The paper contains a final identification theorem for the ‘generic’ $K^*$-groups of finite Morley rank.

In this paper, we give a lower bound of the p-part of the reduced Euler characteristic of the order complex of the centric p-radical subgroups by studying vertices of indecomposable summands of the reduced Lefschetz module. This bound is in fact best possible for at least some groups, and also provides a uniform explanation of the observed phenomenon on the reduced Euler characteristic for some sporadic simple groups.

It is well known that there are bounded domains Ω⊂ℝn whose boundaries ∂Ω are not smooth enough for there to exist a bounded linear extension for the Sobolev space W1p(Ω) into W1p(ℝn), but the embedding W1p(Ω)⊂ Lp(Ω) is nevertheless compact. For the Lipγ boundaries (0<γ<1) studied in [3, 4], there does not exist in general an extension operator of W1p(Ω) into W1p(ℝn) but there is a bounded linear extension of W1p(Ω) into Wγp(ℝn) and the smoothness retained by this extension is enough to ensure that the embedding W1p(Ω)⊂ Lp(Ω) is compact. It is natural to ask if this is typical for bounded domains which are such that W1p(Ω)⊂ Lp(Ω) is compact, that is, that there exists a bounded extension into a space of functions in ℝn which enjoy adequate smoothness. This is the question which originally motivated this paper. Specifically we study the ‘extension by zero’ operator on a space of functions with given ‘generalized’ smoothness defined on a domain with an irregular boundary, and determine the target space with respect to which it is bounded.

A full characterization is given of those compact Lie groups G with the property that every G-map X→X on a finite-dimensional G-complex X of finite orbit type, XG = Ø, is (non-equivariantly) essential. For arbitrary G, conditions are given on the G-space X which guarantee this property. Finally, conditions are given for the non-existence of a G-map X→Y inducing a homotopy equivalence XG≃YG on the fixed point sets. These results have applications to critical point theory of almost G-invariant functionals.

For a hyperbolic knot, the excellent component of the character curve is the one containing the complete hyperbolic structure on the complement of the knot. The paper explains a method of computing the excellent component of the character variety of tunnel number 1 knots.

The paper calculates the average density of the normalized Hausdorff measure on the fractal set generated by a conformal iterated function system. It equals almost everywhere a positive constant given by a truncated generalized s-energy integral, where s is the corresponding Hausdorff dimension. As a main tool a conditional Gibbs measure is determined. The appendix proves an appropriate extension of Birkhoff's ergodic theorem which is also of independent interest.

The cohomology of [Mscr ](n, d), the moduli space of stable holomorphic bundles of coprime rank n and degree d and fixed determinant, over a Riemann surface Σ of genus g [ges ] 2, has been widely studied from a wide range of approaches. Narasimhan and Seshadri [17] originally showed that the topology of [Mscr ](n, d) depends only on the genus g rather than the complex structure of Σ. An inductive method to determine the Betti numbers of [Mscr ](n, d) was first given by Harder and Narasimhan [7] and subsequently by Atiyah and Bott [1]. The integral cohomology of [Mscr ](n, d) is known to have no torsion [1] and a set of generators was found by Newstead [19] for n = 2, and by Atiyah and Bott [1] for arbitrary n. Much progress has been made recently in determining the relations that hold amongst these generators, particularly in the rank two, odd degree case which is now largely understood. A set of relations due to Mumford in the rational cohomology ring of [Mscr ](2, 1) is now known to be complete [14]; recently several authors have found a minimal complete set of relations for the ‘invariant’ subring of the rational cohomology of [Mscr ](2, 1) [2, 13, 20, 25].

Unless otherwise stated all cohomology in this paper will have rational coefficients.

In this paper a formula is derived for the number of conjugacy classes of cyclic subgroups of finite order in those arithmetic Fuchsian groups which are of minimal covolume in their commensurability class. The formula is entirely in terms of the number theoretic data defining the commensurability class of the arithmetic group so that, in particular, any two such groups of minimal covolume in the class, will be isomorphic.

In this paper, we study certain polynomial invariants of links (singular or non-singular) that are related to the Homfly polynomial and Vassiliev's invariants. The Homfly polynomial HL [3] (also known as the Flypmoth polynomial) satisfies the well-known skein relation xHL+ +yHL− +zHL0=0. The Vassiliev invariants [1, 2] (of order 1) satisfy the relations VL× =VL+ −VL− and VL××=0. The invariants that we study satisfy the skein relations

formula here

Let $K$ be a function field of genus $g$ with a finite constant field ${\mathbb{F}}_q$. Choose a place $\infty$ of $K$ of degree $\delta$ and let ${\mathbb{C}}$ be the arithmetic Dedekind domain consisting of all elements of $K$ that are integral outside $\infty$. An explicit formula is given (in terms of $q$, $g$ and $\delta$) for the minimum index of a non-congruence subgroup in SL$_2({\mathcal{C}})$. It turns out that this index is always equal to the minimum index of an arbitrary proper subgroup in SL$_2({\mathcal{C}})$. The minimum index of a normal non-congruence subgroup is also determined.

We establish a Weitzenböck formula for harmonic morphisms between Riemannian manifolds and show that under suitable curvature conditions, such a map is totally geodesic. As an application of the Weitzenböck formula we obtain some non-existence results of a global nature for harmonic morphisms and totally geodesic horizontally conformal maps between compact Riemannian manifolds. In particular, it is shown that the only harmonic morphisms from a Riemannian symmetric space of compact type to a compact Riemann surface of genus at least 1 are the constant maps.

The paper studies the connective complex K-theoretic Euler class of a certain bundle associated to a Euclidean immersion of the lens space L(2k)2n+1> to show that this manifold cannot be immersed in ℝ4n−2α(n) if k [ges ] α(n). The non-immersion is best possible in many cases. This suggests a close relationship between the immersion problem for complex projective spaces and that for ‘high’ 2-torsion lens spaces.

Patchworking of singular hypersurfaces is used to construct projective hypersurfaces with prescribed singularities. For all $n\,{\geq}\, 2$, an asymptotically proper existence result is deduced for hypersurfaces in $\P^n$ with singularities of corank at most 2 prescribed up to analytical or topological equivalence. In the case of $T$-smooth hypersurfaces with only simple singularities, the result is even asymptotically optimal, that is, the leading coefficient in the sufficient existence condition cannot be improved, which is new even in the case of plane curves. Furthermore, an asymptotically proper existence result is proved for hypersurfaces in $\P^n$ with quasihomogeneous singularities. The estimates substantially improve all known (general) existence results for hypersurfaces with these singularities.

That vector potentials have a direct significance to quantum particles moving in magnetic fields is known as the A–B (Aharonov–Bohm) effect. We study scattering by two solenoidal magnetic fields (point-like magnetic fields) in two dimensions and analyze the asymptotic behavior of the scattering amplitude in the semiclassical limit. The corresponding classical mechanical system has a trajectory oscillating between the centers of the two fields. We derive the asymptotic formula with the first three terms and make clear how such a trapping trajectory is reflected in the asymptotic formula through the A–B effect. We also make a brief comment on an extension to the scattering by many solenoidal fields. The result depends on the location of the centers of the fields. In particular, the A–B effect appears strongly when the centers are on an even line.

Acyclic groups of low dimension are considered. To indicate the results simply, let $G^{\prime}$ be the nontrivial perfect commutator subgroup of a finitely presentable group $G$. Then $\mathrm{def}(G)\leq1$. When $\mathrm{def}(G)=1$, $G^{\prime}$ is acyclic provided that it has no integral homology in dimensions above 2 (a sufficient condition for this is that $G^{\prime}$ be finitely generated); moreover, $G/G^{\prime}$ is then $Z$ or $Z^{2}$. Natural examples are the groups of knots and links with Alexander polynomial 1. A further construction is given, based on knots in $S^{2}\times S^{1}$. In these geometric examples, $G^{\prime}$ cannot be finitely generated; in general, it cannot be finitely presentable. When $G$ is a 3-manifold group it fails to be acyclic; on the other hand, if $G^{\prime}$ is finitely generated it has finite index in the group of a $\mathbb{Q}$-homology 3-sphere.

Goldston and Montgomery proved that the strong pair correlation conjecture and two second moments of primes in short intervals are equivalent to each other under the Riemann hypothesis. The paper obtains the second main terms for each of the above, and it is shown that they are almost equivalent to each other.

Thompson's famous theorems on singular values–diagonal elements of the orbit of an n×n matrix A under the action (1) U(n) [otimes ] U(n) where A is complex, (2) SO(n) [otimes ] SO(n), where A is real, (3) O(n) [otimes ] O(n) where A is real are fully examined. Coupled with Kostant's result, the real semi-simple Lie algebra [sfr ][ofr ]n, n yields (2) and hence (3) and the sufficient part (the hard part) of (1). In other words, the curious subtracted term(s) are well explained. Although the diagonal elements corresponding to (1) do not form a convex set in [Copf ]n, the projection of the diagonal elements into ℝn (or iℝn) is convex and the characterization of the projection is related to weak majorization. An elementary proof is given for this hidden convexity result. Equivalent statements in terms of the Hadamard product are also given. The real simple Lie algebra [sfr ][ufr ]n, n shows that such a convexity result fits into the framework of Kostant's result. Convexity properties and torus relations are studied. Thompson's results on the convex hull of matrices (complex or real) with prescribed singular values, as well as Hermitian matrices (real symmetric matrices) with prescribed eigenvalues, are generalized in the context of Lie theory. Also considered are the real simple Lie algebras [sfr ][ofr ]p, q and [sfr ][ofr ]p, q, p < q, which yield the rectangular cases. It is proved that the real part and the imaginary part of the diagonal elements of complex symmetric matrices with prescribed singular values are identical to a convex set in ℝn and the characterization is related to weak majorization. The convex hull of complex symmetric matrices and the convex hull of complex skew symmetric matrices with prescribed singular values are given. Some questions are asked.

It is known that the group of homeomorphisms of a non-compact, connected, metrizable manifold acts transitively on certain spaces of measures defined, among other properties, by their behaviour at the manifold's set of ‘ends’. As a generalization of previous work of Fathi in the realm of compact manifolds, it is shown that the given action admits a continuous section when restricted to ‘biregular’ homeomorphisms and measures. This result exhibits the isotropy groups of measure-preserving homeomorphisms as deformation retracts of the corresponding acting groups.

Let

(formula here)

be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.

Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.

Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.

For each α ∈ A, we define a rational number dα by

(formula here)

where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.

In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.

Let G be a permutation group on a set Ω, and let m and k be integers where 0<m<k. For a subset Γ of Ω, if the cardinalities of the sets Γg\Γ, for g∈G, are finite and bounded, then Γ is said to have bounded movement, and the movement of Γ is defined as move (Γ) =maxg∈G[mid ]Γg\Γ[mid ]. If there is a k-element subset Γ such that move (Γ)[les ]m, it is shown that some G-orbit has length at most (k2−m)\(k−m). When combined with a result of P. M. Neumann, this result has the following consequence: if some infinite subset Γ has bounded movement at most m, then either Γ is a G-invariant subset with at most m points added or removed, or Γ nontrivially meets a G-orbit of length at most m2+m+1. Also, if move (Γ)[les ]m for all k-element subsets Γ and if G has no fixed points in Ω, then either [mid ]Ω[mid ][les ]k+m (and in this case all permutation groups on Ω have this property), or [mid ]Ω[mid ][les ]5m−2. These results generalise earlier results about the separation of finite sets under group actions by B. J. Birch, R. G. Burns, S. O. Macdonald and P. M. Neumann, and groups in which all subsets have bounded movement (by the author).