Let $G$ be a semisimple Lie group of $\mathbb{R}$-rank at least 2 and $\varGamma$ a discrete subgroup of $G$. We consider the limit set of $\varGamma$ in the geometric boundary of the symmetric space associated with $G$. We define the notion of conical and horospherical limit points. In the case of irreducible non-uniform lattices, by using the two Tits building structures, we distinguish the location of their conical limit points. The limit sets of generalized Schottky groups contained in Hilbert modular groups are studied.

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.

2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

In the late 1980s, Vassiliev introduced new graded numerical invariants of knots, which are now called Vassiliev invariants or finite-type invariants. Since he made this definition, many people have been trying to construct Vassiliev type invariants for various mapping spaces. In the early 1990s, Arnold and Goryunov introduced the notion of first order (local) invariants of stable maps. In this paper, we define and study {\it first order semi-local invariants} of stable maps and those of stable fold maps of a closed orientable 3-dimensional manifold into the plane. We show that there are essentially eight first order semi-local invariants. For a stable map, one of them is a constant invariant, six of them count the number of singular fibers of a given type which appear discretely (there are exactly six types of such singular fibers), and the last one is the Euler characteristic of the Stein factorization of this stable map. Besides these invariants, for stable fold maps, the Bennequin invariant of the singular value set corresponding to definite fold points is also a first order semi-local invariant. Our study of unstable fold maps with codimension 1 provides invariants for the connected components of the set of all fold maps.

Let $K$ be the quotient field of a Dedekind domain $R$. We characterize the $R$-orders $\Lambda$ in a separable $K$-algebra for which every $R$-projective $\Lambda$-module decomposes into $\Lambda$-lattices. Butler, Campbell and Kovács have recently shown that the latter holds for the integral group ring of a cyclic group of prime order, as well as for lattice-finite orders over a complete discrete valuation domain.

Let $\{d_j(n)\}_{j=1}^{\tau(n)}$ denote the increasing sequence of divisors of an integer $n$, and write $\Lambda_k(d)$ for the natural density of the set of those integers $n$ with $d_k(n)=d$. Answering a question raised by Erd\H os, we show that $\Lambda_k(d)$ attains its maximum for $d=K^{1/2+o(1)}$, where $K:=k^{(\log \log k)/\log 2}$ (which, by a theorem of Ramanujan, is roughly the size of the smallest integer having $k$ divisors). A key step of the proof consists in establishing a sublinearity property of the counting function $\Psi_1(x,y)$ of $y$-friable square-free integers not exceeding $x$. This is achieved via a complete study of $\Psi_1(x,y)$ using the saddle-point method, which, in particular, also enables a precise description of the Gaussian behaviour of this function in certain ranges.

2000 Mathematical Subject Classification: 11N25, 11N37.

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.

The main purpose of this paper is to study the validity of the Hausdorff–Young inequality for vector-valued functions defined on a non-commutative compact group. As we explain in the introduction, the natural context for this research is that of operator spaces. This leads us to formulate a whole new theory of Fourier type and cotype for the category of operator spaces. The present paper is the first step in this program, where the basic theory is presented, the main examples are analyzed and some important questions are posed.

Let $K$ be a non-archimedean local field of characteristic $0$ with residue class field not of characteristic $2$. In this note we will determine the minimal covolume $\mu(\mbox{SL}_2(K)/\Gamma)$ of a lattice $\Gamma$ in $\mbox{SL}_2(K)$. Furthermore, we describe all lattices of minimal covolume in terms of their associated graphs of groups and hence up to isomorphism.

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.

The variation of the Hodge structure of a Calabi--Yau 3-fold induces a canonical K\"ahler metric on its Kuranishi moduli space, known as the Weil--Petersson metric. Similarly, {\it special} pseudo-K\"ahler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classification of homogeneous special pseudo-K\"ahler manifolds of semisimple groups with compact stabilizer. 1991 Mathematics Subject Classification: 53C30, 53C25.

The cohomology ring of the moduli space of stable holomorphic vector bundles of rank $n$ and degree $d$ over a Riemann surface of genus $g > 1$ has a standard set of generators when $n$ and $d$ are coprime. When $n = 2$ the relations between these generators are well understood, and in particular a conjecture of Mumford, that a certain set of relations is a complete set, is known to be true. In this article generalisations are given of Mumford's relations to the cases when $n > 2$ and also when the bundles are parabolic bundles, and these are shown to form complete sets of relations.

For a topological group $G$ we define $\cal N$ to be the set of all normal subgroups modulo which $G$ is a finite-dimensional Lie group. Call $G$ a pro-Lie group if, firstly, $G$ is complete, secondly, $\cal N$ is a filter basis, and thirdly, every identity neighborhood of $G$ contains some member of $\cal N$. It is easy to see that every pro-Lie group $G$ is a projective limit of the projective system of all quotients of $G$ modulo subgroups from $\cal N$. The converse implication emerges as a difficult proposition, but it is shown here that any projective limit of finite-dimensional Lie groups is a pro-Lie group. It is also shown that a closed subgroup of a pro-Lie group is a pro-Lie group, and that for any closed normal subgroup $N$ of a pro-Lie group $G$, for any one parameter subgroup $Y \colon \mathbb{R} \to G/N$ there is a one parameter subgroup $X \colon \mathbb{R}\to G$ such that $X(t) N = Y(t)$ for any real number $t$. The category of all pro-Lie groups and continuous group homomorphisms between them is closed under the formation of all limits in the category of topological groups and the Lie algebra functor on the category of pro-Lie groups preserves all limits and quotients.

In this paper we classify families of square matrices up to the following natural equivalence. Thinking of these families as germs of smooth mappings from a manifold to the space of square matrices, we allow arbitrary smooth changes of co-ordinates in the source and pre- and post- multiply our family of matrices by (generally distinct) families of invertible matrices, all dependent on the same variables. We obtain a list of all the corresponding simple mappings (that is, those that do not involve adjacent moduli). This is a non-linear generalisation of the classical notion of linear systems of matrices. We also make a start on an understanding of the associated geometry.

Let $G$ be a simply-connected, semisimple algebraic group over $k$, an algebraically closed field of characteristic zero. Let ${\cal O}_{\epsilon}[G]$ be the quantized function algebra of $G$ at a primitive $\ell$th root of unity $\epsilon$, and let $\overline{{\cal O}_{\epsilon}[G]}$ be the `restricted' quantized function algebra at $\epsilon$, a finite-dimensional $k$-algebra obtained from ${\cal O}_{\epsilon}[G]$ by factoring out a centrally generated ideal. It is known that $\overline{{\cal O}_{\epsilon}[G]}$ is a Hopf algebra. We study the cohomology ring $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$, a graded commutative algebra, and, for any finite-dimensional $\overline{{\cal O}_{\epsilon}[G]}$-module $M$, the $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$. We prove that for sufficiently large $\ell$ there is an isomorphism of graded algebras\[ \mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k) \cong k[X_1,\ldots ,X_{2N}],\] where each $X_i$ is homogeneous of degree $2$, and $2N$ equals the number of roots associated to $G$. Moreover we show that in this case $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$is a finitely generated $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$-module. We also show under much less restrictive conditions on $\ell$ that $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,k)$ continues to be a finitely generated graded commutative algebra over which $\mbox{Ext}_{\overline{{\cal O}_{\epsilon}[G]}}^*(k,M)$ is a finitely generated module.\vspace{6mm}\noindent 1991 Mathematics Subject Classification: 16W30, 17B37, 17B56.

Given an algorithm for resolution of singularities that satisfies certain conditions (‘a good algorithm’), natural notions of simultaneous algorithmic resolution, and of equi-resolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are defined. It is proved that these notions are equivalent. Something similar is done for families of sheaves of ideals, where the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed sets $T_j$, such that the induced family on each part $T_j$ is equi-resolvable. In particular, this can be applied to the Hilbert scheme of a smooth projective variety; in fact, our result shows that, in characteristic zero, the underlying topological space of any Hilbert scheme parametrizing embedded schemes can be naturally stratified in equi-resolvable families.

A lemma of Tits establishes a connection between the simple connectivity of an incidence geometry and the universal completion of an amalgam induced by a sufficiently transitive group of automorphisms of that geometry. In the present paper, we generalize this lemma to intransitive geometries, thus opening the door for numerous applications. We treat ourselves some amalgams related to intransitive actions of finite orthogonal groups, as a first class of examples.

Let $p_n$ be the $n$th prime. Then this paper is concerned with proving the following result on the distribution of consecutive primes.

Theorem.}\begin{equation}\sum_{p_{n+1}-p_n>x^{\frac 12},\ x \leq p_n \leq 2x} (p_{n+1}-p_n) \llx^{\frac{25}{36}+\epsilon}.\end{equation}

The exponent of $x$ in this theorem improves on the work of Heath-Brown who proved $(1)$ with exponent $\frac 34$. Under the Riemann hypothesis one can prove$(1)$ with exponent $\frac 12$.The proof of the theorem starts with the Heath-Brown--Linnik identity which leads to a formula giving the number of primes in an interval in terms of coefficients of certain Dirichlet series. I then estimate the coefficients by using, among other things, the information which can be gained from Montgomery's mean value theorem and Huxley's version of the Hal\' asz lemma. Furthermore, by using familiar sieve arguments I am able to discard some of the coefficients allowing us to gain an improvement over the previous result of Heath-Brown.

1991 Mathematics Subject Classification: 11N05.

We determine the relationship between the multiplicities of the zeros of certain rational functions defined on the ${\rm SL}_2(\mathbb C)$ and ${\rm PSL}_2(\mathbb C)$ character varieties of knot exteriors. This leads to a formula for the Culler--Shalen seminorms associated to small Seifert Dehn fillings of such manifolds. 2000 Mathematics Subject Classification: 57M05, 57M27, 57R65.

Non-trivial estimates for fractional moments of smooth cubic Weyl sums are developed. Complemented by bounds for such sums of use on both the major and minor arcs in a Hardy--Littlewood dissection, these estimates are applied to derive an upper bound for the $s$th moment of the smooth cubic Weyl sum of the expected order of magnitude as soon as $s\ge 7.691$. Related arguments demonstrate that all large integers $n$ are represented as the sum of eight cubes of natural numbers, all of whose prime divisors are at most $\exp (c(\log n\log \log n)^{1/2})$, for a suitable positive number $c$. This conclusion improves a previous result of G. Harcos in which nine cubes are required. 1991 Mathematics Subject Classification: 11P05, 11L15, 11P55.

Let $X=\operatorname{Spec} B$ be an affine varietyover a field of arbitrary characteristic, and supposethat there exists an action of a unipotent group(possibly neither smooth nor connected). Thefundamental results are as follows. (1) An algorithm for computing invariants is given, by means of introducing a degree in the ring of functions of the variety, relative to the action.Therefore an algorithmic construction of the quotient, in a certain open set, is obtained. In the case of aGalois extension, $k\hookrightarrow B=K$, which iscyclic of degree $p=\text{ch} (k)$ (that is, such thattheunipotent group is $G={\Bbb Z}/p {\Bbb Z}$),an element of minimal degree becomes anArtin--Schreier radical, and the method for computing invariants gives, in particular, theexpression for any element of $K$ in terms of theseradicals, with an explicit formula. This replacesthe well-known formula of Lagrange (which is validonly when the degree of the extension and thecharacteristic are relatively prime) in the case of an extension of degree $p=\text{ch}(k)$. (2)In this paper we give an effectiveconstruction of a stable open subset where there is a quotient. In this sense we obtain an algebraic localcriterion for the existence of a quotient in aneighbourhood. It is proved (provided the variety isnormal) that, in the following cases, such an openset is the greatest one that admits a quotient:\begin{enumerate}\item[(a)] when the action is such that the orbits havedimension less than or equal to 1 (arbitrary characteristic) and, in particular, for any actionof the additive group $G_a$;\item[(b)] in characteristic 0, when the action is proper(obtained from the results of Fauntleroy) or the group is abelian. 1991 Mathematics Subject Classification:primary 14L30; secondary 14D25, 14D20.