We define the notion of a Kirby element of a ribbon category $\mathcal{C}$ (not necessarily semisimple). Kirby elements lead to 3-manifold invariants. We characterize a class of Kirby elements, the algebraic Kirby elements, in terms of the structure maps of a Hopf algebra in $\mathcal{C}$. This class is sufficiently large to recover the quantum invariants of 3-manifolds of Reshetikhin and Turaev, of Hennings, Kauffman and Radford, and of Lyubashenko when these are well defined. The cases of a semisimple ribbon category and of a category of representations are explored in detail.

Let $H$ be a Hopf algebra and let $\mathcal{M}_s (H)$ be the category of all left $H$-modules and right $H$-comodules satisfying appropriate compatibility relations. An object in $\mathcal{M}_s (H)$ will be called a stable anti-Yetter–Drinfeld module (over $H$) or a SAYD module, for short. To each $M \in \mathcal{M}_s (H)$ we associate, in a functorial way, a cyclic object $\mathrm{Z}_\ast (H, M)$. We show that our construction can be used to compute the cyclic homology of the underlying algebra structure of $H$ and the relative cyclic homology of $H$-Galois extensions.

Let $K$ be a Hopf subalgebra of $H$. For an arbitrary $M \in \mathcal{M}_s (K)$ we define a right $H$-comodule structure on $\mathrm{Ind}_K^H M := H \otimes_K M$ so that $\mathrm{Ind}_K^H M$ becomes an object in $\mathcal{M}_s (H)$. Under some assumptions on $K$ and $M$ we compute the cyclic homology of $\mathrm{Z}_\ast (H, \mathrm{Ind}_K^H M)$. As a direct application of this result, we describe the relative cyclic homology of strongly graded algebras. In particular, we calculate the cyclic homology of group algebras and quantum tori.

Finally, when $H$ is the enveloping algebra of a Lie algebra $\mathfrak{g}$, we construct a spectral sequence that converges to the cyclic homology of $H$ with coefficients in a given SAYD module $M$. We also show that the cyclic homology of almost symmetric algebras is isomorphic to the cyclic homology of $H$ with coefficients in a certain SAYD module.

A regular map of type $\{m,n\}$ is a 2-cell embedding of a graph in an orientable surface, with the property that for any two directed edges $e$ and $e'$ there exists an orientation-preserving automorphism of the embedding that takes $e$ onto $e'$, and in which the face length and the vertex valence are $m$ and $n$, respectively. Such maps are known to be in a one-to-one correspondence with torsion-free normal subgroups of the triangle groups $T(2,m,n)$. We first show that some of the known existence results about regular maps follow from residual finiteness of triangle groups. With the help of representations of triangle groups in special linear groups over algebraic extensions of ${\mathbb Z}$ we then constructively describe homomorphisms from $T(2,m,n)=\langle y,z|\ y^m=z^n=(yz)^2=1\rangle$ into finite groups of order at most $c^r$ where $c=c(m,n)$, such that no non-identity word of length at most $r$ in $x,y$ is mapped onto the identity. As an application, for any hyperbolic pair $\{m,n\}$ and any $r$ we construct a finite regular map of type $\{m,n\}$ of size at most $C^r$, such that every non-contractible closed curve on the supporting surface of the map intersects the embedded graph in more than $r$ points. We also show that this result is the best possible up to determining $C=C(m,n)$. For $r\ge m$ the graphs of the above regular maps are arc-transitive, of valence $n$, and of girth $m$; moreover, if each prime divisor of $m$ is larger than $2n$ then these graphs are non-Cayley. 2000 Mathematics Subject Classification: 05C10, 05C25, 20F99, 20H25.

In this paper the authors prove the following result.

Let $\alpha$ be an irrational number. Then for any $\varepsilon > 0$, there are infinitely many prime numbers p such that $\| \alpha p \| < p^{-16/49 +\varepsilon}$.

The exponent $\frac{16}{49}$ improves on $\frac{9}{28}$, which was obtained recently by the second author [Sci. China Ser. A 43 (2000) 703-721]. The result is very close to the exponent $\frac{1}{3}$, which can be obtained under the Generalized Riemann Hypothesis.

Previous approaches to this problem have all used the same basic estimates for the trigonometric sums that arise. However, the present proof uses new bounds, which depend on the Kloosterman sum and also on a counting problem in the geometry of numbers. In addition new techniques for the sieve method are applied. The most significant feature of the new approach is that, unlike previous methods, the exponential sum estimates remain non-trivial for the exponent $\frac{1}{3}$. This gives one hope for an unconditional result as good as that available under the Generalized Riemann Hypothesis.

2000 Mathematical Subject Classification: 11N36, 11J71.

In this paper, we study applications of Atiyah–Hirzebruch spectral sequences for motivic cobordism recently found by Hopkins and Morel. For example, we compute the mod 2 motivic cohomology of a reduced Cech complex of a splitting variety according to Orlov, Vishik and Voevodsky. The cobordism ring $MGL / 2^{*,*} (Spec(\mathbb{R}))$ is computed and compared with the results of the real cobordism theory of Hu and Kriz. Moreover, we study algebraic cobordism of the classifying spaces $BG$ for algebraic groups over $\mathbb{C}$. For example, the Chow ring $CH^* (BG_2)_(2)$ for the exceptional Lie group $G_2$ is determined by using motivic cobordism and motivic cohomology.

Let $F$ be a non-archimedean local field of residual characteristic different from $2$, and let $G$ be a unitary, symplectic or orthogonal group, considered as the fixed point subgroup in $\tilde G={\rm GL}(N,F)$ of an involution $\sigma$. We generalize the notion of a {\it simple character} for $\tilde G$, which was introduced by Bushnell and Kutzko [Annals of Mathematics Studies 129 (Princeton University Press, 1993)], to define semisimple characters. Given a semisimple character $\theta$ for $\tilde G$ fixed by $\sigma$, we transfer it to a character $\theta_{-}$ for $G$ and calculate its intertwining. If the torus associated to $\theta_{-}$ is maximal compact, we obtain supercuspidal representations of $G$, which are new if the torus is split only over a wildly ramified extension. 2000 Mathematics Subject Classification: 22E50.

We make the category $\textbf{BGrb}_M$ of bundle gerbes on a manifold $M$ into a $2$-category by providing $2$-cells in the form of transformations of bundle gerbe morphisms. This description of $\textbf{BGrb}_M$ as a $2$-category is used to define the notion of a bundle $2$-gerbe. To every bundle $2$-gerbe on $M$ is associated a class in $H^4(M ; \mathbb{Z})$. We define the notion of a bundle $2$-gerbe connection and show how this leads to a closed, integral, differential $4$-form on $M$ which represents the image in real cohomology of the class in $H^4(M ; \mathbb{Z})$. Some examples of bundle $2$-gerbes are discussed, including the bundle $2$-gerbe associated to a principal $G$ bundle $P \to M$. It is shown that the class in $H^4(M ; \mathbb{Z})$ associated to this bundle $2$-gerbe coincides with the first Pontryagin class of $P$: this example was previously considered from the point of view of $2$-gerbes by Brylinski and McLaughlin.

Let $F$ be a non-Archimedean local field, with the ring of integers $\mathfrak{o}_F$. Let $G = \mathrm{GL}_N(F)$, $K = \mathrm{GL}_N (\mathfrak{o}_F)$, and $\pi$ be a supercuspidal representation of $G$. We show that there exists a unique irreducible smooth representation $\tau$ of $K$, such that the restriction to $K$ of a smooth irreducible representation $\pi '$ of $G$ contains $\tau$ if and only if $\pi '$ is isomorphic to $\pi \otimes \chi \circ \det$, where $\chi$ is an unramified quasicharacter of $F^{\times}$. Moreover, we show that $\pi$ contains $\tau$ with the multiplicity 1. As a corollary we obtain a kind of inertial local Langlands correspondence.

The mod $p$ representation associated to an elliptic curve is called split or non-split dihedral if its image lies in the normaliser of a split or non-split Cartan subgroup of $\GL_2(\f_p)$, respectively. Let $\xsplit$ and $\xnonsplit$ denote the modular curves which classify elliptic curves with split and non-split dihedral mod $p$ representation, respectively. We call such curves split and non-split{\it Cartan modular curves}. The curve $\xsplit$ is isomorphic to the curve $X_0^+(p^2)$. Using the Selberg trace formula for Hecke operators, we verify that the jacobian of $\xnonsplit$ is isogenous to the new part of the jacobian of $X_0^+(p^2)$.

1991 Mathematics Subject Classification: primary 11G18; secondary 11F72.

Given a classical Weyl group $W$, that is, a Weyl group of type $A$, $B$ or $D$, one can associate with it a polynomial with integral coefficients $Z_W$ given by the ratio of the Hilbert series of the invariant algebras of the natural action of $W$ and $W^t$ on the ring of polynomials ${\bf C}[x_1, \ldots , x_n]^{\otimes t}$. We introduce and study several statistics on the classical Weyl groups of type $B$ and $D$ and show that they can be used to give an explicit formula for $Z_{D_n}$. More precisely, we define two Mahonian statistics, that is, statistics having the same distribution as the length function, $Dmaj$ and $ned$ on $D_n$. The statistic $Dmaj$, defined in a combinatorial way, has an analogous algebraic meaning to the major index for the symmetric group and the flag-major index of Adin and Roichman for $B_n$; namely, it allows us to find an explicit formula for $Z_{D_n}$. Our proof is based on the theory of $t$-partite partitions introduced by Gordon and further studied by Garsia and Gessel.

Using similar ideas, we define the Mahonian statistic $ned$ also on $B_n$ and we find a new and simpler proof of the Adin–Roichman formula for $Z_{B_n}$.

Finally, we define a new descent number $Ddes$ on $D_n$ so that the pair $(Ddes,Dmaj)$ gives a generalization to $D_n$ of the Carlitz identity on the Eulerian–Mahonian distribution of descent number and major index on the symmetric group.

The main result of this paper is the resolvent similarity criterion which says that linear growth of the resolvent towards the spectrum is sufficient for a Hilbert space contraction with finite rank defect operators and spectrum not covering the unit disc to be similar to a normal operator. Similar results are proved for operators having a spectral set bounded by a Dini-smooth Jordan curve; in particular, a dissipative operator with finite rank imaginary part is similar to a normal operator if and only if its resolvent grows linearly towards the spectrum. Relevant results on the insufficiency of linear resolvent growth not accompanied by smallness of defect operators are presented. Also it is proved that there is no restriction on the spectrum, other than finiteness, which together with linear resolvent growth implies similarity to a normal operator. The construction of corresponding examples depends on a characterization of well-known Ahlfors curves as curves of linear length growth with respect to linear fractional transformations.

Theorem 1 asserts that \emph{in a finitely generated prosoluble group, every subgroup of finite index is open}. This generalises an old result of Serre about pro-$p$ groups. It follows by a standard argument from Theorem 2: \emph{in a $d$-generator finite soluble group, every element of the derived group is equal to a product of $72d^2 +46d$ commutators}. This result about finite soluble groups is proved by induction on the order of the group, and is elementary though rather intricate. The essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field. Corollaries include the following. \emph{Let $\Gamma$ be a finitely generated prosoluble group. Then each term of the lower central series of $\Gamma$ and each power subgroup $\Gamma ^n$ is closed}. 1991 Mathematics Subject Classification: 20E18, 20D10.

Given a fibred, compact, orientable 3-manifold with single boundary component, we show that a fibration with fiber surface of negative Euler characteristic can be perturbed to yield taut foliations realizing an open interval of boundary slopes about the boundary slope of the fibration. These taut foliations extend to taut foliations in the corresponding surgery manifolds. 2000 Mathematics Subject Classification: primary 57M25; secondary 57R30.

Let $f(x, y)$ be a binary cubic form with integral rational coefficients, and suppose that the polynomial $f(x, y)$ is irreducible in $\mathbb{Q}[x, y]$ and no prime divides all the coefficients of $f$. We prove that the set $f(\mathbb{Z}^{2})$ contains infinitely many primes unless $f(a, b)$ is even for each $(a, b)$ in $\mathbb{Z}^{2}$, in which case the set $\frac{1}{2}f(\mathbb{Z}^{2})$ contains infinitely many primes.

2000 Mathematical Subject Classification: primary 11N32; secondary 11N36, 11R44.

Let $k$ be an algebraically closed field, and let $A$ be a finite-dimensional $k$-algebra. The first aim of classsical representation theory was to classify all indecomposable $A$-modules, but the homomorphisms between modules were not considered. One big step forward, made around 1975, was the invention of the Auslander--Reiten quiver of $A$. The Auslander--Reiten quiver yields a description of the category of $A$-modules modulo its infinite radical ${\rm rad}_A^\omega$. But it contains no information about the homomorphisms in ${\rm rad}_A^\omega$. We denote by mod-$A$ the category of finite-dimensional right modules over $A$. Let ${\rm rad}_A$ be the radical of mod-$A$. Following Prest, we define ${\rm rad}_A^\alpha$ for each ordinal number $\alpha$ as follows: if $J$ is an ideal in mod-$A$ and $n \geq 1$ is a natural number, then $J^n$ is the $n$-fold product of the ideal $J$. For a limit number $\beta$ define ${\rm rad}_A^\beta = \bigcap_{\alpha < \beta} {\rm rad}_A^\alpha$. Finally, let $\alpha$ be an arbitrary infinite ordinal number. Thus $\alpha = \beta + n$ for some limit number $\beta$ and some natural number $n \geq 0$. In this case, define ${\rm rad}_A^\alpha = ({\rm rad}_A^\beta)^{n+1}$. We denote the first limit number by $\omega$, the second by $\omega 2$, and so on. The minimal limit number, which is not of the form $\omega n$ for some natural number $n \geq 1$, is denoted by $\omega^2$. {\sc Theorem 1.} {\it For each ordinal number $\alpha$ with $\omega < \alpha < \omega^2$ there exists a finite-dimensional $k$-algebra $A$ with ${\rm rad}_A^\alpha \not= 0$ and ${\rm rad}_A^{\alpha + 1} = 0$. Let $Q$ be a quiver, and let $I$ be an admissible ideal of the path algebra $kQ$. We call $kQ/I$ {\it special biserial} if the following hold. Any vertex of $Q$ is the starting point of at most two arrows and also the end point of at most two arrows. Given an arrow $\beta$, there is at most one arrow $\alpha$ with $e(\alpha) = s(\beta)$ and $\alpha \beta \notin I$ and at most one arrow $\gamma$ with $e(\beta) = s(\gamma)$ and $\beta \gamma \notin I$. {\sc Theorem 2.} {\it Let $A$ be a special biserial algebra. Then the following are equivalent: \begin{enumerate} \item[{\rm (i)}] $A$ is domestic; \item[{\rm (ii)}] ${\rm rad}_A^{\omega^2} = 0$. \end{enumerate} } 1991 Mathematics Subject Classification: 16G20, 16G60, 16N40

A family $\mathbf{A}_\alpha$ of differential operators depending on a real parameter $\alpha \ge 0$ is considered. This family was suggested by Smilansky as a model of an irreversible quantum system. We find the absolutely continuous spectrum $\sigma_{a.c.}$ of the operator $\mathbf{A}_\alpha$ and its multiplicity for all values of the parameter. The spectrum of $\mathbf{A}_0$ is purely absolutely continuous and admits an explicit description. It turns out that for $\alpha < \sqrt 2$ one has $\sigma_{a.c.}(\mathbf{A}_\alpha) = \sigma_{a.c.}(\mathbf{A}_0)$, including the multiplicity. For $\alpha \ge \sqrt2$ an additional branch of the absolutely continuous spectrum arises; its source is an auxiliary Jacobi matrix which is related to the operator $\mathbf{A}_\alpha$. This birth of an extra branch of the absolutely continuous spectrum is the exact mathematical expression of the effect that was interpreted by Smilansky as irreversibility.

A finite permutation group is said to be innately transitive if it contains a transitive minimal normal subgroup. In this paper, we give a characterisation and structure theorem for the finite innately transitive groups, as well as describing those innately transitive groups which preserve a product decomposition. The class of innately transitive groups contains all primitive and quasiprimitive groups.

Let V be a pencil of curves in ${\bf P}^2$ with one place at infinity, and $X \longrightarrow {\bf P}^2$ the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariski's unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity.

2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

The topological zeta function and Igusa's local zeta function are respectively a geometrical invariant associated to a complex polynomial $f$ and an arithmetical invariant associated to a polynomial $f$ over a $p$-adic field.

When $f$ is a polynomial in two variables we prove a formula for both zeta functions in terms of the so-called log canonical model of $f^{-1} \{ 0 \}$ in $\Bbb A^2$. This result yields moreover a conceptual explanation for a known cancellation property of candidate poles for these zeta functions. Also in the formula for Igusa's local zeta function appears a remarkable non-symmetric ‘$q$-deformation’ of the intersection matrix of the minimal resolution of a Hirzebruch-Jung singularity.

1991 Mathematics Subject Classification: 32S50 11S80 14E30 (14G20)

We construct a homological approximation to the partition complex, and identify it as the Tits building. This gives a homological relationship between the symmetric group and the affine group, leads to a geometric tie between symmetric powers of spheres and the Steinberg idempotent, and allows us to use the self-duality of the Steinberg module to study layers in the Goodwillie tower of the identity functor. 2000 Mathematics Subject Classification: 55N25, 55S15, 20B30, 55P25.