A description is given of methods that have been used to analyze the spectrum of non-self-adjoint differential operators, emphasizing the differences from the self-adjoint theory. It transpires that even in cases in which the eigenfunctions can be determined explicitly, they often do not form a basis; this is closely related to a high degree of instability of the eigenvalues under small perturbations of the operator.

Free entropy is the analogue of entropy in free probability theory. The paper is a survey of free entropy, its applications to von Neumann algebras and its connections to random matrix theory, as well as a discussion of open problems and of a basic variational problem, connected to random multimatrix models.

This paper is concerned with the chromatic polynomials of ‘bracelets’: specifically, graphs constructed by taking n copies of a complete graph and linking them together in a ring. Using a sieve method, explicit formulae for the dominant and subdominant terms of the chromatic polynomial are obtained. Finally, a simple description of the effect of twisting the links is obtained.

It has recently been shown by Mo and Negreiros that a necessary condition for an invariant almost complex structure on the complex full flag manifold ${\bb F}(n)$ to admit a (1, 2)-symplectic invariant metric is that its associated tournament be cone-free. In this paper, a canonical stair-shaped form is given for such tournaments, and this is applied to show that the condition is also sufficient; in the process, all the associated (1, 2)-symplectic metrics are described.

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.

A survey is given of several key themes that have characterised mathematics in the 20th century. The impact of physics is also discussed, and some speculations are made about possible developments in the 21st century.

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 authors show that any $k$ -Osserman Lorentzian algebraic curvature tensor has constant sectional curvature, and give an elementary proof that any local 2-point homogeneous Lorentzian manifold has constant sectional curvature. They also show that a Szabó Lorentzian covariant derivative algebraic curvature tensor vanishes.

A counterexample has been constructed to show that a conjectured global solution structure for bifurcation of non-trivial solutions from a simple eigenvalue of the linearization at zero really can occur. In addition, new results and counterexamples have been obtained for bifurcation from an eigenvalue of geometric multiplicity 1 and odd algebraic multiplicity.

This paper concerns systems of r homogeneous diagonal equations of degree k in s variables, with integer coefficients. Subject to a suitable non-singularity condition, it is shown that the expected asymptotic formula holds for the number of such systems inside a box [−P,P]s, provided only that s > (3r+1)2k−2. By way of comparison, classical methods based on the use of Hua's lemma would establish a similar conclusion, provided instead that s > r2k.

For any x ∈ (0, 1], let the series [sum ]∞n=1 1/dn(x) be the Sylvester expansion of x. Galambos has shown that the Lebesgue measure of the set

[formula here]

is 1 when α = e, the base of the natural logarithm. This paper provides a proof that for any α [ges ] 1, A(α) has Hausdorff dimension 1 when α ≠ e.

Let F be a field of characteristic other than 2. Let F(2) denote the compositum over F of all quadratic extensions of F, let F(3) denote the compositum over F(2) of all quadratic extensions of F(2) that are Galois over F, and let F{3} denote the compositum over F(2) of all quadratic extensions of F(2). This paper shows that F(3) = F{3} if and only if F is a rigid field, and that F(3) = K(3) for some extension K of F if and only if F is Pythagorean and K = F(√−1). The proofs depend mainly on the behavior of quadratic forms over quadratic extensions, and the corresponding norm maps.

In this paper, the authors study intersections of a special class of curves with algebraic subgroups of the multiplicative group of complex dimension at least 2. They show how results of Khovanskii on fewnomials can be used to derive finiteness results and bounds for the degrees of algebraic points for such intersections from more general results on intersections of curves with non-algebraic subgroups. They thereby generalise their earlier results, and recover in some cases, using different methods, more uniform bounds than those given in related work of Bombieri, Masser and Zannier.

This paper proves that for every Lipschitz function $f:{\bb R}^n\longrightarrow {\bb R}^m,\;m < n$ , there exists at least one point of $\varepsilon$ -differentiability of $f$ which is in the union of all $m$ -dimensional affine subspaces of the form $q_0+{\rm span}\{q_1,q_2,\ldots,q_m\},\;{\rm where}\;q_j(j=0,1,\ldots,m)$ are points in ${\bb R}^n$ with rational coordinates.

A sufficient condition is provided for a compact set to be removable for analytic functions in the Zygmund spaces in terms of a porosity. As a consequence, it is shown that the classes of removable sets for the Zygmund space and the little Zygmund space are different.

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.

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d-dimensional packing lattice of balls one can find a four- dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

A new property of a complete intersection in a graded Noetherian ring is proved; this result, in conjunction with the Jacobian criterion, is then applied to the description of the loci of singularities in a family of affine varieties introduced in a well-known paper on forms of many variables.

Let Z be the centre of the group algebra of a symmetric group [Sscr ](n) over a field F characteristic p. One of the principal results of this paper is that the image of the Frobenius map z → zp, for z ∈ Z, lies in span Zp′ of the p-regular class sums. When p = 2, the image even coincides with Z2′. Furthermore, in all cases Zp′ forms a subalgebra of Z. Let pt be the p-exponent of [Sscr ](n). Then jpt = 0, for each element j of the Jacobson radical J of Z. It is shown that there exists j ∈ J such that jpt−1 ≠ 0. Most of the results are formulated in terms of the p-blocks of [Sscr ](n).

A planar set $G \subset {\bb R}^2$ is constructed that is bilipschitz equivalent to ( $G, d^z$ ), where ( $G, d$ ) is not bilipschitz embeddable to any uniformly convex Banach space. Here, $z \in (0, 1)$ and $d^z$ denotes the $z$ th power of the metric $d$ . This proves the existence of a strong $A_{\infty}$ weight in ${\bb R}^2$ , such that the corresponding deformed geometry admits no bilipschitz mappings to any uniformly convex Banach space. Such a weight cannot be comparable to the Jacobian of a quasiconformal self-mapping of ${\bb R}^2$ .