Despite a true antipathy to the subject, Hardy contributed deeply to modern probability. His work with Ramanujan begat probabilistic number theory. His work on Tauberian theorems and divergent series has probabilistic proofs and interpretations. Finally, Hardy spaces are a central ingredient in stochastic calculus. This paper reviews his prejudices and accomplishments, through these examples.

It is shown that a necessary condition for the existence of a bicolored Steiner triple system of order $n$ is that $n$ can be written in the form $A^2+3B^2$ for integers $A$ and $B$ . In the case when $n=q$ is either a prime congruent to 1 mod 3, or the square of a prime congruent to 2 mod 3, it is shown that the numbers of colored vertices in the triple system would be unique, and are given by the number of points on specific twists of the CM elliptic curve $y^2=x^3-1$ over the finite field ${\bb F}_q$ .

The presentation of the quantum cohomology of the moduli space of stable vector bundles of rank two and odd degree with fixed determinant over a Riemann surface of genus $g>2$ is obtained. The argument avoids the use of gauge theory, providing an alternative proof to that given by the author in Duke Math. J. 98 (1999) 525–540.

Let ${\cal L}$ be a uniformly elliptic operator in divergence form on ${\bb R}^d$ , and let $p(t,x,y)$ be the fundamental solution to the heat equation for ${\cal L}$ . A new proof is given of Aronson's upper bound: \[ p(t,x,y)\le c_1t^{-d/2}\exp(-c_2|x-y|^2/t). \]

This paper provides examples of closed manifolds having a Morse number different from the stable Morse number.

The authors prove in this paper that, given any knot $k$ of genus $g$ , $k$ fails to be strongly $n$ -trivial for all $n$ with $n\ge 6g-3$ .

Let $W$ be a finite-dimensional ${\bb Z}/p$ -module over a field, ${\bf k}$ , of characteristic $p$ . The maximum degree of an indecomposable element of the algebra of invariants, ${\bf k}[W]^{{\bb Z}/p}$ , is called the Noether number of the representation, and is denoted by $\beta(W)$ . A lower bound for $\beta(W)$ is derived, and it is shown that if $U$ is a ${\bb Z}/p$ submodule of $W$ , then $\beta(U)\le \beta(W)$ . A set of generators, in fact a SAGBI basis, is constructed for ${\bf k}[V_2\oplus V_3]^{{\bb Z}/p}$ , where $V_n$ is the indecomposable ${\bb Z}/p$ -module of dimension $n$ .

A proof is given that the full symmetric group over any infinite set is the product of finitely many Abelian subgroups. In fact, 289 subgroups suffice. Sharp bounds are also obtained on the minimal number $k$ , such that the finite symmetric group $S_n$ is the product of $k$ Abelian subgroups. Using this, $S_n$ is proved to be the product of $72n^{1/2}(\log n)^{3/2}$ cyclic subgroups.

If the subset $E$ of ${\bb T}$ supports a synthesizable $p$ -pseudofunction $T$ , then the restriction $A_p(E)$ to $E$ of the Herz algebra on ${\bb T}$ is not Arens regular. The proof is direct, by brute force, and does not show the existence of Day points for $A_p(E)$ .

The main purpose of this paper is to use the properties of Gaussian sums and the analytic method to study the problem of calculating one kind of character sums modulo $q$ , and to give an interesting identity.

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 $G$ .

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.

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.