Bulletin of the London Mathematical Society - current issue

Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics

Asok, A., Doran, B., Kirwan, F. — 2008-07-18

Atiyah and Bott used equivariant Morse theory applied to the Yang–Mills functional to calculate the Betti numbers of moduli spaces of vector bundles over a Riemann surface, rederiving inductive formulae obtained from an arithmetic approach which involved the Tamagawa number of SL_n. This article attempts to survey and extend our understanding of this link between Yang–Mills theory and Tamagawa numbers, and to explain how methods used over the last three decades to study the singular cohomology of moduli spaces of bundles on a smooth projective curve over C can be adapted to the setting of A¹-homotopy theory to study the motivic cohomology of these moduli spaces over an algebraically closed field.

The probability of generating a nilpotent subgroup of a finite group

Wilson, J. S. — 2008-07-18

It is proved that for finite groups G the probability that two randomly chosen elements of G generate a nilpotent subgroup tends to 0 as the index of the Fitting subgroup of G tends to infinity.

Norm-attaining operators between Marcinkiewicz and Lorentz spaces

Acosta, M. D., Kaminska, A. — 2008-07-18

Bishop and Phelps proved that the set of norm-attaining functionals on any Banach space is dense in the topological dual. After that, the study of the same kind of problems for operators was initiated by Lindenstrauss, and several general positive results were proved. It was then consistently continued for different classes of spaces including L¹(µ) or C(K). Here a similar problem is studied in the context of classical interpolation Marcinkiewicz and Lorentz spaces, M_W⁰ and _{1, v}, in both the real and the complex cases. We show that if wv L¹ then the identity operator between these spaces is bounded, but it is not possible to approximate it by norm-attaining operators. We also prove that every compact operator from M_W⁰ to _{1, v} can be approximated by finite-rank norm-attaining operators.

Some power of an element in a Garside group is conjugate to a periodically geodesic element

Lee, E.-K., Lee, S.-J. — 2008-07-18

We show that for each element g of a Garside group, there exists a positive integer m such that g^m is conjugate to a periodically geodesic element h, an element with |hⁿ|_D = |n| · |h|_D for all integers n, where |g|_D denotes the shortest word length of g with respect to the set D of simple elements. We also show that there is a finite-time algorithm that computes, given an element of a Garside group, its stable super summit set.

A Favard-type problem for 3d convex bodies

Campi, S., Gronchi, P. — 2008-07-18

A theorem due to Favard states that among all plane sets of given area and perimeter, the symmetric lens has maximum circumradius. This paper deals with the higher-dimensional problem of finding the convex body in R³ of given volume and mean width with the largest possible diameter. It is shown that the solution is the convex hull of a surface of revolution with constant Gauss curvature and a segment lying on the axis of revolution. Such a body is conjectured also to maximize the circumradius in the same class.

On minimal distances in Krull monoids with infinite class group

Chapman, S. T., Schmid, W. A., Smith, W. W. — 2008-07-18

Let H be a Krull monoid with infinite class group such that each divisor class contains a prime divisor. It is shown that for every positive integer n, there exists a divisor closed submonoid S of H such that min (S) = n.

On the Mahler measure in several variables

Amoroso, F. — 2008-07-18

If the total degree of a polynomial in n ≥ 2 variables of dimension n is bounded by a double exponential function in n, then we show that its Mahler measure is bounded from below by an absolute constant greater than 1.

The eigenvalues of limits of radial Toeplitz operators

Suarez, D. — 2008-07-18

Let A² be the Bergman space on the unit disk. A bounded operator S on A² is called radial if Szⁿ = _n zⁿ for all n ≥ 0, where _n is a bounded sequence of complex numbers. We characterize the eigenvalues of radial operators that belong to the Toeplitz algebra.

Anneaux de definition des dg-algebres propres et lisses

Toen, B. — 2008-07-18

RÈSUMÈ

Soit k = colim_i k_i une colimite filtrante d’anneaux commutatifs. On montre que la théorie homotopique des dg-algèbres propres et lisses sur k est la colimite des théories homotopiques des dg-algèbres propres et lisses sur les k_i. Nous en déduisons en particulier que toute dg-algèbre propre et lisse est définissable sur une Z-algèbre commutative de type fini.

Let k = colim_i k_i be a filtered colimit of commutative rings. We show that the homotopy theory of smooth and proper dg-algebras over k is the colimit of the homotopy theories of smooth and proper dg-algebras over the k_i. We deduce, in particular, that every smooth and proper dg-algebra can be defined over a commutative Z-algebra of finite type.

A note on attractors with finite fractal dimension

Czaja, R., Efendiev, M. — 2008-07-18

Conditions for the finite fractal dimension of precompact invariant sets are formulated and exponential attractors for discrete and continuous dynamical systems are constructed.

Construction of larger Riemannian metrics with bounded sectional curvatures and applications

Chen, X., Wang, F.-Y. — 2008-07-18

For any (not necessarily complete) Riemannian manifold, we construct a larger Riemannian metric which is complete and with bounded sectional curvatures. As an application, log-Sobolev inequalities are established on arbitrary Riemannian manifolds with reference measures having smooth and strictly positive densities. In particular, a conjecture of the second-named author (see [J. Math. Anal. Appl. 300 (2004) 426–435]) is solved.

The third homotopy module of a 2-complex

Mannan, W. H. — 2008-07-18

Given a connected 2-complex X with fundamental group G, we show how ₃(X) may be computed as a module over Z[G]. Further, if X is a finite connected 2-complex with G (=₁(X)) finite of odd order, then the stable class of ₃(X) is determined by G. In this case ₃(X) Q is free over Q[G].

On the direct product conjecture for sigma invariants

Schutz, D. — 2008-07-18

We show that the direct product conjecture for ⁿ(G; Z), where G is the direct product of two groups of type FP_n, holds for n = 3 and give counterexamples for n ≥ 4. Previously, counter-examples were known only for a related conjecture involving the homotopical -invariants, where the conjecture already fails for n ≥ 3.

Frame duality properties for projective unitary representations

Han, D., Larson, D. — 2008-07-18

Let be a projective unitary representation of a countable group G on a separable Hilbert space H. If the set B of Bessel vectors for is dense in H, then for any vector x H the analysis operator _x makes sense as a densely defined operator from B to ²(G)-space. Two vectors x and y are called -orthogonal if the range spaces of _x and _y are orthogonal, and they are -weakly equivalent if the closures of the ranges of _x and _y are the same. These properties are characterized in terms of the commutant of the representation. It is proved that a natural geometric invariant (the orthogonality index) of the representation agrees with the cyclic multiplicity of the commutant of (G). These results are then applied to Gabor systems. A sample result is an alternate proof of the known theorem that a Gabor sequence is complete in L²(R ^d) if and only if the corresponding adjoint Gabor sequence is ²-linearly independent. Some other applications are also discussed.

Periodic points of endomorphisms on solenoids and related groups

Miles, R. — 2008-07-18

This paper investigates the problem of finding the possible sequences of periodic point counts for endomorphisms of solenoids. For an ergodic epimorphism of a solenoid, a closed formula is given that expresses the number of points of any given period in terms of sets of places of finitely many algebraic number fields and distinguished elements of those fields. The result extends to more general epimorphisms of compact abelian groups.

Languages of k-interval exchange transformations

Ferenczi, S., Zamboni, L. Q. — 2008-07-18

This paper gives a complete characterization of those sequences of subword complexity (k – 1)n + 1 that are natural codings of orbits of k-interval exchange transformations, thereby answering an old question of Rauzy.

Artin HNN-extensions virtually embed in Artin groups

Hsu, T., Leary, I. J. — 2008-07-18

An Artin HNN-extension is an HNN-extension of an Artin group in which the stable letter conjugates a pair of suitably chosen subsets of the standard generating set. We show that some finite index subgroup of an Artin HNN-extension embeds in an Artin group. We also obtain an analogous result for Coxeter groups.

Book Reviews

Baird, P. — 2008-07-18

Linear algebra in action * (Graduate Studies in Mathematics 78) * By Harry Dym: 541 pp., US$79.00, ISBN 978-0-821838-13-6 * (American Mathematical Society, Providence, RI, 2007)

Partington, J. R. — 2008-07-18

Algebraic analysis of singular perturbation theory * (Translations of Mathematical Monographs 227) * By Takahiro Kawai and Yoshitsugu Takei (Translated by Goro Kato): * 130 pp., US$29.00 ISBN 0-8218-3547-5 * (American Mathematical Society, Providence, RI, 2005)

Novokshenov, V. — 2008-07-18