International Mathematics Research Notices - recent issues

2009-06-28

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2009-06-28

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2009-06-28

Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

Juillet, N. — 2009-06-28

We prove that no curvature-dimension bound CD(K,N) holds in any Heisenberg group . On the contrary, the measure contraction property MCP(0, 2n + 3) holds and is optimal for the dimension 2n + 3. For the nonexistence of a curvature-dimension bound, we prove that the generalized "geodesic" Brunn–Minkowski inequality is false in . We also show in a new and direct way (and for all ), that the general "multiplicative" Brunn–Minkowski inequality with dimension N > 2n + 1 is false.

Fusion Algebras for Superconformal Field Theories through Coinvariants II : Ramond Sector

Iohara, K., Koga, Y. — 2009-06-28

We determine the fusion rules for the minimal series representations over the super-Virasoro algebras including the Ramond sector.

Non-adic Formal Schemes

Yasuda, T. — 2009-06-28

Our purpose is to make a contribution to the foundation of the theory of formal scheme. We are interested particularly in non-Noetherian or non-adic formal schemes, which have been little studied. We redefine the formal scheme as a proringed space and study its basic properties. We also find several examples of non-adic formal schemes.

Rational Points of Definable Sets and Results of Andre-Oort-Manin-Mumford type

Pila, J. — 2009-06-28

We prove some simple special cases, partly new, of results of André–Oort–Manin–Mumford type using an extension to algebraic points of bounded degree of a result of Pila–Wilkie on the density of rational points on sets definable in an o-minimal structure. The strategy follows that of a recent new proof of the Manin–Mumford conjecture by Pila–Zannier, and a proof of a special (but new) case of Pink's relative Manin–Mumford conjecture by Masser–Zannier.

Regularity Criteria for the Viscous Camassa-Holm Equations

Zhou, Y., Fan, J. — 2009-06-28

In this paper, we consider the viscous n-dimensional Camassa–Holm equations in the whole space. Various regularity criteria for the strong solution are established. As a corollary, we show the existence of a global smooth solution when .

Frobenius Map for Quintic Threefolds

Shapiro, I. — 2009-06-28

We calculate the matrix of the Frobenius map on the middle-dimensional cohomology of the one-parameter family that is related by mirror symmetry to the family of all quintic threefolds.

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2009-06-16

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2009-06-16

Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator

Gallagher, I., Gallay, T., Nier, F. — 2009-06-16

Initially motivated by a problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator on , where f is a real-valued function and is a small parameter. We define as the infimum of the real part of the spectrum of , and as the supremum of the norm of the resolvent of along the imaginary axis. Under appropriate conditions on f, we show that both quantities and go to infinity as , and we give precise estimates of the growth rate of . We also provide an example where if is small. Our main results are established using variational "hypocoercive" methods, localization techniques, and semiclassical subelliptic estimates.

A Quiver Construction of Symmetric Crystals

Enomoto, N. — 2009-06-16

In the papers [4–6] with Masaki Kashiwara, the author introduced the notion of symmetric crystals and presented the Lascoux–Leclerc–Thibon–Ariki-type conjectures for the affine Hecke algebras of type B. Namely, we conjectured that certain composition multiplicities and branching rules for the affine Hecke algebras of type B are described by using the lower global basis of symmetric crystals of . In the present paper, we prove the existence of crystal bases and global bases of for any symmetric quantized Kac–Moody algebra by using a geometry of quivers (with a Dynkin diagram involution). This is analogous to George Lusztig's geometric construction of U^–_v and its lower global basis.

On Knot Floer Homology and Cabling: 2

Hedden, M. — 2009-06-16

We continue our study of the knot Floer homology invariants of cable knots. For large | n|, we prove that many of the filtered subcomplexes in the knot Floer homology filtration associated to the (p, pn+1) cable of a knot, K, are isomorphic to those of K. This result allows us to obtain information about the behavior of the Ozsváth–Szabó concordance invariant under cabling, which has geometric consequences for the cabling operation. Applications considered include quasipositivity in the braid group, the knot theory of complex curves, smooth concordance, and L-space surgeries.

Minimal Microlocal Gevrey Regularity for "Sums of Squares"

Albano, P., Bove, A., Chinni, G. — 2009-06-16

A theorem of minimal microlocal Gevrey regularity is proved for operators that are sums of squares of vector fields with real analytic coefficients, thus providing a microlocal version of a well-known theorem of Derridj and Zuily ("Régularité analytique et Gevrey d'opérateurs elliptiques dégénérés," Journal de Mathématiques Pures et Appliquées 52 (1973): 65–80).

Explicit Reduction Modulo p of Certain Two-Dimensional Crystalline Representations

Buzzard, K., Gee, T. — 2009-06-16

In this paper, we use the p-adic local Langlands correspondence for to explicitly compute the reduction modulo p of certain two-dimensional crystalline representations of small slope, and give applications to modular forms.

Relations Between Multizeta Values for

Thakur, D. S. — 2009-06-16

Despite the failure of naive analogs of the sum shuffle or integral shuffle relations, and despite the lack of understanding of analogs of many classical structures that exist in the corresponding theory in the number field case, the multizeta values defined by the author are proved (and conjectured) to satisfy many interesting and combinatorially involved identities. The connections of these multizeta values with iterated extensions of Carlitz–Tate t-motives, analogs of Ihara power series, and Deligne–Soulé cocycles, etc., make it an interesting challenge to understand all the identities and discover the other relevant underlying structures.

The Existence of Convex Body with Prescribed Curvature Measures

Guan, P., Lin, C., Ma, X.-N. — 2009-05-29

Discrete Koenigs Nets and Discrete Isothermic Surfaces

Bobenko, A. I., Suris, Y. B. — 2009-05-29

We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual if their corresponding sides are parallel, and their noncorresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by coplanarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.

Equivariant Genera of Complex Algebraic Varieties

Cappell, S. E., Maxim, L., Shaneson, J. L. — 2009-05-29

Equivariant Hirzebruch genera of a variety X acted upon by a finite group of algebraic automorphisms are defined by combining the group action with the information encoded by the Hodge filtration in cohomology. For smooth manifolds, Atiyah and Meyer studied contributions of monodromy to usual signatures. While for a projective manifold equivariant genera can by computed by the Atiyah–Singer holomorphic Lefschetz theorem, we derive a Atiyah–Meyer-type formula for such genera even when X is not necessarily smooth or compact, but just fibers equivariantly (in the complex topology) over an algebraic manifold. These results apply to computing Hirzebruch invariants of orbit spaces. We also obtain results comparing equivariant genera of the range and domain of an equivariant morphism in terms of its singularities.

Multizeta Values for , Their Period Interpretation, and Relations between Them

Anderson, G. W., Thakur, D. S. — 2009-05-29

We provide a period interpretation for multizeta values (in the function field context) in terms of explicit iterated extensions of tensor powers of Carlitz motives (mixed Carlitz–Tate t-motives). We give examples of combinatorially involved relations that these multizeta values satisfy.

Determinant Form of the Complex Phase Function of the Steepest Descent Analysis of Riemann-Hilbert Problems and Its Application to the Focusing Nonlinear Schrodinger Equation

Tovbis, A., Venakides, S. — 2009-05-29

We derive a determinant formula for the g-function that plays a key role in the steepest descent asymptotic analysis of the solution of matrix Riemann–Hilbert problems (RHPs) and is closely related to a hyperelliptic Riemann surface. We formulate a system of transcendental equations in determinant form (modulation equations), that govern the dependence of the branchpoints of the Riemann surface on a set of external parameters. We prove that, subject to the modulation equations, is identically zero for all the branchpoints. Modulation equations are also obtained in the form of ordinary differential equations with respect to external parameters; some applications of these equations to the semiclassical limit of the focusing nonlinear Schrödinger equation (NLS) are discussed.

Quantum D-modules, Elliptic Braid Groups, and Double Affine Hecke Algebras

Jordan, D. — 2009-05-29

We build representations of the elliptic braid group from the data of a quantum D-module M over a ribbon Hopf algebra U. The construction is modelled on, and generalizes, similar constructions by Lyubashenko and Majid ("Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity," Communications in Mathematical Physics 172 (1995): 467–516; "Braided Groups and Quantum Fourier Transform," Journal of Algebra 166 (1994): 506–28), and also certain geometric constructions of Calaque, Enriquez, and Etingof ("Universal KZB equations I: The elliptic case," (2007): preprint arXiv:math/0702670) concerning trigonometric Cherednik algebras. In this context, the former construction is the special case where M is the basic representation, while the latter construction can be recovered as a quasi-classical limit of , as . In the latter case, we produce representations of the double affine Hecke algebra of type , for each n.

Courant Algebroids and Poisson Geometry

Li-Bland, D., Meinrenken, E. — 2009-05-29

Given a manifold M with an action of a quadratic Lie algebra , such that all stabilizer algebras are coisotropic in , we show that the product becomes a Courant algebroid over M. If the bilinear form on is split, the choice of transverse Lagrangian subspaces of defines a bivector field on M, which is Poisson if is a Manin triple. In this way, we recover the Poisson structures of Lu–Yakimov, and in particular the Evens–Lu Poisson structures on the variety of Lagrangian Grassmannians and on the de Concini–Procesi compactifications. Various Poisson maps between such examples are interpreted in terms of the behavior of Lagrangian splittings under Courant morphisms.

International Mathematics Research Notices - recent issues

Contents

Subscriptions

Editors

Geometric Inequalities and Generalized Ricci Bounds in the Heisenberg Group

Fusion Algebras for Superconformal Field Theories through Coinvariants II : Ramond Sector

Non-adic Formal Schemes

Rational Points of Definable Sets and Results of Andre-Oort-Manin-Mumford type

Regularity Criteria for the Viscous Camassa-Holm Equations

Frobenius Map for Quintic Threefolds

Editors

Subscriptions

Contents

Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator

A Quiver Construction of Symmetric Crystals

On Knot Floer Homology and Cabling: 2

Minimal Microlocal Gevrey Regularity for "Sums of Squares"

Explicit Reduction Modulo p of Certain Two-Dimensional Crystalline Representations

Relations Between Multizeta Values for

The Existence of Convex Body with Prescribed Curvature Measures

Discrete Koenigs Nets and Discrete Isothermic Surfaces

Equivariant Genera of Complex Algebraic Varieties

Multizeta Values for , Their Period Interpretation, and Relations between Them

Determinant Form of the Complex Phase Function of the Steepest Descent Analysis of Riemann-Hilbert Problems and Its Application to the Focusing Nonlinear Schrodinger Equation

Quantum D-modules, Elliptic Braid Groups, and Double Affine Hecke Algebras

Courant Algebroids and Poisson Geometry