Suppose that $K$ is a field of characteristic zero, $K_a$ is its algebraic closure, and that $f(x) \in K[x]$ is an irreducible polynomial of degree $n \ge 5$, whose Galois group coincides either with the full symmetric group $\Sn$ or with the alternating group $\An$. Let $p$ be an odd prime, $\Z[\zeta_p]$ the ring of integers in the $p$th cyclotomic field $\Q(\zeta_p)$. Suppose that $C$ is the smooth projective model of the affine curve $y^p\,{=}\,f(x)$ and $J(C)$ is the jacobian of $C$. We prove that the ring $\End(J(C))$ of $K_a$-endomorphisms of $J(C)$ is canonically isomorphic to $\Z[\zeta_p]$

We present an algebraic formulation of genus 2 hyperelliptic functions which exploits the underlying covariance of the family of genus 2 curves. This allows a simple interpretation of all identities in representation theoretic terms. We show how the classical theory is recovered when one branch point is moved to infinity.

We give an algebraic and topological interpretation of essential coordinate components of characteristic varieties and illustrate their importance with an example.

We discuss and compare several necessary criteria for the local extremality of the Koebe mapping in extremal problems for univalent functions. These criteria are applied to study Robertson type inequalities and also to investigate a conjecture of Bombieri.

This paper looks at some results concerning the Monodromy Conjecture. The conjecture states that for a nonconstant regular function $f$ on a surface germ $(S,0)$ with $f(0)=0$, if a rational number $s_\circ$ is a pole of the topological zeta function $Z_\mathrm{top}(f,s)$, then $e^{2\pi is_\circ}$ is an eigenvalue of the local monodromy of $f$ at some point of $f^{-1}\!\{0\}$. First we give our own, very elementary and conceptual, proof of the conjecture for the well-known case where the germ $(S,0)$ is nonsingular. This proof is not only included for its simplicity, but also because we will need exactly the same arguments in the second part. There we explain what can, and what cannot, be expected in the singular case.

We study conformal deformations of uniform Loewner spaces, verify the Gehring-Hayman Inequality, demonstrate uniform continuity of the identity mapping, and provide results which describe the boundary of the deformed space. In addition, we establish two connections between Hausdorff content and modulus.

We identify the class of smooth boundary conditions that yield an initial-boundary value problem admitting a unique smooth solution for the case of a dispersive linear evolution PDE of arbitrary order, in one spatial dimension, defined on a finite interval.

This result is obtained by an application of a spectral transform method, introduced by Fokas, which allows us to reduce the problem to the study of the singularities of the set of functions arising as the unique solution of a certain linear system.

In this paper we give an asymptotic expansion including error terms for the number of closed orbits in the homology classes for homological full Anosov flows. In particular we obtain formulae concerning the coefficients of error terms which depend on the homology classes.

We study functions $f$ of one complex variable with values in a complete locally convex space $E$. In [13] we had shown that for $f$ to be holomorphic it suffices that it is locally bounded and that there is a separating subset $H$ of $E'$ such that $x'\circ f$ is holomorphic for each functional $x'\in H$. We give here a new and considerably shorter proof of this result, and we extend it to a weak criterion for holomorphic continuability.

We prove that the boundary extension of a quasi-isometry of hyperbolic space which conjugates two Kleinian groups maps the Myrberg limit set of the first group bijectively onto the Myrberg limit set of the second group. As a first application of this observation we obtain new geometric proofs of Mostow's Rigidity Theorem. Furthermore, we recover the classical result that motions in the Teichmüller space of a compact hyperbolic surface induce singular boundary maps.

Three kinds of proper actions of increasing strength are defined. We prove that the three definitions specialize to the definitions by Bourbaki, by Palais and by Baum, Connes and Higson in their respective settings. The third of these, which thus turns out to be the strongest, originally only concerns actions of second countable locally compact groups on metrizable spaces. In this situation, it is shown to coincide with the other two definitions if the total space locally has the Lindelöf property and the orbit space is regular.

The purpose of this note is to call attention to an error in the paper [1]. The main result was a new proof of the main theorem in [2]. In line 1 of page 446 of [1] one finds the following remark: “This is the result obtained in [2]”. However, a comparison of the two results shows that the generator which is called $d_1 d^{-1}_2$ in the notation of [1] appears in [2] but is missing from the list in [1].

This paper considers various kinds of multiplicative transfers, namely Evens transfer, Kozlowski transfer and normalized Evens transfer in singular and Galois cohomologies, which have been introduced as tools for solving the problem of determining characteristic classes of direct image bundles for covering maps (Fulton–MacPherson) or Stiefel–Whitney classes of induced Galois representation for finite separable extension of commutative fields of characteristic not 2 (Kahn).

We show that normalized Evens transfer is covariant, when restricted to homogeneous elements or Stiefel–Whitney classes, in the case of singular cohomology. We deduce that, in the case of Galois cohomology, Kozlowski transfer and the transfer involved by Kahn coïncide on invertible elements. We conclude that the Kahn formula can be viewed as an inflation of the Fulton–MacPherson formula.

Let $f:M \to N$ be a smooth map of a closed $n$-dimensional manifold $M$ into a $p$-dimensional manifold $N (n \geq p)$. When $f$ has only definite fold singular points, we call it a special generic map. Porto and Furuya introduced the notion of regular equivalence for such maps. In this paper, we define another equivalence relation for special generic maps, called weak regular equivalence and we classify special generic maps up to these equivalences in the case where $M$ is a connected orientable closed $n$-dimensional manifold with $n \geq 3$ and $N$ is the plane. We also show the existence of a pair of special generic maps which are $C^0$ right-left equivalent but are not $C^\infty$ right-left equivalent.

Using Fuchsian techniques, a large family of Gowdy vacuum spacetimes have been constructed for which one has detailed control over asymptotic behaviour. In this paper we formulate a condition on initial data yielding the same form of asymptotics.