Let $C_A/\Q$ be the curve $y^2 = x^5 + A$, and let $L(s, J_A)$ denote the $L$-series of its Jacobian. Under the assumption that the sign in the functional equation for $L(s, J_A)$ is $+1$, the critical value $L(1, J_A)$ is evaluated in terms of the value of a theta series for $\Q(\sqrt{5})$ depending on $A$ at a complex multiplication point coming from $\Q(\zeta_5)$.

The paper explores dualizing differential graded (DG) modules over DG algebras. The focus is on DG algebras that are commutative local, and finite. One of the main results established is that, for this class of DG algebras, a finite DG module is dualizing precisely when its Bass number is 1. As a corollary, one obtains that the Avramov–Foxby notion of Gorenstein DG algebras coincides with that due to Frankild and Jørgensen. One other key result is that, under suitable hypotheses, any two dualizing DG modules are quasiisomorphic up to a suspension. In addition, it is established that a number of naturally occurring DG algebras possess dualizing DG modules.

The number of crepant valuations of an isolated canonical singularity $0\in X:(f\,{=}\,0)\,{\subset}\,\mathbb{C}^{n}$, which is assumed to be nondegenerate with respect to its Newton polyhedron, is calculated in terms of weightings and the Newton polyhedron of $f$.

Introducing the framework of pseudo-motivic homology, the paper finishes the proof that the Brauer–Manin obstruction is the only obstruction to the local–global principle for zero-cycles on a Severi–Brauer fibration of squarefree index over a smooth projective curve over a number field, provided that the Tate–Shafarevich group of the Jacobian of the base curve is finite. More precisely, for such a variety the Chow group of global zero-cycles is dense in the subgroup of collections of local cycles that are orthogonal to the (cohomological) Brauer group of the variety.

A partition of the class of all Hermite–Biehler entire functions of finite order into subclasses ${\cal P}_\kappa$ is introduced. A given function $E(z)$ belongs to ${\cal P}_\kappa$ if and only if $-z^{-1}\log E(z)\in{\mc N}_\kappa$, where the class ${\mc N}_\kappa$ is a well studied family of meromorphic functions on the upper half plane, which originates from operator theoretic problems. It is also proved that the subclasses ${\cal P}_\kappa$ are stable under bounded type perturbation.

Let $G$ be a group and $K$ a field. For any finite-dimensional $KG$-module $V$ and any positive integer $n$, let $L^n(V)$ denote the $n$th homogeneous component of the free Lie $K$-algebra generated by (a basis of) $V$. Then $L^n(V)$ can be considered as a $KG$-module, called the $n$th Lie power of $V$. The paper is concerned with identifying this module up to isomorphism. A simple formula is obtained which expresses $L^n(V)$ in terms of certain linear functions on the Green ring. When $n$ is not divisible by the characteristic of $K$ these linear functions are Adams operations. Some results are also obtained which clarify the relationship between Adams operations defined by means of exterior powers and symmetric powers and operations introduced by Benson. Some of these results are put into a more general setting in an appendix by Stephen Donkin.

The first example of a non-finitely based variety of centre-by-abelian-by-nilpotent groups was found in 1995. This example has an infinite exponent but it can be seen that the method used to construct it also gives a variety that has an exponent of 128. Later it was proved that there exists a variety with similar properties but with an exponent of 32. The paper constructs a non-finitely based variety of centre-by-abelian-by-nilpotent groups which has an exponent of 8.

A new technique is presented which gives conditions under which perturbations of certain base potentials are uniquely determined from the location of eigenvalues and resonances in the context of a Schrödinger operator on a half line. The method extends to complex-valued potentials and certain potentials whose first moment is not integrable.Research supported in part by the US National Science Foundation under grants DMS-9970299 and DMS-0107492 and by the UK EPSRC.

The coefficients in asymptotics of regularized traces and associated trace (spectral) distributions for Schrödinger operators with short and long range potentials are computed. A kernel expansion for the Schrödinger semigroup is derived, and a connection with non-commutative Taylor formulas is established.

By critical point theory, a new approach is provided to study the existence of periodic and subharmonic solutions of the second order difference equation $$\Delta^2 x_{n-1} +f(n, x_n)\,{=}\,0,$$ where $f\in C({\bf R}\times {\bf R}^m, {\bf R}^m), f(t+M,z)=f(t,z)$ for any $(t, z)\in {\bf R}\times{\bf R}^m$ and $M$ is a positive integer. This is probably the first time critical point theory has been applied to deal with the existence of periodic solutions of difference systems.This project is supported by TRATOYT of China and by the State Education Commission Trans-Century Training Program Foundation for the Talents.

The heat content asymptotics on a Riemannian manifold with smoooth boundary defined by Dirichlet, Neumann, transmittal and transmission boundary conditions are studied.

The cohomology groups of some sequence algebras are considered. In particular, the amenability of the James space $J_p$ is investigated; this space was first studied by R. C. James and was shown to be a Banach algebra for the pointwise product. Then a full classification of the closed ideals in $J_p$ is given, extending a result of N. J. Laustsen. Finally, some related algebras $\alp$ are considered, and it is established when these Banach algebras are amenable.

This research was partially supported by the National Security Agency. It is well known from work of Bruhat and Tits that an affine building has a spherical building at infinity. The paper studies the structure at infinity for an affine {\em twin} building. It is shown that the twinning restricts the structure at infinity in a natural way, producing two smaller spherical buildings that are canonically isomorphic to one another. In the process it is shown that to each wall and panel of these buildings at infinity is attached a twin tree.

The integral cohomology rings of the configuration spaces of $n$-tuples of distinct points on arbitrary surfaces (not necessarily orientable, not necessarily compact and possibly with boundary) are studied. It is shown that for punctured surfaces the cohomology rings stabilize as the number of points tends to infinity, similarly to the case of configuration spaces on the plane studied by Arnold, and the Goryunov splitting formula relating the cohomology groups of configuration spaces on the plane and punctured plane to arbitrary punctured surfaces is generalized. Moreover, on the basis of explicit cellular decompositions generalizing the construction of Fuchs and Vainshtein, the first cohomology groups for surfaces of low genus are given.

For each Artin group, the reduced $\eltwo$-cohomology of (the universal cover of) its ‘Salvetti complex’ is computed. This is a CW-complex which is conjectured to be a model for the classifying space of the Artin group. In the many cases when this conjecture is known to hold the calculation describes the reduced $\eltwo$-cohomology of the Artin group.

The relationship between the connections on two Riemannian manifolds that are related by a Riemannian submersion are used to characterise parallel translation on shape space in terms of its behaviour on the pre-shape sphere and so to derive a concrete procedure for the unrolling, or development, of curves in shape space onto tangent spaces. This will enable one to adapt to shape spaces the standard statistical procedures for fitting curves to Euclidean data.

The integrability of a control function of the almost sure convergence $$ \displaystyle\lim_{n \to \infty}\frac{X_1(\omega)+X_2(\omega)+\ldots+X_n(\omega)}{T_n}, $$ where $T_n \,{\uparrow}\, \infty $ as $n \to \infty$ and $X_1,X_2,\cdots,X_n,\cdots $ are independent identically distributed random variables, is studied.