Minimal Submanifolds in a Locally Symmetric and Conformally Flat Space

We study minimal submanifolds in the locally symmetric and conformally flat Riemannian manifold and generalize Yau's result obtained in J. Amer. Math. 97 (1975), 76–100.     

Three-dimensional FC Artin Groups are CAT(0)

Building upon earlier work of T. Brady, we construct locally CAT(0) classifying spaces for those Artin groups which are three-dimensional and which satisfy the FC (flag complex) condition. The approach is to verify the ‘link condition’ by applying gluing arguments for CAT(1) spaces and by using the curvature testing techniques of Elder and McCammond [Expositio Math. 11(1) (2002), 143–158].     

On a Conjectured Formula for Quiver Varieties

In A.S. Buch and W. Fulton [Invent. Math. 135 (1999), 665–687] a formula for the cohomology class of a quiver variety is proved. This formula writes the cohomology class of a quiver variety as a linear combination of products of Schur polynomials. In the same paper it is conjectured that all of the coefficients in this linear combination are non-negative, and given by a generalized Littlewood-Richardson rule, which states that the coefficients count certain sequences of tableaux called factor sequences. In this paper I prove some special cases of this conjecture. I also prove that the general conjecture follows from a stronger but simpler statement, for which substantial computer evidence has been obtained. Finally I will prove a useful criterion for recognizing factor sequences.     

Anti-tori in Square Complex Groups

An anti-torus is a subgroup 〈a,b 〉 in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering space X such that a and b do not have any commuting nontrivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups Γ_p,l originally studied by Mozes [Israel J. Math. 90(1–3) (1995), 253–294]. It turns out that anti-tori in Γ_p,l directly correspond to non commuting pairs of Hamilton quaternions. Moreover, free anti-tori in Γ_p,l are related to free groups generated by two integer quaternions, and also to free subgroups of . As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not free.     

On Lagrangian–Grassmannian codes

Using the Lagrangian–Grassmannian, a smooth algebraic variety of dimension n(n + 1)/2 that parametrizes isotropic subspaces of dimension n in a symplectic vector space of dimension 2n, we construct a new class of linear codes generated by this variety, the Lagrangian–Grassmannian codes. We explicitly compute their word length, give a formula for their dimension and an upper bound for the minimum distance in terms of the dimension of the Lagrangian–Grassmannian variety.     

Vandermonde Matrices, NP-Completeness, and Transversal Subspaces

Let K be an infinite field. We give polynomial time constructions of families of r-dimensional subspaces of K ⁿ with the following transversality property: any linear subspace of K ⁿ of dimension n–r is transversal to at least one element of the family. We also give a new NP-completeness proof for the following problem: given two integers n and m with n \leq m and a n × m matrix A with entries in Z, decide whether there exists an n × n subdeterminant of A which is equal to zero.     

A Class of Orthogonal Refinable Functions and Wavelets

We give a construction, for any n 2, of a space S of spline functions of degree n – 1 with simple knots in (1/4)Z which is generated by a triple of refinable, orthogonal functions with compact support. Indeed, the result holds more generally by replacing the B-spline of degree n – 1 with simple knots at the integers by any continuous refinable function whose mask is a Hurwitz polynomial of degree n. A simple construction is also given for the corresponding wavelets.     

The structure of weighing matrices having large weights

We examine the structure of weighing matricesW(n, w), wherew=n–2,n–3,n–4, obtaining analogues of some useful results known for the casen–1. In this setting we find some natural applications for the theory ofsigned groups and orthogonal matrices with entries from signed groups, as developed in [3]. We construct some new series of Hadamard matrices from weighing matrices, including the following:W(n, n–2) implies an Hadamard matrix of order2n ifn0 mod 4 and order 4n otherwise;W(n, n–3) implies an Hadamard matrix of order 8n; in certain cases,W(n, n–4) implies an Hadamard matrix of order 16n. We explicitly derive 117 new Hadamard matrices of order 2 ^t p, p<4000, the smallest of which is of order 2³·419.Supported by an NSERC grant     

Lefschetz Complex Conditions for Complex Manifolds

For any compact complex manifold M with a compatible symplectic form, we consider the homomorphisms L _1,0: H ^1,0(M) H ^{{n, n–1}(M) and L _{0, 1}: H ^{0, 1}(M) H ^{n – 1, n} (M) given by the cup product with [] ^{n – 1}, n being the complex dimension of M andH ^{*, *}(M) the Dolbeault cohomology of M. We say that Mhas Lefschetz complex type (1, 0) (resp. (0, 1)) if L _{1, 0} (resp.L _{0, 1}) is injective. Such conditions can be considered as complexversions of the (real) Lefschetz condition studied by Benson and Gordonin [Topology 27 (1988), 513–518]for symplectic manifolds. Within the class of compactcomplex nilmanifolds, we prove that the injectivity of L _{1, 0}characterizes those complex structures which are Abelian in the sense ofBarberis et al. [Ann. Global Anal. Geom. 13 (1995), 289–301]. In contrast, complex tori are the only nilmanifolds having Lefschetz complex type (0, 1).     

A bound on the plurigenera of projective varieties

We exhibit a sharp Castelnuovo bound for the i-th plurigenus of a smooth variety of given dimension n and degree d in the projective space P ^r , and classify the varieties attaining the bound, when n2, r2n+1, d>>r and i>>r. When n=2 and r=5, or n=3 and r=7, we give a complete classification, i.e. for any i1. In certain cases, the varieties with maximal plurigenus are not Castelnuovo varieties, i.e. varieties with maximal geometric genus. For example, a Castelnuovo variety complete intersection on a variety of dimension n+1 and minimal degree in P ^r , with r>(n ² +3n)/(n–1), has not maximal i-th plurigenus, for i>>r. As a consequence of the bound on the plurigenera, we obtain an upper bound for the self-intersection of the canonical bundle of a smooth projective variety, whose canonical bundle is big and nef. Mathematics Subject Classification (2000):Primary 14J99; Secondary 14N99     

Motivic Serre invariants of curves

In this article, we generalize the theory of motivic integration on formal schemes topologically of finite type and the notion of motivic Serre invariant, to a relative point of view. We compute the relative motivic Serre invariant for curves defined over the field of fractions of a complete discrete valuation ring R of equicharacteristic zero. One aim of this study is to understand the behavior of motivic Serre invariants under ramified extension of the ring R. Thanks to our constructions, we obtain, in particular, an expression for the generating power series, whose coefficients are the motivic Serre invariant associated to a curve, computed on a tower of ramified extensions of R. We give an interpretation of this series in terms of the motivic zeta function of Denef and Loeser.     

Induced Projective Modules Over Group von Neumann Algebras

Let G be a group and k a subring of the field of complex numbers. In this paper we study the additive map in reduced K-theory, which is associated with the inclusion of the group algebra kG into the group von Neumann algebra G, and obtain necessary and sufficient conditions for it to be identically zero (or zero modulo torsion). Our results complete the work of Eckmann [Comm. Math. Helvet. 71 (1996), 453–462; Arch. Math. 76 (2001), 241–249] and Schafer [K-theory 19 (2000), 211–217], while reducing to Swan’s theorem on induced representations [Ann. Math. 71 (1960), 552–578], in the case where the group G is finite. (Received: January 2005) Research supported by University of Athens grant Pythagoras 70/3/7298.     

Weak invariance principle for solutions of stochastic recurrence equations in a banach space

We show that, in a Banach space, continuous random processes constructed by using solutions of the difference equationX _n =A _n X _n+1+V _n , n=1, 2,..., converge in distribution to a solution of the corresponding operator equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 114–117, January, 1995.     

On rational embeddings of curves in the second Garcia–Stichtenoth tower

In 1995, Garcia and Stichtenoth explicitly constructed a tower of projective curves over a finite field with q² elements which reaches the Drinfeld–Vlăduţ bound. These curves are given recursively by covers of Artin–Schreier type where the curve on the nth level of the tower has a natural model in . In this paper, for q an even prime power, we use point projections in order to embed these curves into projective space of the lowest possible dimension.     

Large sieve inequalities for -forms in the conductor aspect

Duke and Kowalski in [A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)] derive a large sieve inequality for automorphic forms on GL(n) via the Rankin–Selberg method. We give here a partial complement to this result: using some explicit geometry of fundamental regions, we prove a large sieve inequality yielding sharp results in a region distinct to that in [Duke and Kowalski, A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations, Invent. Math. 139(1) (2000) 1–39 (with an appendix by Dinakar Ramakrishnan)]. As an application, we give a generalization to GL(n) of Duke's multiplicity theorem from [Duke, The dimension of the space of cusp forms of weight one, Internat. Math. Res. Notices (2) (1995) 99–109 (electronic)]; we also establish basic estimates on Fourier coefficients of GL(n) forms by computing the ramified factors for GL(n)×GL(n) Rankin–Selberg integrals.     

Laurent–Padé Approximants to Four Kinds of Chebyshev Polynomial Expansions. Part I. Maehly Type Approximants

Laurent Padé–Chebyshev rational approximants, A _m (z,z ^–1)/B _n (z,z ^–1), whose Laurent series expansions match that of a given function f(z,z ^–1) up to as high a degree in z,z ^–1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm and Common [1]. All of these methods require knowledge of Chebyshev coefficients of f up to degree m+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions between f(z,z ^–1)B _n (z,z ^–1) and A _m (z,z ^–1). The derivation was relatively simple but required knowledge of Chebyshev coefficients of f up to degree m+2n. In the present paper, Padé–Chebyshev approximants are developed not only to first, but also to second, third and fourth kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for developing the Padé–Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8] but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m,n) Padé–Chebyshev approximant, of degree m in the numerator and n in the denominator, is matched to the Chebyshev series up to terms of degree m+n, based on knowledge of the Chebyshev coefficients up to degree m+2n. Numerical tests are carried out on all four Padé–Chebyshev approximants, and results are outstanding, with some formidable improvements being achieved over partial sums of Laurent–Chebyshev series on a variety of functions. In part II of this paper [7] Padé–Chebyshev approximants of Clenshaw–Lord type will be developed for the four kinds of Chebyshev series and compared with those of the Maehly type.     

Logarithmic Kodaira dimension and the poles of the Hodge and motivic zeta functions for surfaces

To any polynomial f[x ₁,,x _n ] with f(0)=0 one associates the well-known motivic zeta function and its specialization to the level of Hodge polynomials. These zeta functions can be given very explicitly in terms of an embedded resolution of f ^–1{0} in ⁿ . In this paper, where we work with polynomials in three variables, i.e., n=3, we find a geometric condition for having a nonzero contribution to the residue at a candidate pole. More precisely, for a given embedded resolution h we fix an exceptional surface E with h(E)={0}, which induces in a canonical way a candidate pole q of the motivic zeta function. Then we prove that, when the surface E is non-rational and we are in a generic situation, the maximality of the logarithmic Kodaira dimension of E implies the non-vanishing of the contribution of E to the residue at q. Here E denotes the part of E that doesn't belong to any other irreducible component of h ^–1(f ^–1{0}). The same result is already true on the level of Hodge polynomials.Postdoctoral Fellow of the Fund for Scientific Research – Flanders (Belgium). Mathematics Subject Classificaton (2000):14B05, 14E15, 14J17 (32S45)     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

A case of the 3-dimensional problem of Apollonius

Summary Inn-dimensions the problem of Apollonius is to determine the (n–1)-spheres tangent ton+1 given (n–1)-spheres. In case no two of the given (n–1)-spheres intersect and no three have the property that one separates the other two, the expected number of solutions is 2 ⁿ⁺¹. Whenn=2 this special problem does indeed always have 8 solutions, but for higher dimensions it turns out that the number of solutions becomes dependent on the relative size and location of the given (n–1)-spheres. We describe in detail the dependence of the number of solutions in the case of the 3-dimensional problem of Apollonius on the 6 inversively invariant parameters that describe configurations of 4 given spheres. We find that the number of solutions, if finite, can be any integer from 0 to 16 and, if infinite, can be a one-, two- or three-fold infinity where the stated multiplicity refers to the number of one-parameter families of solutions that are present.