This paper aims to provide a survey on the subject of representations of fundamental groups of 2-manifolds, or in other guises flat connections on orientable 2-manifolds or moduli spaces parametrizing holomorphic vector bundles on Riemann surfaces. It emphasizes the relationships between the different descriptions of these spaces. The final two sections of the paper outline results of the author and Kirwan on the cohomology rings of certain of the spaces described earlier (formulas for intersection numbers that were discovered by Witten (Commun. Math. Phys. 141 (1991) 153–209 and J. Geom. Phys. 9 (1992) 303–368) and given a mathematical proof by the author and Kirwan (Ann. of Math. 148 (1998) 109–196)).

Let $(a_n)_{n\ge 0}$ be a sequence of complex numbers, and, for $n\,{\ge}\,0$, let \[ b_n=\sum_{k=0}^n{n\choose k}a_k\quad\hbox{and}\quad c_n=\sum_{k=0}^n{n\choose k}(-1)^{n-k}a_k. \] A number of results are proved relating the growth of the sequences $(b_n)$ and $(c_n)$ to that of $(a_n)$. For example, given $p\,{\ge}\,0$, if $b_n\,{=}\,O(n^p)$ and $c_n\,{=}\,O(e^{\epsilon\sqrt{n}})$ for all $\epsilon\gt0$, then $a_n\,{=}\,0$ for all $n\gt p$. Also, given $0\lt\rho\lt1$, then $b_n,c_n\,{=}\,O(e^{\epsilon n^\rho})$ for all $\epsilon\gt0$ if and only if $n^{1/\rho-1}|a_n|^{1/n}\to 0$. It is further shown that, given $\beta\gt1$, if $b_n,c_n\,{=}\,O(\beta^n)$, then $a_n\,{=}\,O(\alpha^n)$, where $\alpha\,{=}\,\sqrt{\beta^2-1}$, thereby proving a conjecture of Chalendar, Kellay and Ransford. The principal ingredients of the proofs are a Phragmén–Lindelöf theorem for entire functions of exponential type zero, and an estimate for the expected value of $e^{\phi(X)}$, where $X$ is a Poisson random variable.

The existence of Salem numbers of every trace is proved; the nontrivial part of this result concerns Salem numbers of negative trace. The proof has two main ingredients. The first is a novel construction, using pairs of polynomials whose zeros interlace on the unit circle, of polynomials of specified negative trace having one factor a Salem polynomial, with any other factors being cyclotomic. The second is an upper bound for the exponent of a maximal torsion coset of an algebraic torus in a variety defined over the rationals. This second result, which may be of independent interest, has enabled the construction to be refined so as to avoid cyclotomic factors, giving a Salem polynomial of any specified trace, with a trace-dependent bound for its degree. It is also shown how this new interlacing construction can be easily adapted to produce Pisot polynomials, giving a simpler, and more explicit, construction for Pisot numbers of arbitrary trace than was previously known.

This paper proves the rationality of Poincaré series whose coefficients are defined using the number of conjugacy classes in a system of finite groups defined as finite images of linear groups over $\Bbb{Z}_{p}$ modulo natural congruence subgroups.

It is proved that Epstein's zeta-function $\zeta_Q(s)$, related to a positive definite integral binary quadratic form, has a zero $ 1/2 +i\gamma $ with $T \leq \gamma \leq T+T^{5/11+\varepsilon }$ for sufficiently large positive numbers $T$. This improves a classical result of H. S. A. Potter and E. C. Titchmarsh (Proc. London Math. Soc. (2) 39 (1935) 372–384).

Shelah has recently proved that an uncountable free group cannot be the automorphism group of a countable structure. In fact, he proved a more general result: an uncountable free group cannot be a Polish group. A natural question is: can an uncountable $\aleph _{1}$-free group be a Polish group? A negative answer is given here; indeed, it is proved that an $\aleph _{1}$-free group cannot be a homomorphic image of a Polish group. In fact, a stronger result is proved, involving a non-commutative analogue of the notion of separable group.

Given a set $\pi$ of odd primes and a $\pi$-solvable finite group $G$, one can choose three (not necessarily distinct) Hall $\pi$-subgroups of $G$ whose intersection is $O_{\pi}(G)$. This statement relies on a result concerning long orbits in coprime actions of groups of odd order.

A proof is given that the quasivariety of groups generated by finite and torsion-free groups does not contain the class of periodic groups. This result is related to (and inspired by) the solvability of equations over groups. The proof uses the Feit–Thompson theorem on the solvability of finite groups of odd order as well as Kostrikin–Zelmanov results on the restricted Burnside problem, and applies technical details of a recent construction of weakly finitely presented periodic groups.

It is proved that two random elements of the Nottingham group generate a free group with probability 1. This answers a question of A. Shalev.

The above-titled paper of mine appeared in the Bulletin of the London Mathematical Society, 32 (2000) 297–304. Regrettably, there is a careless error in the proofs of Theorems 6 and 8. In line 6 of the proof of Theorem 6, it is claimed that a certain subset must be a subgroup. For this to hold, the subset must contain the zero element. This need not be the case; the true deduction is that the subset is a coset, say $M+h$, of a subgroup $M$. Now $M$ and $M+h$ contain the same number of elements, and so the deduction that $M$ has $p$ elements is still correct.

In the context of Lebesgue integration the Gauss–Green theorem is proved for bounded vector fields with substantial sets of singularities with respect to continuity and differentiability. The resulting integration by parts is applied to removable sets for the Cauchy–Riemann, Laplace, and minimal surface equations. A simple connection between the Gauss–Green theorem and distributional divergence is established.

Smale's mean value conjecture asserts that $\min_\theta |P(\theta)/\theta| \le K|P'(0)|$ for every polynomial $P$ of degree $d$ satisfying $P(0)\,{=}\,0$, where $K\,{=}\,(d-1)/d$ and the minimum is taken over all critical points $\theta$ of $P$. A stronger conjecture due to Tischler asserts that \[ \min_\theta\left|\frac12-\frac{P(\theta)}{\theta\cdot P'(0)}\right| \le K_1 \] with $K_1=\frac12-1/d$. Tischler's conjecture is known to be true: (i) for local perturbations of the extremum $P_0(z)=z^d-dz$, and (ii) for all polynomials of degree $d\le 4$. In this paper, Tischler's conjecture is verified for all local perturbations of the extremum $P_1(z)=(z-1)^d-(-1)^d$, but counterexamples to the conjecture are given in each degree $d\ge 5$. In addition, estimates for certain weighted $L^1$- and $L^2$-averages of the quantities $\frac12-{P(\theta)}/{\theta\cdot P'(0)}$ are established, which lead to the best currently known value for $K_1$ in the case $d=5$.

A norm on a Banach space $X$ is called maximal when no equivalent norm has a larger group of isometries. If, besides this, there is no equivalent norm with the same isometries (apart from its scalar multiples), the norm is said to be uniquely maximal, which is equivalent to the convex-transitivity of $X$: the convex hull of the orbits under the action of the isometry group on the unit sphere is dense in the unit ball of $X$. The main result of the paper is that the complex $C_0(\Omega)$ is convex-transitive in its natural supremum norm if $\Omega$ is a connected manifold (without boundary). As a complement, it is shown that if $\Omega$ is a connected manifold of dimension at least two, then the diameter norm is convex transitive on the corresponding space of real functions.

In this paper, the existence problem is studied for extremals of the Sobolev trace inequality $W^{1,p}(\Omega)\to L^{p_*}(\partial\Omega)$, where $\Omega$ is a bounded smooth domain in $\RR^N$, $p_*=p(N-1)/(N-p)$ is the critical Sobolev exponent, and $1<p<N$.

Hyperplane arrangements in a three-dimensional vector space are considered in this paper. A characterization of the freeness of such an arrangement is given in terms of the characteristic polynomial and a restricted multiarrangement. As an application, the freeness of cones over certain two-dimensional affine arrangements is proved.

Let $G$ be a compact Lie group, and let $\pi$ be any prime or set of primes. A ‘$\pi$-perfection map’ is constructed: that is, a continuous function from the space of conjugacy classes of all closed subgroups of $G$ to the space of conjugacy classes of $\pi$-perfect subgroups with finite index in their normalizer. This is used to show that the idempotent elements of the Burnside ring of $G$ localized at $\pi$ are in bijective correspondence with the open and closed subsets of the space of conjugacy classes of $\pi$-perfect subgroups of $G$ with finite index in their normalizer.

In this paper, some identities in laws involving ladder processes for random walks and Lévy processes are extended and unified.

Wilfred Halliday Cockcroft only became known as such after he was knighted in 1983, for he was always ‘Bill’ to everybody. A large Yorkshireman born and bred, son of a proudly skilled plumber, he moved in 1941 from Keighley Grammar School as an Exhibitioner to Balliol College, Oxford. A year later he was called into the Royal Air Force, to become a technical officer in the Far East. He returned to Oxford in 1946 with a group of talented ex-servicemen, and completed his undergraduate work before becoming a research student under Henry Whitehead. Having married Rhona, he needed a job, and so he joined the staff at Aberdeen University, where he completed his doctorate. His first promotion created a vacancy for me, and I personally found it a great general education to be with him, as well as a great luxury to have another topologist to talk to, at a time when topologists (and indeed, people with a ‘modern’ vocabulary) were rare. He was strongly extrovert, very sociable in Oxford, and got on well with the Aberdeen Professor, E. M. Wright, at a time when relations between junior staff and professors tended to be stiff; his relaxed manner with Henry Whitehead (and indeed everybody else) amazed those of us raised among the conventionally introverted. It was thus that Bill had begun to acquire some of his extensive savoir faire and confidence. Besides research, he was naturally drawn to reflect on the problem of teaching the huge ‘First Ordinary’ class in the (to us) strange traditions of a Scottish university. At that time, G. H. Hardy had been writing about ‘Youth’ in a way that was to damage women and the over-thirties, and this aroused fears in Bill that he might soon be too old for research in mathematics, so it was interesting to hear him make the surprising remark (for 1952), that he would want to encourage ‘old’ lecturers to write good school texts if the Hardyesque social climate was making them feel that their pukka research days were over. (In spite of some splendid examples, writing almost any mathematics book in those days was often regarded as creative death, though not by Hardy.) Bill seemed to be fascinated by textbooks (and listed such reading as one of his recreations); he read texts with ease, and thereby widened his mathematical knowledge, in contrast to Cambridge contemporaries of the time, who were brought up on a higher standard of lecturing that left many of them narrow.