We consider cyclic graphs, that is, graphs with cyclic orders at the vertices, corresponding to 2-cell embeddings of graphs into orientable surfaces, or combinatorial maps. We construct a three variable polynomial invariant of these objects, the cyclic graph polynomial, which has many of the useful properties of the Tutte polynomial. Although the cyclic graph polynomial generalizes the Tutte polynomial, its definition is very different, and it depends on the embedding in an essential way. 2000 Mathematical Subject Classification: 05C10.

The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995) on a mean value containing the fourth power of the zeta function. 2000 Mathematical Subject Classification: 11N05.

Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for descendent integrals on the moduli space of curves, is a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one ‘$\lambda$-class’), and show that, after a non-trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber. We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented proves all such recursions. As examples we present surprisingly simple new recursions in genus 0 to 3. Similar recursions should exist for all genera. As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self-contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics. 2000 Mathematical Subject Classification: primary 14H10, 81T40; secondary 05C30, 58D29.

In an earlier work, the second author proved a general formula for the equivariant Poincaré polynomial of a linear transformation g which normalises a unitary reflection group G, acting on the cohomology of the corresponding hyperplane complement. This formula involves a certain function (called a Z-function below) on the centraliser CG(g), which was proved to exist only in certain cases, for example, when g is a reflection, or is G-regular, or when the centraliser is cyclic. In this work we prove the existence of Z-functions in full generality. Applications include reduction and product formulae for the equivariant Poincaré polynomials. The method is to study the poset L(CG(g)) of subspaces which are fixed points of elements of CG(g). We show that this poset has Euler characteristic 1, which is the key property required for the definition of a Z-function. The fact about the Euler characteristic in turn follows from the ‘join-atom’ property of L(CG(g)), which asserts that if {X1,...,Xk} is any set of elements of L(CG(g)) which are maximal (set theoretically) then their setwise intersection $\bigcap_{i=1}^kX_i$ lies in L(CG(g)). 2000 Mathematical Subject Classification: primary 14R20, 55R80; secondary 20C33, 20G40.

We prove that, if $M > 4(1+2\sqrt{3})$ and $\varepsilon > 0$, if $\mathcal{V}$ and $\mathcal{W}$ are complex JBW*-triples (with preduals $\mathcal{V}_{*}$ and $\mathcal{W}_{*}$, respectively), and if $U$ is a separately weak*-continuous bilinear form on $\mathcal{V} \times \mathcal{W}$, then there exist norm-one functionals $\varphi_{1},\varphi_{2}\in \mathcal{V}_{*}$ and $\psi_{1},\psi_{2}\in \mathcal{W}_{*}$ satisfying |U(x,y)| \leq M \,\|U\| ( \|x\|_{\varphi_{2}}^{2} + \varepsilon^{2} \, \|x\|_{\varphi_{1}}^{2} )^{\frac 12} ( \|y\|_{\psi_{2}}^{2} + \varepsilon^{2} \, \|y\|_{\psi_{1}}^{2} )^{\frac 12} for all $(x,y)\in \mathcal{V} \times \mathcal{W}$. Here, for a norm-one functional $\varphi$ on a complex JB*-triple $\mathcal{V}$, $\|\cdot\|_{\varphi}$ stands for the prehilbertian seminorm on $\mathcal{V}$ associated to $\varphi$ given by $\|x\|_{\varphi}^{2} := \varphi \{x,x,z\}$ for all $x\in \mathcal{W}$, where $z\in \mathcal{V}^{**}$ satisfies $\varphi (z) = \|z\|=1$. We arrive at this form of ‘Grothendieck's inequality’ through results of C.-H. Chu, B. Iochum, and G. Loupias, and an amended version of the ‘little Grothendieck's inequality’ for complex JB*-triples due to T. Barton and Y. Friedman. We also obtain extensions of these results to the setting of real JB*-triples. 2000 Mathematical Subject Classification: 17C65, 46K70, 46L05, 46L10, 46L70.

We prove that abelian subgroups of fundamental groups of Haken 3-manifolds and closed hyperbolic n-orbifolds are separable. 2000 Mathematical Subject Classification: 20E26, 51M10, 57M05.

We study modules for the general linear group (over an infinite field of arbitrary characteristic) which are direct summands of tensor products of exterior powers and symmetric powers of the natural module. These modules, which we call listing modules, include the tilting modules and the injective modules for Schur algebras. The modules are studied via their relationship to linear source modules for symmetric groups on the one hand, and simple modules for Schur superalgebras on the other. Listing modules are parametrized by certain pairs of partitions. They are used to describe, by generators and relations, the Grothendieck ring of polynomial functors generated by the symmetric and exterior powers. We also (continuing work of J. Grabmeier) describe the vertices and sources of linear source modules for symmetric groups. 2000 Mathematical Subject Classification: 20G05, 20C30.

Let K be an algebraically closed field endowed with a complete non-archimedean norm with valuation ring R. Let $f\colon Y \to X$ be a map of K-affinoid varieties. In this paper we study the analytic structure of the image $f(Y) \subset X$; such an image is a typical example of a subanalytic set. We show that the subanalytic sets are precisely the $\mathbf D$-semianalytic sets, where $\mathbf D$ is the truncated division function first introduced by Denef and van den Dries. This result is most conveniently stated as a Quantifier Elimination result for the valuation ring R in an analytic expansion of the language of valued rings. To prove this we establish a Flattening Theorem for affinoid varieties in the style of Hironaka, which allows a reduction to the study of subanalytic sets arising from flat maps, that is, we show that a map of affinoid varieties can be rendered flat by using only finitely many local blowing ups. The case of a flat map is then dealt with by a small extension of a result of Raynaud and Gruson showing that the image of a flat map of affinoid varieties is open in the Grothendieck topology. Using Embedded Resolution of Singularities, we derive in the zero characteristic case, a Uniformization Theorem for subanalytic sets: a subanalytic set can be rendered semianalytic using only finitely many local blowing ups with smooth centres. As a corollary we obtain the fact that any subanalytic set in the plane R2 is semianalytic. 2000 Mathematical Subject Classification: 32P05, 32B20, 13C11, 12J25, 03C10.

We define two canonical cohomology theories for Hopf C*-algebras and for Hopf von Neumann algebras (with coefficients in their comodules). We then study the situations when these cohomologies vanish. The cases of locally compact groups and compact quantum groups are considered in more detail. E-mail: c.k.ng@qub.ac.uk 2000 Mathematical Subject Classification: primary 46L05, 46L55; secondary 43A07, 22D25.

Let $\varGamma =\varGamma _{\tau ,z}$ be one of the $N^2$-dimensional bicovariant first order differential calculi for the quantum groups $\mathrm{GL}_q(N)$, $\mathrm{SL}_q(N)$, $\mathrm{SO}_q(N)$, or $\mathrm{Sp}_q(N)$, where $q$ is a transcendental complex number and $z$ is a regular parameter. It is shown that the de Rham cohomology of Woronowicz' external algebra $\varGamma^\land $ coincides with the de Rham cohomologies of its left-coinvariant, its right-coinvariant and its (two-sided) coinvariant subcomplexes. In the cases $\mathrm{GL}_q(N)$ and $\mathrm{SL}_q(N)$ the cohomology ring is isomorphic to the coinvariant external algebra $\varGamma ^\land _{\scriptscriptstyle{\mathrm{Inv}}}$ and to the vector space of harmonic forms. We prove a Hodge decomposition theorem in these cases. The main technical tool is the spectral decomposition of the quantum Laplace-Beltrami operator. 2000 Mathematical Subject Classification: 46L87, 58A12, 81R50.