Upper bounds are obtained for the heat content and the torsional rigidity of a complete Riemannian manifold $M$, assuming a Hardy inequality for the Dirichlet Laplacian on $M$.

It is well known that no non-trivial Killing vector field exists on a compact Riemannian manifold of negative Ricci curvature; analogously, no non-trivial harmonic one-form exists on a compact manifold of positive Ricci curvature. One can consider the following, more general, problem. By reducing the assumption on the Ricci curvature to one on the scalar curvature, such vanishing theorems cannot hold in general. This raises the question: “What information can we obtain from the existence of non-trivial Killing vector fields (or, respectively, harmonic one-forms)?” This paper gives answers to this problem; the results obtained are optimal.

The author recently proved that 1-cotilting modules are pure-injective; however, the problem of determining whether n-cotilting modules are pure-injective remained open. In this paper, necessary and sufficient conditions are given for the pure-injectivity of n-cotilting modules.

The class of nilpotent groups of class $c$ and prime exponent $p\,{>}\,c$ with additional predicates $P_c\,{\subseteq}\,P_{c-1}\,{\subseteq}\,\ldots\,{\subseteq}\,P_1$ for suitable subgroups has the amalgamation property. Hence the Fraïssé limit $D$ of the finite groups of this class exists. $\langle 1\rangle\,{\subseteq}\,P_c(D)\,{\subseteq}\,\ldots\,{\subseteq}\,P_2(D)\,{\subseteq}\,P_1(D)\,{=}\,D$ is the lower and the upper central series of $D$. In this extended language, $D$ is ultrahomogeneous. The elementary theory of $D$ allows the elimination of quantifiers and it is $\aleph_0$-categorical. For $c\,{=}\,2$ this was proved by Baudisch in Bull. London Math. Soc. 33 (2001) 513–519.

This paper presents an algebraic approach to Mislin's theorem that control of $p$-fusion is equivalent to inducing a mod-$p$ cohomology isomorphism. There are consequences for the cohomology of $p$-permutation modules.

An ovoid of ${\rm PG}(3,q)$ can be defined as a set of $q^2+1$ points with the property that every three points span a plane, and at every point there is a unique tangent plane. In 2000, M. R. Brown proved that if an ovoid of ${\rm PG}(3,q)$, $q$ even, contains a conic, then the ovoid is an elliptic quadric. Generalising the definition of an ovoid to a set of $(n-1)$-spaces of ${\rm PG}(4n-1,q)$, J. A. Thas introduced the notion of pseudo-ovoids or eggs: a set of $q^{2n}+1$$(n-1)$-spaces in ${\rm PG}(4n-1,q)$, with the property that any three egg elements span a $(3n-1)$-space and at every egg element there is a unique tangent $(3n-1)$-space. In this paper, a proof is given that an egg in ${\rm PG}(4n-1,q)$, $q$ even, contains a pseudo-conic (that is, a pseudo-oval arising from a conic of ${\rm PG}(2,q^n)$) if and only if the egg is classical (that is, arising from an elliptic quadric in ${\rm PG}(3,q^n)$).

Based on the classification of contact structures on the 3-torus and classical contact geometric methods, it is shown that there exist essential loops in the space of contact structures on torus bundles over the circle.

Holomorphic almost modular forms are holomorphic functions of the complex upper half plane that can be approximated arbitrarily well (in a suitable sense) by modular forms of congruence subgroups of large index in ${\rm SL}(2,{\mathbb Z})$. It is proved that such functions have a rotation-invariant limit distribution when the argument approaches the real axis. An example of a holomorphic almost modular form is the logarithm of $\prod_{n=1}^\infty (1-\exp(2\pi\i n^2 z))$. The paper is motivated by the author's previous studies [Int. Math. Res. Not. 39 (2003) 2131–2151] on the connection between almost modular functions and the distribution of the sequence $n^2x$ modulo one.

Let $G\subseteq \mathbb{C}^{2}$ be the open symmetrized bidisc, namely $G= \{(\lambda_{1}+\lambda_{2},\lambda_{1}\lambda_{2}):|\lambda_{1}|<1,|\lambda_{2}|<1\}$. In this paper, a proof is given that $G$ is not biholomorphic to any convex domain in $\mathbb{C}^{2}$. By combining this result with earlier work of Agler and Young, the author shows that $G$ is a bounded domain on which the Carathéodory distance and the Kobayashi distance coincide, but which is not biholomorphic to a convex set.

There is a Weil cohomology theory with coefficients in a pseudofinite field of characteristic zero.

Transformation formulas are obtained for generalized Dedekind eta functions; these are simpler to apply than Schoeneberg's formulas. As an application, a list is given of the generators of all the function fields associated with torsion-free genus zero congruence subgroups of $\hbox{PSL}_2(\mathbb R)$.

In this paper, a number of results are deduced on the arithmetic structure of products of integers in short intervals. By way of an example, work of Saradha and Hanrot, and of Saradha and Shorey, is completed by the provision of an answer to the question of when the product of $k$ out of $k+1$ consecutive positive integers can be an ‘almost’ perfect power. The main new ingredient in these proofs is what might be termed a practical method for resolving high-degree binomial Thue equations of the form $ax^n - by^n = \pm 1$, based upon results from the theory of Galois representations and modular forms.

In his curriculum vitae, John Hawkes lists his research interests as geometric measure theory, probability (Lévy processes), and potential theory (probabilistic). In fact, he made significant contributions to all three areas, and there are strong relationships between them. He used both geometric measure theory and potential theory as tools for his study of the trajectories of particular Lévy processes, but in many cases he needed to develop the tool before it was ready to be used. We will summarise his research later, but first we discuss what is known of his life history.