In as elementary a way as possible, we place Wiles' proof of Fermat's last theorem into the context of a general description of reciprocity conjectured to obtain between algebraic varieties defined over Q and Hecke eigenvectors in the homology of the spaces of lattices in Rn.

Nous donnons une majoration de la somme d'exponentielles SM = [sum ]Mm=1e(f(m)) lorsque la dérivée cinquième de f est d'un ordre de grandeur constant petit, noté λ, en fonction de M et de λ, améliorant un résultat ancien de Van Der Corput. La démonstration utilise un théorème de moyenne des puissances sixièmes de sommes d'exponentielles qui fait l'objet d'un article indépendant [6].

We give a bound for the exponential sum SM = [sum ]Mm=1e(f(m)) where f is a real-valued function whose fifth derivative is of a constant small size, say λ, by means of M and λ, improving an old result of Van Der Corput. The proof relies on a mean value theorem for sixth powers of exponential sums which is treated independently in [6].

Let G be a residually-finite virtually soluble minimax group. We give upper and lower linear bounds for the degree of subgroup growth of G in terms of the Hirsch length of G.

In this paper, branching rules for the fundamental representations of the symplectic groups in positive characteristic are found. The submodule structure of the restrictions of the fundamental modules for the group Sp2n(K) to the naturally embedded subgroup Sp2n−2(K) is determined. As a corollary, inductive systems of fundamental representations for Sp∞(K) are classified. The submodule structure of the fundamental Weyl modules is refined.

In this paper we obtain the precise structure of right nonsingular semiperfect right CS-rings. It is shown that such rings are direct sums of certain specific block matrix rings.

We show that the norms of the sequence of spectral projections of certain non-self-adjoint ordinary differential operators cannot be polynomially bounded. The proof involves the investigation of ‘tame’ bi-orthogonal systems in Banach spaces.

Let f[ratio ]C→Ĉ be a function which is either transcendental meromorphic or rational with degree at least 2. We discuss the uniform perfectness of the attracting or parabolic cycle of stable domains of f(z), and include a proof that the Julia set of a meromorphic function of finite type is uniformly perfect.

Analogues of the Funk–Hecke formula for spherical harmonics are proved for Dunkl's h-harmonics associated to the reflection groups, and for orthogonal polynomials related to h-harmonics on the unit ball. In particular, an analogue and its application are discussed for the weight function (1−[mid ]x[mid ]2)μ−1/2 on the unit ball in ℝd.

We examine the range of universal Taylor series. We prove that every universal Taylor series on the unit disc assumes every complex number, with one possible exception, infinitely often. On the other hand, we prove that on any simply connected domain there exist universal functions that omit one value.

Using basic harmonic analysis on the n-torus, it is shown that the non-commutative n-tori are the only primitive C*-algebras (up to isomorphism) on which the n-torus [ ]n acts freely and ergodically. An explicit construction recovers the generating unitaries in a natural way from the action.

We prove weak laws of large numbers for freely independent, uniformly bounded and non-identically distributed random variables belonging to non-commutative Ciach and, in particular, Lorentz Lp-spaces.

We prove upper and lower heat kernel bounds for the Laplacian on weighted graphs which include the case that the weights have no strictly positive lower bound. Our estimates give rise to a very explicit probabilistic interpretation, and can be formulated in terms of a weighted metric. Interestingly, this metric is not equivalent to the intrinsic metric.

Philip Holgate was born in Chesterfield on 8 December 1934. His family moved from Derbyshire to Devon in 1945, and he was educated at Newton Abbot Grammar School from 1945 to 1952, and the University College of the South-West (now Exeter University, but which then awarded London degrees) from 1952 to 1955. He qualified as both a teacher (King's College, London, 1955–56) and a statistician (University College, London, 1956–57).

After teaching mathematics and physics at Burgess Hill School, in Hampstead and then in Borehamwood, Philip joined the Statistics Section at Rothamsted Experimental Station (1961–62). He then spent five years in the Biometrics Section of the Nature Conservancy. His first publications date from this period, and the interests he acquired then were to develop into what became his most enduring, and distinctive, scientific interests.

Philip joined the Department of Statistics, Birkbeck College, University of London, in 1967, and remained at Birkbeck for the rest of his career, until his death from a heart attack on 13 April 1993.