It has recently been shown by Mo and Negreiros that a necessary condition for an invariant almost complex structure on the complex full flag manifold ${\bb F}(n)$ to admit a (1, 2)-symplectic invariant metric is that its associated tournament be cone-free. In this paper, a canonical stair-shaped form is given for such tournaments, and this is applied to show that the condition is also sufficient; in the process, all the associated (1, 2)-symplectic metrics are described.

The authors show that any $k$ -Osserman Lorentzian algebraic curvature tensor has constant sectional curvature, and give an elementary proof that any local 2-point homogeneous Lorentzian manifold has constant sectional curvature. They also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.

A sufficient condition is provided for a compact set to be removable for analytic functions in the Zygmund spaces in terms of a porosity. As a consequence, it is shown that the classes of removable sets for the Zygmund space and the little Zygmund space are different.

A planar set $G \subset {\bb R}^2$ is constructed that is bilipschitz equivalent to ( $G, d^z$ ), where ( $G, d$ ) is not bilipschitz embeddable to any uniformly convex Banach space. Here, $z \in (0, 1)$ and $d^z$ denotes the $z$ th power of the metric $d$ . This proves the existence of a strong $A_{\infty}$ weight in ${\bb R}^2$ , such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of ${\bb R}^2$ .

An example is provided of a space that does not have the homotopy type of an H-space, and yet has the same $n$ -type as a double loop space for each natural number $n$ . It is also locally homotopy equivalent to a double loop space at each prime $p$ .

This paper solves a problem posed by Colin Adams, showing that any link $L$ in ${\bb R}^3$ is isotopic to a smooth link $\hat{L}$ that has no parallel or antiparallel tangents.

Let $D$ be a bounded domain in ${\bb R}^n$ . For a function $f$ on the boundary $\partial D$ , the Dirichlet solution of $f$ over $D$ is denoted by $H_D f$ , provided that such a solution exists. Conditions on $D$ for $H_D$ to transform a Hölder continuous function on $\partial D$ to a Hölder continuous function on $D$ with the same Hölder exponent are studied. In particular, it is demonstrated here that there is no bounded domain that preserves the Hölder continuity with exponent 1. It is also also proved that a bounded regular domain $D$ preserves the Hölder continuity with some exponent $\alpha$ , $0 < \alpha < 1$ , if and only if $\partial D$ satisfies the capacity density condition, which is equivalent to the uniform perfectness of $\partial D$ if $n = 2$ .

Let $K$ and $L$ be two convex bodies in ${\bb R}^n$ . The volume ratio ${\rm vr}(K, L)$ of $K$ and $L$ is defined by ${\rm vr}(K, L) = \inf(\vert K\vert/\vert T(L)\vert)^{1/n}$ , where the infimum is over all affine transformations $T$ of ${\bb R}^n$ for which $T(L) \subseteq K$ . It is shown in this paper that ${\rm vr}(K, L) \leqslant c \sqrt{n} \log n$ , where $c > 0$ is an absolute constant. This is optimal up to the logarithmic term.

Some generalizations of the classical Hurewicz formula are obtained for the extension dimension and $C$ -spaces. A characterization is also given of the class of metrizable spaces that are absolute neighborhood extensors for all metrizable $C$ -spaces.

In this paper, a growth result is obtained for normalized $k$ -fold symmetric convex biholomorphic mappings on the unit ball of a complex Banach space.

A Hurwitz group is any non-trivial finite group that can be (2,3,7)-generated; that is, generated by elements $x$ and $y$ satisfying the relations $x^2 = y^3 = (xy)^7 = 1$ . In this short paper a complete answer is given to a 1965 question by John Leech, showing that the centre of a Hurwitz group can be any given finite abelian group. The proof is based on a recent theorem of Lucchini, Tamburini and Wilson, which states that the special linear group ${\rm SL}_n(q)$ is a Hurwitz group for every integer $n \geqslant 287$ and every prime-power $q$ .

The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups $U(2, 1)$ and $SU(2, 1)$ are calculated. The results are obtained using the identification of these moduli spaces with moduli spaces of Higgs bundles, and Morse theory, following Hitchin's programme. This requires a careful analysis of critical submanifolds, which turn out to have a description using either symmetric products of the surface or moduli spaces of Bradlow pairs.

In this paper, the authors give sharp estimates on the norms in the trace class of wavelet multipliers $P_{\sigma\varphi}:L^2({\bb R}^n) \longrightarrow L^2({\bb R}^n)$ in terms of the symbols $\sigma$ and the admissible wavelets $\varphi$ .

Mary Warner, as she was mainly known in the mathematical world, died in April 1998. At a time when few women mathematicians reached the top in their profession, she succeeded in doing so through her ability and determination. Her research contributions were commemorated at a recent international conference on fuzzy topology, the field in which she was one of the pioneers and recognized as one of the leading figures for the past thirty years. She was also an outstanding teacher. But to understand her achievements properly it is necessary to know something of her life.