometric quantities related to the level sets of the solution. From this, we shall see that a geometric PDE holds on the level sets of stable solutions. We shall study in detail the particular case of local minimizers of a Ginzburg–Landau–Allen–Cahn-type phase transition model and provide for them some one-dimensional symmetry results.

Functoriality for the Classical Groups over Function Fields

Lomeli, L. A. — Tue, 27 Oct 2009 09:33:21 PDT

Langlands' functoriality for generic representations from the split classical groups to an appropriate GL_N is established. The functorial lift or transfer to GL_N is obtained with the help of a converse theorem once the analytic properties of L-functions are studied using the Langlands–Shahidi approach. This paper is mostly devoted to understanding L-functions for the classical groups over a global function field, since the Langlands–Shahidi method has only been developed over number fields. To overcome many difficulties, stability of -factors under twists by highly ramified characters is used together with multiplicativity. Finally, by analyzing the image of functoriality, a proof of the Ramanujan conjecture for generic representations is obtained.

Nonunitarizable Representations and Random Forests

Epstein, I., Monod, N. — Tue, 27 Oct 2009 09:33:21 PDT

We establish a connection between Dixmier's unitarizability problem and the expected degree of random forests on a group. As a consequence, a residually finite group is nonunitarizable if its first L²-Betti number is nonzero or if it is finitely generated with nontrivial cost. Our criterion also applies to torsion groups constructed by Osin, thus providing the first examples of nonunitarizable groups without free subgroups.

Addendum to: Complete Conformal Metrics of Negative Ricci Curvature on Compact Manifolds with Boundary

Guan, B. — Tue, 27 Oct 2009 09:33:21 PDT

International Mathematics Research Notices - recent issues

Subscriptions

Editors

Contents

Frame Stabilizers for Framed Vertex Operator Algebras Associated to Lattices Having 4-Frames

Symmetric Waves Are Traveling Waves

Moduli of Bundles over Rational Surfaces and Elliptic Curves II: Nonsimply Laced Cases

Rational Curves on Smooth Cubic Hypersurfaces

Stable String Operations Are Trivial

Holomorphic Line Bundles on Projective Toric Manifolds from Lagrangian Sections of their Mirrors by SYZ Transformations

Integral Chow Rings of Toric Stacks

The Fate of the Landau Levels under Perturbations of Constant Sign

Subscriptions

Editors

Contents

Circular Jacobi Ensembles and Deformed Verblunsky Coefficients

Action Selectors and Maslov Class Rigidity

Moduli Spaces of Semistable Sheaves on Singular Genus 1 Curves

Desingularization of Orbifolds Obtained from Symplectic Reduction at Generic Coadjoint Orbits

Local Structure of the Moduli Space of K3 Surfaces in Positive Characteristic

Foliations on Hypersurfaces in Holomorphic Symplectic Manifolds

Subscriptions

Editors

Contents

Rigidity, Locally Symmetric Varieties, and the Grothendieck-Katz Conjecture

Quantum Alpha-Determinants and q-Deformed Hypergeometric Polynomials

Fusion Subcategories of Representation Categories of Twisted Quantum Doubles of Finite Groups

L2-Betti Numbers and Non-Unitarizable Groups without Free Subgroups

Geometric PDEs in the Grushin Plane: Weighted Inequalities and Flatness of Level Sets

Functoriality for the Classical Groups over Function Fields

Nonunitarizable Representations and Random Forests

Addendum to: Complete Conformal Metrics of Negative Ricci Curvature on Compact Manifolds with Boundary