As the first step of research on functional equations for multiple zeta-functions, we present a candidate of the functional equation for a class of two variable double zeta-functions of the Hurwitz–Lerch type, which includes the classical Euler sum as a special case.

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 $\Z[\zeta_p]$

A language-theoretic definition of word hyperbolic semigroup is given, which coincides with the original definition for a group. Word problems of semigroups are also considered.

We use hypercovers to study the homotopy theory of simplicial presheaves. The main result says that model structures for simplicial presheaves involving local weak equivalences can be constructed by localizing at the hypercovers. One consequence is that the fibrant objects can be explicitly described in terms of a hypercover descent condition, and the fibrations can be described by a relative descent condition. We give a few applications for this new description of the homotopy theory of simplicial presheaves.

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.

In this paper we describe the relationship between characters of finitely generated torsion-free nilpotent groups of class 2 and their completions. In particular we give a detailed description of this connection for the discrete Heisenberg group.

Let ${\frak M}_{g}$ be the moduli space of smooth, integral curves of genus $g$ over the complex field ${\mathbb C}$ and let $\overline{{\frak M}}_{g}$ be its Mumford–Deligne compactification. If $C$ is a smooth, integral curve of genus $g$ we denote by [$C$] its isomorphism class in ${\frak M}_{g}$.

Let $C$ be a curve of genus $g$ and $E$ a generic (semistable) vector bundle of rank $r$ and degree $d$. Fix a rank $r'\,{<}\,r$ and a degree $d'$ for subsheaves $E'$ of $E$. If $r'd-rd'=r'(r-r')(g-1)$, the number of such subbundles is finite. We shall denote this number by $m(r,d,r',g)$.

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.

We prove a conjecture made by Frank Peterson on the global structure of the Dickson algebras arising as odd primary general linear group invariants. The Dickson algebra $W_{n}$ of invariants in a rank $n$ polynomial algebra over $\mathbb{F}_{p}$ is an unstable algebra over the mod $p$ Steenrod algebra. We prove that $W_{n}$ is a free unstable algebra on a certain cyclic module, modulo just one additional relation. The result is both similar to and different from the corresponding result we previously obtained with Frank Peterson at the prime 2. We also extend our characterization to the algebras of invariants under the special linear groups.

We show that a non-elementary Möbius group acting in higher dimensions is discrete if and only if its elementary subgroup whose elements fix every limit point is finite and every two-generator subgroup is discrete. This result gives an improvement over earlier results of Abikoff and Haas [1], Fang and Nai [5], Martin [11] and the author [9]. We also give an example to show that the first condition is necessary when dimension $n\ge 3$.

We show that projective unitary groups, or more generally, central quotients of products of unitary groups are totally $N$-determined when considered as $p$-compact groups at odd primes $p$.

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 give a closed formula for the multivariable Conway potential function of any graph link in a homology sphere. As corollaries, we answer three questions by Walter Neumann [8] about graph links.

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 use the tangle model to study the action of the site-specific recombinase Gin, an enzyme that can introduce topological changes on circular DNA molecules. Gin and its bound DNA are modelled as a 2-string tangle which undergoes changes during recombination, thereby changing the topology of the DNA substrate. We show that the tangles involved in the analysis are all rational tangles. This technique allows us to prove that, under the model's assumptions, there is a unique topological description of the enzymatic action. The Gin system is one of the few to date where tangle analysis can be carried out systematically and rigorously, yielding a single, biologically reasonable solution.

Existence varieties (briefly e-varieties) of regular semigroups were introduced in [1] as classes of regular semigroups closed under the formation of homomorphic images, regular subsemigroups and direct products. Existence varieties of orthodox semigroups, which are simply the sub-e-varieties of the e-variety $\hbox{\sp O}$ of all orthodox semigroups, were independently introduced in [4] under the name of bivarieties. Moreover, in [4], the notions of bifree objects, biidentities and biinvariant congruences were introduced within $\hbox{\sp O}$ in such a way that a theory properly generalizing the theory of varieties of inverse semigroups arose. Existence varieties of orthodox semigroups include all varieties of orthodox completely regular semigroups, and hence, in particular, all varieties of idempotent semigroups, that is, all subvarieties of the variety $\hbox{\sp B}$ of all bands. These varieties form a countable lattice whose full description is well known.

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.

We prove that every closed, connected contact 3-manifold can be obtained from $S^3$ with its standard contact structure by contact (${\pm}1$)-surgery along a Legendrian link. As a corollary, we derive a result of Etnyre and Honda about symplectic cobordisms (in a slightly stronger form).