Richard Rado was born in Berlin; he was the second son of Leopold Rado, from Budapest. At one stage of his education he had to decide whether to become a concert pianist or a mathematician. He chose the latter in the belief that he could continue with music as a hobby, but that he could never treat mathematics in that way. He studied at the University of Berlin, but also spent some time in Göttingen. He took a DPh at Berlin with his thesis ‘Studien zur Kombinatorik’ [3] under Issai Schur in 1933. During this period he was also influenced by Erhard Schmidt.

On 16 March 1933, he married Luise Zadek, the elder daughter of Hermann Zadek, whom he had earlier come to know when he needed a partner to play piano duets. It was indeed a remarkable partnership.

As Hitler came to power in 1933, the Rados, being Jewish, made their way to England, Richard having obtained a scholarship of £300 p.a. from Sir Robert Mond through the recommendation of Professor Lindemann (later Lord Cherwell), who had interviewed him in Berlin, to enable him to study at Cambridge.

Let M and N be closed non-positively curved manifolds, and let f[ratio ]M→N be a smooth map. Results of Eells and Sampson [1] show that f is homotopic to a harmonic map ϕ, and Hartman [6] showed that this ϕ is unique when N is negatively curved and f∗(π1M) is not cyclic. Lawson and Yau conjectured that if f[ratio ]M→N is a homotopy equivalence, where M and N are negatively curved, then the unique harmonic map ϕ homotopic to f would be a diffeomorphism. Counterexamples to this conjecture appeared in [2], and later in [7] and [5].

There remains the question of whether a ‘topological’ Lawson–Yau conjecture holds.

Let Ratk(CPn) denote the space of based holomorphic maps of degree k from the Riemannian sphere S2 to the complex projective space CPn. The basepoint condition we assume is that f(∞)=[1, …, 1]. Such holomorphic maps are given by rational functions:

Ratk(CPn) ={(p0(z), …, pn(z))[ratio ]each pi(z) is a monic, degree-k polynomial and such that there are no roots common to all pi(z)}. (1.1)

The study of the topology of Ratk(CPn) originated in [10]. Later, the stable homotopy type of Ratk(CPn) was described in [3] in terms of configuration spaces and Artin's braid groups. Let W(S2n) denote the homotopy theoretic fibre of the Freudenthal suspension E[ratio ]S2n→ ΩS2n+1. Then we have the following sequence of fibrations: Ω2S2n+1→ W(S2n)→S2n→ ΩS2n+1. A theorem in [10] tells us that the inclusion Ratk(CPn)→ Ω2kCPn≃ Ω2S2n+1 is a homotopy equivalence up to dimension k(2n−1). Thus if we form the direct limit Rat∞(CPn)= limk→∞ Ratk(CPn), we have, in particular, that Rat∞(CPn) is homotopy equivalent to Ω2S2n+1.

If we take the results of [3] and [10] into account, we naturally encounter the following problem: how to construct spaces Xk(CPn), which are natural generalizations of Ratk(CPn), so that X∞(CPn) approximates W(S2n). Moreover, we study the stable homotopy type of Xk(CPn).

The purpose of this paper is to give an answer to this problem. The results are stated after the following definition.

The purpose of this note is to give a short proof of the Gross logarithmic Sobolev inequality using the asymptotics of the sharp L2 Sobolev constant and the product structure of Euclidean space. Let F∈Lr(IRn) for some positive r with ∥F∥r=1.

John Charles Burkill, born on 1 February 1900, was the only child of Hugh Roberson Burkill (1867–1951) and Bertha (née Bourne, 1866–1937). His father came from a family which had farmed in Lincolnshire for generations, whereas his mother came from a background of prosperous farming and building. On neither side was there a strong academic tradition, but Charles was soon to show evidence of intellectual distinction by winning a scholarship to St Paul's school at the age of 14. There he profited fully from the excellent teaching that the school offered and which was reflected not only by his mathematical prowess, which led to a scholarship to Trinity College, Cambridge, in 1918, but also in his ability in classical studies in which he maintained a lifelong interest. He was also a formidable chess player, and had a mischievous sense of humour which he retained, albeit in a more restrained mode, in later life. A striking example of his grasp of the essence of a practical joke is recorded in the story of how, as a boy on a visit to a house-proud aunt, he saw the comic potential of a trail of corn from the chicken run through the front door and upstairs to the bedrooms.

We consider a simple shift procedure which transforms any mapping f[ratio ]{0, …, n}→{0, …, n} into a permutation of 0, …, n. We study this shift under the assumption that the values f(0), …, f(n) are uniformly and independently chosen random numbers in {0, …, n}. The exact and asymptotic distributions of the required shift sizes are derived, and a connection to queueing theory is exhibited.

The purpose of this note is to provide a new proof for the explicit formulas of the heat kernel on hyperbolic space. By definition, the hyperbolic space IHn is a (unique) simply connected complete n-dimensional Riemannian manifold with a constant negative sectional curvature −1.

John Arthur Todd was one of the last survivors of the school of classical algebraic geometry that flourished in Cambridge under H. F. Baker, FRS. But, unlike most of his contemporaries, with the notable exception of W. V. D. Hodge, FRS, Todd made seminal contributions to the more modern theory, and his name is now enshrined in the literature and widely known.

The aim of this note is to give a sharp upper bound on the ratio

formula here

where ϕ is a nonconstant eigenfunction for the Laplace–Beltrami operator on a connected compact Riemannian manifold without boundary. This ratio is always positive, since maxϕ>0 and minϕ<0 for every nonconstant eigenfunction. We assume that maxϕ[ges ]−minϕ, in order to simplify the notation. For the case of a two-dimensional manifold with nonnegative Ricci curvature, our theorem implies that the above ratio is less than the ratio of the maximum divided by the absolute value of the minimum of the Bessel function of order zero.

The proof is based on a gradient estimate from a previous paper of the author (see [5]), which in turn was proved using the maximum principle technique. In contrast to the standard applications of gradient estimates, which are based on integration along geodesics, we arrive at a contradiction by integrating the gradient estimate over small spheres centred at a point where the absolute value of the eigenfunction attains its maximum.

The main motivation for our work is that the ratio of the maximum and the minimum of an eigenfunction plays a role in estimates of the corresponding eigenvalues (see [5] and [7]). More precisely, our theorem implies that there are minimizing sequences of compact manifolds such that the first eigenvalues of the manifolds approach the corresponding lower bound for the first eigenvalue obtained in [5, Theorem 2] for every possible ratio of the maximum and the minimum of the corresponding eigenfunction.