Radial Components, Prehomogeneous Vector Spaces, and Rational Cherednik Algebras

Let be a finite-dimensionalrepresentation of a connected reductive complex Lie group . Denote by G the derived subgroup of and assume that the categorical quotient is one dimensional, i.e. for a nonconstant polynomial f. In this situationthere exists a homomorphism , the radial component map, where is the first Weyl algebra. We show that theimage of is isomorphicto the spherical subalgebra of a rational Cherednik algebrawhose multiplicity function is defined by the roots of the Bernstein–Satopolynomial of f. In the case where is also multiplicity free we describe the kernel of and prove a Howe duality result between representationsof G occurring in andlowest-weight modules over the Lie algebra generated by f andthe "dual" differential operator ; this extends results of H. Rubenthaler obtained when is a parabolic prehomogeneous vector space.If satisfies a Capelli-typecondition, some applications are given to holonomic and equivariantD-modules on V. These applications are related to results provedby Muro [34, 37, 38] or Nang [40, 42, 44] in special cases ofthe representation .     

Equations Defining Symmetric Varieties and Affine Grassmannians

Suppose be a simpleinvolution of a semisimple algebraic group and suppose H is the subgroup of G of pointsfixed by . If the restrictedroot system is of type or and G is simplyconnected, or if the restricted root system is of type and G is of adjoint type, then we describe astandard monomial theory and the equations for the coordinatering using the standardmonomial theory and the Plücker relations of an appropriate(maybe infinite-dimensional) Grassmann variety.     

Decaying Long-Time Asymptotics for the Focusing NLS Equation with Periodic Boundary Condition

We consider the initial boundary value problem for the focusingnonlinear Schrödinger equation in the quarter plane , inthe case of decaying initial data (for , as )and Dirichlet boundary data (for ) approaching a periodic (single-frequency) background as .We first provide admissibility conditions for the normal derivativeof the solution on the boundary, under the assumption that itbehaves asymptotically in a similar (single-frequency) manner.We then show that for the range , the long-time asymptotics of the solution inside the quarterplane exhibits decaying oscillations of Zakharov–Manakovtype.     

Zeta Functions for G2 and Their Zeros

The exceptional group G₂ has two maximal parabolic subgroups corresponding to theso-called long root and short root. In this paper, the secondnamed author introduces two zeta functions associated with and respectively, and the first named author proves that these zetassatisfy the Riemann hypothesis.     

Operations on the A-theoretic nil-terms

For a space X, we define Frobenius and Verschiebung operationson the nil-terms inthe algebraic K-theory of spaces, in three different ways. Twoapplications are included. Firstly, we show that the homotopygroups of are eithertrivial or not finitely generated as abelian groups. Secondly,the Verschiebung operation defines a -module structure on the homotopy groups of , with the multiplicative monoid. We also give a calculation of the homotopy type of the nil-terms after p-completion foran odd prime p and their homotopy groups as -modules up to dimension 4p – 7. We obtainnon-trivial groups only in dimension 2p – 2, where itis finitely generated as a -module, and in dimension 2p – 1, where it is not finitelygenerated as a -module. Received April 12, 2007.     

A p-Adic Quasi-Quadratic Time Point Counting Algorithm

In this article, we give an algorithm for the computation ofthe number of rational points on the Jacobian variety of a genericordinary hyperelliptic curve defined over a finite field of cardinality q with time complexity O(n^2+o(1))and space complexity O(n²), where n = log(q). In the lattercomplexity estimate the genus and the characteristic are assumedas fixed. Our algorithm forms a generalization of both, theAGM algorithm of J.-F. Mestre and the canonical lifting methodof T. Satoh. We canonically lift a certain arithmetic invariantof the Jacobian of the hyperelliptic curve in terms of thetaconstants. The theta null values are computed with respect toa semi-canonical theta structure of level 2p where >0 isan integer and . Theresults of this paper suggest a global positive answer to thequestion whether there exists a quasi-quadratic time algorithmfor the computation of the number of rational points on a genericordinary abelian variety defined over a finite field.     

Non-Degeneracy of Pollard Rho Collisions

The Pollard algorithm is a widely used algorithm for solvingdiscrete logarithms on general cyclic groups, including ellipticcurves. Recently the first nontrivial runtime estimates wereprovided for it, culminating in a sharp bound for the collision time on a cyclic groupof order n [4, 5]. In this paper, we show that for n satisfyinga mild arithmetic condition, the collisions guaranteed by theseresults are nondegenerate with high probability: that is, thePollard algorithm successfully finds the discrete logarithm.     

The integral Novikov conjectures for linear groups containing torsion elements

In this paper, we show that for any global field k, the generalizedintegral Novikov conjecture in both K- and L-theories holdsfor every finitely generated subgroup of GL(n, k). This impliesthat the conjecture holds for every finitely generated subgroupof , where is the algebraic closure of . We also show that for every linear algebraicgroup defined over k, every S-arithmetic subgroup satisfiesthis generalized integral Novikov conjecture. We note that theintegral Novikov conjecture implies the stable Borel conjecture,in particular, the stable Borel conjecture holds for all theabove torsion-free groups. Most of these subgroups are not discretesubgroups of Lie groups with finitely many connected components,and some of them are not finitely generated. When the fieldk is a function field such as , and the k-rank of is positive, many of these S-arithmeticsubgroups such as donot admit cofinite universal spaces for proper actions. Received February 15, 2007.     

Families Index for Manifolds with Hyperbolic Cusp Singularities

Manifolds with fibered hyperbolic cusp metrics include hyperbolicmanifolds with cusps and locally symmetric spaces of -rank 1. We extend Vaillant's treatment of Dirac-typeoperators associated to these metrics by weakening the hypotheseson the boundary families through the use of Fredholm perturbationsas in the family index theorem of Melrose and Piazza, and bytreating the index of families of such operators. We also extendthe index theorem of Moroianu and Leichtnam–Mazzeo–Piazzato families of perturbed Dirac-type operators associated tofibered cusp metrics (sometimes known as fibered boundary metrics).     

Representing Dehn Twists with Branched Coverings

We show that any homologically nontrivial Dehn twist of a compactsurface F with boundary is the lifting of a half-twist in thebraid group , with respectto a suitable branched covering p : F B². In particular, weallow the surface to have disconnected boundary. As a consequence,any allowable Lefschetz fibration on B² is a branched coveringof B² x B².     

Constructing infinitely many smooth structures on small 4-manifolds

The purpose of this article is two-fold. First we outline ageneral construction scheme for producing simply connected minimalsymplectic -manifolds with small Euler characteristics. Usingthis scheme, we illustrate how to obtain irreducible symplectic-manifolds homeomorphic but not diffeomorphic to for k = 1, ..., 4, or to for l = 1, ..., 6. Secondly, for each of thesehomeomorphism types, we show how to produce an infinite familyof pairwise nondiffeomorphic nonsymplectic 4-manifolds belongingto it. In particular, we prove that there are infinitely manyexotic irreducible nonsymplectic smooth structures on, and . Received April 17, 2007. Revised September 26, 2007.     

Norm attaining polynomials

If L 0 is a continuous symmetric n-linear form on a Banachspace and is the associatedcontinuous n-homogeneous polynomial, the ratio always lies between 1 and nⁿ/n!. At one extreme,if L is defined on Hilbert space, then . If L attains norm on Hilbert space, then also attains norm; in this case, we give anexplicit construction to provide a unit vector x₀ with . At the other extreme, if and L attains norm, then attains norm. We prove that in general the converseis not true.     

Traces and extensions of matrix-weighted Besov spaces

Let V be a matrix weight on ⁿ⁺¹ and let W be a matrix weighton ⁿ, satisfying, for example, the matrix A_p condition. Definethe trace, or restriction, operator Tr by Tr (f)(x^')=f(x^', 0),where x^'n and f is a function on ⁿ⁺¹. If –1/p>n (1/p–1)₊+(β–n)/p,where β is the doubling exponent of W, then the trace operatoris bounded from into (matrix-weighted Besov spaces) if and only ifthe weights V and W uniformly satisfy an estimate controllingthe average of on anydyadic cube I ⁿ by the average of on Q(I)=Ix[0, (I)], for all . If V and W satisfy the converse inequality, then there existsa continuous linear map .If both inequalities hold, then Tr Ext is the identity on .     

Hierarchical matrix techniques for low- and high-frequency Helmholtz problems

Wolfgang Hackbusch ^{In this paper, we discuss the application of hierarchical matrixtechniques to the solution of Helmholtz problems with largewave number in 2D. We consider the Brakhage–Werner integralformulation of the problem discretized by the Galerkin boundary-elementmethod. The dense n x n Galerkin matrix arising from this approachis represented by a sum of an -matrix and an 2-matrix, two different hierarchical matrix formats.A well-known multipole expansion is used to construct the 2-matrix. We present a new approach to dealingwith the numerical instability problems of this expansion: theparts of the matrix that can cause problems are approximatedin a stable way by an -matrix. Algebraic recompression methods are used to reducethe storage and the complexity of arithmetical operations ofthe -matrix.Further, an approximate LU decomposition of such a recompressed-matrix is aneffective preconditioner. We prove that the construction ofthe matrices as well as the matrix-vector product can be performedin almost linear time in the number of unknowns. Numerical experimentsfor scattering problems in 2D are presented, where the linearsystems are solved by a preconditioned iterative method.}     

Decay Bounds on Eigenfunctions and the Singular Spectrum of Unbounded Jacobi Matrices

Bounds on the exponential decay of generalized eigenfunctionsof bounded and unbounded selfadjoint Jacobi matrices in are established. Two cases are considered separatelyand lead to different results: (i) the case in which the spectralparameter lies in a general gap of the spectrum of the Jacobimatrix and (ii) the case of a lower semibounded Jacobi matrixwith values of the spectral parameter below the spectrum. Itis demonstrated by examples that both results are sharp. Weapply these results to obtain a "many barriers-type" criterionfor the existence of square-summable generalized eigenfunctionsof an unbounded Jacobi matrix at almost every value of the spectralparameter in suitable open sets. In particular, this leads toexamples of unbounded Jacobi matrices with a spectral mobilityedge, i.e., a transition from purely absolutely continuous spectrumto dense pure point spectrum.     

Spectral properties of banded Toeplitz matrices * By Albrecht Bottcher and Sergei M. Grudsky: * 411 pp., US$ 95.00, ISBN 0-89871-599-7 * (Society for Industrial and Applied Mathematics, Philadelphia, 2005)

Recall that an infinite Toeplitz matrix, , say, is one for which the values t_jk dependonly on j – k, so we may write t_jk = a_{j – k}; thusthe entries are constant on diagonals sloping down from topleft to bottom right:

Sucha matrix is said to be banded if there are only finite manynon-zero a_j, say . The propertiesof such a matrix, when defining a linear operator T on a sequencespace, are often closely related to the properties of the Laurentpolynomial

for zlying on the unit circle in the complex plane. For example,suppose that the matrix acts on the most familiar sequence space,the Hilbert space ²; then it has been known for many years thatthe operator norm of T is the maximum     

Fractional decompositions of dense hypergraphs

A seminal result of Rödl (the Rödl nibble) assertsthat the edges of the complete r-uniform hypergraph can be packed, almost completely, with copiesof , where k is fixed.We prove that the same result holds in a dense hypergraph setting.It is shown that for every r-uniform hypergraph H₀, there existsa constant = (H₀) < 1 such that every r-uniform hypergraphH in which every (r – 1)-set is contained in at least n edges has an H₀-packing that covers |E(H)|(1 – o_n(1))edges. Our method of proof uses fractional decompositions andmakes extensive use of probabilistic arguments and additionalcombinatorial ideas.     

The Cauchy Transform * (Mathematical Surveys and Monographs 125) * By Joseph A. Cima, Alec L. Matheson and William T. Ross: * 272 pp., {pound}43.75 (US$75), ISBN 0-8218-3871-7 * (American Mathematical Society, Providence, RI, 2006).

In the border country between complex analysis, harmonic analysisand differential equations, there can be found many populartransforms, of which those of Fourier and Laplace are probablythe most commonly used. This book is concerned with anotherone, which is more rarely sighted. The Cauchy transform of a finite complex measure µ definedon the unit circle in the complex plane is the analytic functionKµ defined by

forz in the unit disc . Its name derives from the observation thatCauchy's integral formula, when applied to an analytic functionf, is equivalent to the statement that the Cauchy transformof the measure is just     

The Tate Conjecture for Generic Abelian Varieties

Let A V be a Kuga fibre variety of Mumford's Hodge type, definedover a finitely generated subfield of C, and let be the genericpoint of V. We show that any element of which is invariant under , for some finite extension E of k(), is fixed bythe semisimple part of the Hodge group of A. If A V satisfiesthe H₂-condition, then the Hodge and Tate conjectures are equivalentfor A, and the Mumford–Tate conjecture is true.     

On products of quadratic twists and ranks of elliptic curves over large fields

In this paper, we give examples of elliptic curves E/K overa number field K satisfying the property that there exist P₁,P₂ K[t] such that the twists and are of positiverank over K(t). As a consequence of this result on twists, weshow that for those elliptic curves E/K, and for each , the rank of E over the fixed field (K^ab) under is infinite, where K^ab is the maximal abelian extension ofK.