Mary Cartwright with J. E. Littlewood FRS first observed the phenomena which developed into Chaos Theory. Thus she had a significant effect on the modern world. She was the only woman so far to be a president of the London Mathematical Society, one of the first to be a Fellow of the Royal Society, and the first woman to serve on its Council. She was born on 17 December 1900 and died on 3 April 1998.

This paper gives new sufficient conditions for the density of the set of norm attaining multilinear forms in the space of all continuous multilinear forms on a Banach space. The symmetric case is also discussed.

This paper provides a new, simpler proof of the fact that the smallest volume hyperbolic 3-manifold has betti number at least 3. In the process, the best known lower bound on the volume of such a manifold is improved.

The paper shows that the homeomorphism groups of, respectively, Cantor's discontinuum, the rationals and the irrationals have uncountable cofinality. It is well known that the homeomorphism group of Cantor's discontinuum is isomorphic to the automorphism group Aut ${\bb B}$ of the countable, atomless boolean algebra ${\bb B}$ . So also Aut ${\bb B}$ has uncountable cofinality, which answers a question posed earlier by the first author and H. D. Macpherson. The cofinality of a group $G$ is the cardinality of the length of a shortest chain of proper subgroups terminating at $G$ .

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 author considers globally defined $h$ Fourier integral operators $(h\, {\rm FIO})$ with complex-valued phase functions. Symbolic calculus of $h\, {\rm FIO}$ is considered and, using a new complex Gauss transform, the composition of $h$ pseudodifferential operators $(h\, {\rm PDO})$ and $h\, {\rm FIO}$ is considered. For a self-adjoint $h\, {\rm PDO}\, A(h)$ and $h\, {\rm PDO}\, P(h)$ and $Q(h)$ with compactly supported symbols, the results are applied to approximate the kernel of the operator \[ U_{P,Q}(t;h):=P(h)e^{-ih^{-1}tA(h)}Q(h)^{\ast},\quad t\in {\bb R},\;h>0, \] by a single, globally defined $h$ -oscillatory integral.

Let G denote the p-adic group GL(n), and let S(G) denote the Schwartz algebra of G. We construct a Chern character from the K-theory of the reduced C*-algebra of G to the periodic cyclic homology of S(G) which becomes an isomorphism after tensoring over ℤ with [Copf ].

A simple proof that no subset of the plane that meets every line in precisely two points is an Fσ-set in the plane is presented. It was claimed that this result can be generalized for sets that meet every line in either one point or two points. No proof of this assertion is known, however. The main results in this paper form a partial answer to the question of whether the claim is valid. In fact, it is shown that a set that meets every line in the plane in at least one but at most two points must be zero-dimensional, and that if it is σ-compact then it must be a nowhere dense Gδ-set in the plane. Generalizations for similar sets in higher-dimensional Euclidean spaces are also presented.

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 $\varphi$ .

This paper provides a proof that an n-dimensional complete open Riemannian manifold M with sectional curvature KM [ges ] −1 is diffeomorphic to a Euclidean n-space Rn if the volume growth of geodesic balls in M is close to that of the balls in an n-dimensional hyperbolic space Hn(−1) of sectional curvature −1.

This paper deals with an operator theory approach to the corona conjecture for H∞([ ]n). Treil gave a counter-example to this conjecture in the case where n = 1 for operator-valued functions; thus one might hope to use this to disprove the corona conjecture for H∞([ ]n) (for n [ges ] 2). This paper shows that this natural approach towards a negative answer fails. On the other hand, the second result here shows that ‘commutant lifting’ cannot be true for more than two contractions for any constant. This obstructs a natural attempted proof of the corona conjecture for H∞([ ]n) (for n [ges ] 2) by our previous result.

The topological disc (De Paepe's) \[ P=\{(z^2,\bar{z}^2+\bar{z}^3):|z|\le 1\}\subset {\bb C}^2 \]

is shown here to have non-trivial polynomially convex hull. In fact, the authors show that this holds for all discs of the form $X=\{(z^2,f(\bar{z})):|z|\le r\}$ , where $f$ is holomorphic on $|z|\le r$ , and $f(z)=z^2+a_3z^3+\ldots$ , with all coefficients $a_n$ real, and at least one $a_{2n+1}\ne 0$ .

Eric Primrose was appointed to a Lectureship in Pure Mathematics at the University College, Leicester, in 1947 and promoted to a Senior Lectureship in 1954. He came to Leicester direct from Oxford where he had spent the 1946–47 year completing his degree, which had been interrupted by war service. Eric was awarded his PhD in 1957 by the University of London.

Eric's secondary education was at Chigwell School and in 1939 he won an Open Scholarship in Mathematics and went up to St John's College, Oxford. He took ‘shortened finals’ in 1941 and then entered the RAF as a Technical Officer (Radar), reaching the rank of Flight Lieutenant, having served in various locations throughout Europe.