We introduce a variant of the usual K\"{a}hler forms on singular free divisors, and show that it enjoys the same depth properties as K\"{a}hler forms on isolated hypersurface singularities. Using these forms it is possible to describe analytically the vanishing cohomology, and the Gauss--Manin connection, in families of free divisors, in precise analogy with the classical description for the Milnor fibration of an isolated complete intersection singularity, due to Brieskorn and Greuel. This applies in particular to the family $\{D(f_\lambda)\}_{\lambda\in \Lambda}$ of discriminants of a versal deformation $\{f_\lambda\}_{\lambda\in\Lambda}$ of a singularity of a mapping. 1991 Mathematics Subject Classification: 14B07, 14D05, 32S40.

Given an open set $\Omega$ of compact closure in $\mathbb{R}^m$, the classical Dirichlet problem is to extend a given continuous function $\psi : \partial \Omega \to \mathbb{R}$ to a continuous function $\phi : \overline \Omega \to \mathbb{R}$ such that $\phi$ is harmonic (that is, satisfies the Laplace equation) in $\Omega$. The set $\Omega$ is termed regular if the Dirichlet problem has a (necessarily unique) solution for any continuous boundary function $\psi$. For example, every simply connected planar domain is regular (but may have a 'bad' boundary $\partial\Omega$, for instance, a fractal).

In this article it is shown that, when $\Omega$ is regular (in the above sense), every continuous map $\psi$ from $\partial\Omega$ to a simply connected complete Riemannian manifold $(N, h)$ of sectional curvature at most 0 has a unique continuous extension $\phi : \overline\Omega \to N$ which is harmonic in $\Omega$. This is done with $\mathbb{R}^m$ replaced more generally by an $m$-dimensional Riemannian manifold $(M, g)$.

The proof relies on the unique solvability of the corresponding variational Dirichlet problem (for any open set $\Omega \Subset M$). And for that, the above target manifold $N$ can be replaced more generally by any simply connected complete geodesic space $Y$ of curvature at most 0 in the sense of A. D. Alexandrov. Assuming that $M$ satisfies the Poincaré inequality, we show that, for any map $\psi : M \to Y$ of finite energy in the sense of N. J. Korevaar and R. M. Schoen, there exists a unique map $\phi : M \to Y$ with $\phi = \psi$ on $M \setminus \Omega$ such that $\phi$ minimizes the energy of all maps $M \to Y$ which agree with $\psi$ on $M \setminus \Omega$. If $\Omega$ is regular then $\phi$ is continuous at any point of $\partial\Omega$ at which $\psi$ is continuous. For a Lipschitz (and hence regular) domain $\Omega \Subset M$, existence and uniqueness of the variational solution $\phi$ was obtained by Korevaar and Schoen, and earlier for suitable polyhedral targets $Y$ by Gromov and Schoen.

Instead of the Riemannian manifold $M$ our domain space can still more generally be an admissible Riemannian polyhedron (as studied in the recent Cambridge Tract by J. Eells and the present author); the variational solution $\phi$ is then in general only Hölder continuous in $\Omega$.

The proofs of the stated results of this article rely in part on potential theory relative to the fine topology of H. Cartan.

The invariantly harmonic functions in the unit ball ${\Bbb B}^n$ in ${\Bbb C}^n$ are those annihilated by the Bergman Laplacian $\Delta$. The Poisson-Szeg\"o kernel $P(z,\zeta)$ solves the Dirichlet problem for $\Delta$: if $f\in C(S^n)$, the Poisson-Szeg\"o transform of $f$, $P[f](z)=\int_{S^n}P(z,\zeta) f(\zeta)\,d\sigma(\zeta),$ where $d\sigma$ is the normalized Lebesgue measure on $S^n$, is the unique invariantly harmonic function $u$ in ${\Bbb B}^n$, continuous up to the boundary, such that $u=f$ on $S^n$. The Poisson-Szeg\"o transform establishes, loosely speaking, a one-to-one correspondence between function theory in $S^n$ and invariantly harmonic function theory in ${\Bbb B}^n$. When $n\geq 2$, it is natural to consider on $S^n$ function spaces related to its natural non-isotropic metric, for these are the spaces arising from complex analysis. In the paper, different characterizations of such spaces of smooth functions are given in terms of their invariantly harmonic extensions, using maximal functions and area integrals, as in the corresponding Euclidean theory. Particular attention is given to characterization in terms of purely radial or purely tangential derivatives. The smoothness is measured in two different scales: that of Sobolev spaces and that of Lipschitz spaces, including BMO and Besov spaces.

1991 Mathematics Subject Classification: 32A35, 32A37, 32M15, 42B25.

We construct quantum deformations of imaginary Verma modules over $U(A_1^{(1)})$ and show that, for generic $q$, imaginary Verma modules over $U(A_1^{(1)})$ can be deformed to those over the quantum group $U_q(A_1^{(1)})$ in such a way that the dimensions of the weight spaces are invariant under the deformation. We also prove the PBW theorem for $U_q(A_1^{(1)})$ with respect to the triangular decomposition induced from the root partition corresponding to the imaginary Verma modules.

1991 Mathematics Subject Classification: 17B67, 17B65, 17B10.

The existence and continuity for the Calderón projector of the perturbed odd signature operator on a 3-manifold is established. As an application we give a new proof of a result of Taubes relating the modulo 2 spectral flow of a family of operators on a homology 3-sphere with the difference in local intersection numbers of the character varieties coming from a Heegard decomposition.

Suppose that $N$ is a normal $p$-subgroup of a finite group $G$ and let $G^0$ be the set of elements of $G$ whose $p$-part lies in $N$. We prove the existence of a canonical basis ${\rm IBr}(G, N)$ of the space of complex class functions of $G$ defined on $G^0$, such that the restriction $\chi^0$ of any irreducible complex character $\chi$ of $G$ is a linear combination $\sum_{\phi\in{\rm IBr}(G, N)} d_{\chi \phi} \phi$ of the elements of this basis, where the $d_{\chi \phi}$ are non-negative integers. Furthermore, if we write $\Phi_\phi=\sum_{\chi} d_{\chi \phi}\chi$, then the $\Phi_\phi$ form the K\"ulshammer--Robinson ${\Bbb Z}$-basis of the ${\Bbb Z}$-module generated by the characters afforded by the $N$-projective $RG$-modules, where $R$ is a certain complete discrete valuation ring. By using these `decomposition numbers', it is possible to define a linking in the set of the irreducible complex characters of $G$. 1991 Mathematics Subject Classification: 20C15, 20C20.

Lattices in the 3-dimensional Lie group Sol are shown to be not asynchronously automatic. The proof consists of a combination of geometry and logical complexity arguments. 2000 Mathematics Subject Classification: 20F10, 20F16.

We construct a sequence of concordance invariants for classical links, which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group. The terminology stems from the fact that we replace the Magnus expansion in the definition of Milnor's $\bar{\mu}$-invariants by the similar Campbell–Hausdorff expansion. The main point is that we introduce a new universal indeterminacy, which depends only on the number of components of the link. The Campbell–Hausdorff invariants are new, effectively computable and can efficiently distinguish (unordered and unoriented) isotopy types of links, as we indicate on several families of closed braid examples. They also satisfy certain natural dependence relations, which generalize well-known symmetries of the $\bar{\mu}$-invariants.

1991 Mathematics Subject Classification: 81S25, 46L10, 46L50, 47A60.

Suppose that $X$ is a Polish space which is not $\sigma$-compact. We prove that for every Borel colouring of $X^2$ by countably many colours, there exists a monochromatic rectangle with both sides closed and not $\sigma$-compact. Moreover, every Borel colouring of $[X]^2$ by finitely many colours has a homogeneous set which is closed and not $\sigma$-compact. We also show that every Borel measurable function $f:X^2 \rightarrow X$ has a free set which is closed and not $\sigma$-compact. As corollaries of the proofs we obtain two results: firstly, the product forcing of two copies of superperfect tree forcing does not add a Cohen real, and, secondly, it is consistent with ZFC to have a closed subset of the Baire space which is not $\sigma$-compact and has the property that, for any three of its elements, none of them is constructible from the other two. A similar proof shows that it is consistent to have a Laver tree such that none of its branches is constructible from any other branch. The last four results answer questions of Goldstern and Brendle. 2000 Mathematics Subject Classification: 03E15, 26B99, 54H05.

A generalized quadrangle $\cal S$ is laxly embedded in a (finite) projective space {\bf PG}$(d,q)$ if $\cal S$ is a subgeometry of the geometry of points and lines of {\bf PG}$(d,q)$, with the only condition that the points of $\cal S$ generate the whole space {\bf PG}$(d,q)$ (which one can always assume without loss of generality). In this paper, we classify thick laxly embedded quadrangles satisfying some additional hypotheses. The hypotheses are (a combination of) a restriction on the dimension $d$, a restriction on the parameters of $\cal S$, and an assumption on the isomorphism class of $\cal S$. In particular, the classification is complete in the following cases: \begin{enumerate} \item[(1)] for $d\geq 5$; \item[(2)] for $d=4$ and $\cal S$ having `known' order $(s,t)$ with $t\not= s^2$; \item[(3)] for $d\geq 3$ and $\cal S$ isomorphic to a finite Moufang quadrangle distinct from $W(s)$ with $s$ odd. \end{enumerate}

As a by-product, we obtain a new characterization theorem of the classical quadrangle $H(4,s^2)$, and we also show that every generalized quadrangle of order $(s,s+2)$, with $s>2$, has at least one non-regular line.

2000 Mathematics Subject Classification: 51E12.

Let $V$ be a vector space and let $\{ e_1,\hdots,e_r \}$ be a basis of $V$. An algebra structure on $V$ is given by $r^3$ structure constants $c_{ij}^h$ where $e_i\cdot e_j = \sum_h c_{ij}^h e_h$. We require this algebra structure to be associative with unit element $e_1$. This limits the sets of structure constants $(c_{ij}^h)$ to a subvariety of $k^{r^3}$, which we denote by $\mbox{Alg}_r$. Base changes in $V$ (leaving $e_1$ fixed) give rise to the natural transport of structure action on $\mbox{Alg}_r$; isomorphism classes of $r$-dimensional algebras are in one-to-one correspondence with the orbits under this action.

In this paper we classify the smooth closed subvarieties of $\mbox{Alg}_r$ which are invariant under the transport of structure action and study the singularities which may occur. In particular, we show that if $r=n^2$ then the closure of the locus corresponding to the matrix algebra $M_n(k)$ is not smooth for $n \geq 3$. This gives a negative answer to a question of Seshadri on the desingularization of moduli spaces of vector bundles over curves.

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 an earlier paper the second author used the formal, algebraic properties of $2$-dimensional Shintani generating functions to construct a $1$-cocycle on ${\rm PGL}_2(\mathbb{Q})$. We aim to generalise these results by using such functions in dimension $n$ to obtain an $(n-1)$-cocycle on ${\rm PGL}_n(\mathbb{Q})$, presumably related to the Bernoulli and Eisenstein cocycles of R.~Sczech. By improving our methods we achieve this goal for $n=3$. For $n>3$ we encounter obstacles related to degenerate configurations of hyperplanes in $n$-space. Nevertheless, we obtain partial results closely connected to reciprocity laws for certain $n$-dimensional Dedekind sums. 1991 Mathematics Subject Classification: 11F20, 11F75.

Let R be a Seifert surface obtained by applying Seifert's algorithm to a connected diagram D for a link L. In this paper, letting D be almost alternating, we give a practical algorithm to determine whether L is a fibered link and R is a fiber surface. We further show that L is a fibered link and R is a fiber surface for L if and only if R is a Hopf plumbing, that is, a successive plumbing of a finite number of Hopf bands. It has been known for some time that this is true if D is alternating, and we show that it is not always true if D is 2-almost alternating. In the appendix, we partially answer C. Adams's open question concerning almost alternating diagrams.

2000 Mathematical Subject Classification: 57M25.

We determine when a metric measure space admits a uniformizing density which has Ahlfors regular volume growth. A characterization of uniformizing conformal densities in terms of doubling and a lifting procedure are the key ingredients in our presentation. We also furnish an application characterizing metric doubling measures.

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 $a_n$ be an increasing sequence of positive reals with $a_n \rightarrow \infty$ as $n \rightarrow \infty$. Necessary and sufficient conditions are obtained for each of the sequences $[\alpha a_n], [\alpha^{a_n}], [{a_n}^{\alpha}]$ to take on infinitely many prime values for almost all $\alpha > \beta$. For example, the sequence $[\alpha a_n]$ is infinitely often prime for almost all $\alpha > 0$ if and only if there is a subsequence of the $a_n$, say $b_n$, with $b_{n+1} > b_n + 1$ and with the series $\sum 1/{b_n}$ divergent. Asymptotic formulae are obtained when the sequences considered are lacunary. An earlier result of the author reduces the problem to estimating the measure of overlaps of certain sets, and sieve methods are used to obtain the correct order upper bounds.

1991 Mathematics Subject Classification: primary 11N05; secondary 11K99, 11N36.

The moduli space of principally polarised abelian 4-folds can be compactified in several different ways by toroidal methods. Here we consider in detail the Igusa compactification and the (second) Voronoi compactification. We describe in both cases the cone of nef Cartier divisors. The proof depends on a detailed description of the Voronoi compactification, which makes it possible to proceed by induction, using the known description of the nef cone for compactifications of ${\mathcal A}_3$. The Igusa compactification has a non-${\mathbb Q}$-factorial singularity, which is resolved by a single blow-up: this resolution is the Voronoi compactification. The exceptional divisor $E$ is a toric Fano variety (of dimension 9): the other boundary divisor, $D$, corresponds to degenerations with corank~1. After imposing a level structure in order to avoid certain technical complications, we show that the closure of $D$ in the Voronoi compactification maps to the Voronoi compactification of ${\mathcal A}_3$. The toric description of the exceptional divisor allows us to describe the map in sufficient detail to estimate the intersection numbers needed. This inductive process is only valid for the Voronoi compactification: the result for the Igusa compactification is deduced from the Voronoi compactification.

We construct the moduli spaces $M(n)$ of semistable parabolic sheaves of rank $n$ and fixed Euler characteristic on a disjoint union $X$ of integral projective curves with parabolic structures over Cartier divisors on $X$. In the case where $X$ is non-singular, $M$ is a normal projective variety. Suppose that $X$ is the desingularisation of a reducible reduced curve $Y$ with at most ordinary double points as singularities. We show that, for a suitable choice of parabolic structure, $M(n)$ is the normalisation of the moduli space of torsion-free sheaves of rank $n$ and fixed Euler characteristic on $Y$, and it is a desingularisation if semistability coincides with stability. We find explicit descriptions of $M(n)$ for small $n$ in some cases.

The authors consider the system of forced differential equations with variable delays $$x'(t) + \sum^N_{j=1}B_j(t)x(t-\tau_j(t)) = F(t)\eqno(*)$$ where $B_j(t)$ is a continuous $n\times n$ matrix on ${{\Bbb R}^+}$, $F\in C({{\Bbb R}^+, {\Bbb R}^n})$ and $\tau \in C({{\Bbb R}^+, {\Bbb R}^+})$. Using Razumikhin-type techniques and Liapunov's direct method, they establish conditions to ensure the ultimate boundedness and the global attractivity of solutions of $(*)$, and when $F(t) \equiv 0$, the asymptotic stability of the zero solution. Under those same conditions, they also show that $\int^{+\infty}_0\sum_{j=1}^{N}|B_j(t)|\,dt = +\infty$ is a necessary and sufficient condition for all of the above properties to hold. 1991 Mathematics Subject Classification: 34K15, 34C10.