Bulletin of the London Mathematical Society - current issue

The fundamental group of a p-compact group

Dwyer, W. G., Wilkerson, C. W. — 2009-05-21

We compute the fundamental group of a connected p-compact group in terms of the map from the homology of the classifying space of a maximal torus to the homology of the classifying space of its normalizer.

Intermediate convergents and a metric theorem of Khinchin

Haynes, A. K. — 2009-05-21

A landmark theorem in the metric theory of continued fractions begins this way: Select a non-negative real function f defined on the positive integers and a real number x, and form the partial sums s_n of f evaluated at the partial quotients a₁, ..., a_n in the continued fraction expansion for x. Does the sequence {s_n/n} have a limit as n -> ? In 1935 Khinchin proved that the answer is yes for almost every x, provided that the function f does not grow too quickly. In this article we are going to explore a natural reformulation of this problem in which the function f is defined on the rationals and the partial sums in question are over the intermediate convergents to x with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.

Integral means and boundary limits of Dirichlet series

Saksman, E., Seip, K. — 2009-05-21

This paper deals with the boundary behaviour of functions in the Hardy spaces ^p for ordinary Dirichlet series. The main result, answering a question of Hedenmalm, shows that the classical Carlson theorem on integral means does not extend to the imaginary axis for functions in , that is, for the ordinary Dirichlet series in H of the right half-plane. We discuss an important embedding problem for ^p, the solution of which is only known when p is an even integer. Viewing ^p as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.

On theta functions of order 4

Kopeliovich, Y., Pauly, C., Serman, O. — 2009-05-21

We prove that the fourth powers of theta functions with even characteristics form a basis of the space H⁰(A, O_A(4))₊ of even theta functions of order 4 on a principally polarized Abelian variety (A, ) without a vanishing theta-null.

Fields with measure and automorphism

Tomasic, I. — 2009-05-21

We consider a difference field (K, ) such that finite-dimensional definable sets over K can be compared in size, or measured. Let k be the fixed field of the automorphism . We show that curves of genus 1 defined over k are approximately the size of the affine line over k, an ‘approximative version’ of the Riemann hypothesis for curves of genus 1.

The regular part of symmetric forms associated with second-order elliptic differential expressions

Vogt, H. — 2009-05-21

Let be a (not necessarily closable) positive symmetric form associated with a second-order elliptic differential expression. We show that the regular part of (in the sense of B. Simon) can be obtained by modifying the coefficients of suitably; in particular, the regular part is again associated with a second-order elliptic differential expression.

Sufficiency of jets with line singularities

Brodersen, H. — 2009-05-21

Let z: (R ^{n + 1}, 0) -> (R, 0) be an r-jet with a singular set containing a 1-dimensional manifold L. Let be the set of homeomorphism germs h : (R ^{n + 1}, 0) -> (R ^{n + 1}, 0) leaving L invariant. Let be the set of C^r germs, f : (R ^{n + 1}, 0) -> (R , 0), with singular sets containing L. We say that z is sufficient in if any two f and g in with are -equivalent. In this paper we give necessary and sufficient conditions in terms of Lojasiewicz inequalities for such a jet z to be sufficient in .

Stability of projective Poincare and Picard bundles

Biswas, I., Brambila-Paz, L., Newstead, P. E. — 2009-05-21

Let X be an irreducible smooth projective curve of genus g ≥ 3 defined over the complex numbers, and let M denote the moduli space of stable vector bundles on X of rank n and determinant , where is a fixed line bundle of degree d. If n and d have a common divisor, then there is no universal vector bundle on X x M. We prove that there is a projective bundle on X x M with the property that its restriction to X x {E} is isomorphic to P(E) for all E M and that this bundle (called the projective Poincaré bundle) is stable with respect to any polarization; moreover its restriction to {x} x M is also stable for any x X. We also prove stability results for bundles induced from the projective Poincaré bundle by homomorphisms PGL(n) -> H for any reductive H. We further show that there is a projective Picard bundle on a certain open subset M' of M for any d > n(g–1) and that this bundle is also stable. Also, we obtain new results on the stability of the Picard bundle even when n and d are coprime.

Hochschild homology and global dimension

Bergh, P. A., Madsen, D. — 2009-05-21

We prove that, for certain classes of graded algebras (Koszul, local and cellular), infinite global dimension implies that Hochschild homology does not vanish in high degrees, provided that the characteristic of the ground field is zero. Our proof uses Igusa's formula relating the Euler characteristic of relative cyclic homology to the graded Cartan determinant.

The generating condition for coalgebras

Iovanov, M. C. — 2009-05-21

For a ring R, the properties of being (left) self-injective or being a cogenerator for the left R-modules do not imply one another, and the two combined give rise to the important notion of pseudo-Frobenius-rings. For a coalgebra C, (left) self-projectivity implies that C is a generator for right comodules and the coalgebras with this property are called right quasi-co-Frobenius; however, whether the converse implication is true is an open question. We provide an extensive study of this problem. We show that this implication does not hold, by giving a large class of examples of coalgebras having the ‘generating property’. In fact, we show that any coalgebra C can be embedded in a coalgebra C that generates its right comodules, and, if C is local over an algebraically closed field, then C can be chosen local as well. We also give some general conditions under which the implication ‘C-projective (left) C generator for right comodules’ does work, and such conditions are when C is right semiperfect or when C has finite coradical filtration.

'High spots' theorems for sloshing problems

Kulczycki, T., Kuznetsov, N. — 2009-05-21

We investigate several 2D and 3D cases of the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem. In particular, for a domain W R² (canal's cross-section), where W = F B and F (cross-section of the free surface of fluid) is an interval of the x-axis, whereas B (bottom's cross-section) is the graph of a negative function, the following result is proved. The fundamental eigenfunction u₁ of the sloshing problem (the corresponding eigenvalue is simple) has monotonic traces on F and B; moreover, u₁ attains its maximum and minimum values at the end-points of F. It is established that for the 2D (3D) ice-fishing problem with a single (circular) hole, the function u₁ (both fundamental eigenfunctions) attains its maximum value at an inner point of F. A relationship between the high spots and hot spots theorems is considered.

Schwarz lemma for the tetrablock

Edigarian, A., Zwonek, W. — 2009-05-21

We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results from Abouhajar et al. [‘A Schwarz lemma for a domain related to mu-synthesis’, J. Geom. Anal. 17 (2007) 717–750]. The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.

The equivariant Euler characteristic of real Coxeter toric varieties

Henderson, A., Lehrer, G. — 2009-05-21

Let W be a Weyl group, and let T_W be the complex toric variety attached to the fan of cones corresponding to the reflecting hyperplanes of W, and its weight lattice. The real locus T_W(R) is a smooth, connected, compact manifold with a W-action. We give a formula for the equivariant Euler characteristic of T_W(R) as a generalised character of W. In type A_n–1 for n odd, one obtains a generalised character of Sym_n whose degree is (up to sign) the nth Euler number.

On the definition of pseudospectra

Shargorodsky, E. — 2009-05-21

It is well known that the -pseudospectrum of a bounded linear operator defined with the help of a strict inequality is equal to the union of the spectra of all perturbed operators with perturbations that have norms strictly less than . The aim of the paper is to show that the same is not true in the case of non-strict inequalities.

Twisted Alexander polynomials and representation shifts

Silver, D. S., Williams, S. G. — 2009-05-21

For any knot, the following are equivalent. (1) The infinite cyclic cover has uncountably many finite covers; (2) there exists a finite-image representation of the knot group for which the twisted Alexander polynomial vanishes; (3) the knot group admits a finite-image representation such that the image of the fundamental group of an incompressible Seifert surface is a proper subgroup of the image of the commutator subgroup of the knot group.

Capitulation for locally free class groups of orders of group algebras over number fields

Greither, C., Johnston, H. — 2009-05-21

We prove a capitulation result for locally free class groups of orders of group algebras over number fields. This result allows some control over ramification, and so, as a corollary, we obtain an ‘arithmetically disjoint capitulation result’ for the Galois module structure of rings of integers.

Local well-posedness of nonlinear dispersive equations on modulation spaces

Benyi, A., Okoudjou, K. A. — 2009-05-21

By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the nonlinear Schrödinger, nonlinear wave and nonlinear Klein–Gordon equations with Cauchy data in modulation spaces M_0,s^p,1.

Centralizers in domains of finite Gelfand-Kirillov dimension

Bell, J. P. — 2009-05-21

We study centralizers of elements in domains. We extend a result of the author and Small ‘Centralizers in domains of Gelfand–Kirillov dimension 2’, Bull. London Math. Soc. 36 (2004) 779–785, showing that if A is a finitely generated domain of finite Gelfand–Kirillov (GK) dimension and a A is not algebraic over the extended center of A, then the centralizer of a has GK dimension at most one less than the GK dimension of A. In the case that A is a finitely generated noetherian domain of GK dimension 3 over the complex numbers, we show that the centralizer of an element a A that is not algebraic over the extended center of A satisfies a polynomial identity.

The group of automorphisms of a real rational surface is n-transitive

Huisman, J., Mangolte, F. — 2009-05-21

Let X be a rational nonsingular compact connected real algebraic surface. Denote by Aut(X) the group of real algebraic automorphisms of X. We show that the group Aut(X) acts n-transitively on X, for all natural integers n. As an application we give a new and simpler proof of the fact that two rational nonsingular compact connected real algebraic surfaces are isomorphic if and only if they are homeomorphic as topological surfaces.

Classifying spaces of sporadic groups * (Mathematical Surveys and Monographs 147) * By David J. Benson and Stephen D. Smith: 289 pp., US$85.00, * ISBN 978-0821-84474-8 * (American Mathematical Society, Providence, RI, 2008)

Lazarev, A. — 2009-05-21

Quantum groups: A path to current algebra * (Australian Mathematical Society Lecture Series 19) * By Ross Street: 141 pp., {pound}28.99 (US$57.00), ISBN 0-5216-9524-4 * (Cambridge University Press, Cambridge, 2007)

Launois, S. — 2009-05-21

Honorary Members 2008

2009-05-21