It is shown that every sufficiently large integer congruent to $14$ modulo $240$ may be written as the sum of $14$ fourth powers of prime numbers, and that every sufficiently large odd integer may be written as the sum of $21$ fifth powers of prime numbers. The respective implicit bounds $14$ and $21$ improve on the previous bounds $15$ (following from work of Davenport) and $23$ (due to Thanigasalam). These conclusions are established through the medium of the Hardy-Littlewood method, the proofs being somewhat novel in their use of estimates stemming directly from exponential sums over prime numbers in combination with the linear sieve, rather than the conventional methods which `waste' a variable or two by throwing minor arc estimates down to an auxiliary mean value estimate based on variables not restricted to be prime numbers. In the work on fifth powers, a switching principle is applied to a cognate problem involving almost primes in order to obtain the desired conclusion involving prime numbers alone. 2000 Mathematics Subject Classification: 11P05, 11N36, 11L15, 11P55.

Although the theory of singularities of curves - resolution, classification, numerical invariants - goes through with comparatively little change in finite characteristic, pencils of curves are more difficult. Bertini's theorem only holds in a much weaker form, and it is convenient to restrict to pencils such that, when all base points are resolved, the general member of the pencil becomes non-singular. Even here, the usual rule for calculating the Euler characteristic of the resolved surface has to be modified by a term measuring wild ramification.

We begin by describing this background, then proceed to discuss the exceptional members of a pencil. In characteristic 0 it was shown by Há and Lê and by Lê and Weber, using topological reasoning, that exceptional members can be characterised by their Euler characteristics. We present a combinatorial argument giving a corresponding result in characteristic p. We first treat pencils with no base points, and then reduce the remaining case to this.

2000 Mathematical Subject Classification: primary 14H20; secondary 14D05, 14E22, 14F20.

We consider cyclic graphs, that is, graphs with cyclic orders at the vertices, corresponding to 2-cell embeddings of graphs into orientable surfaces, or combinatorial maps. We construct a three variable polynomial invariant of these objects, the cyclic graph polynomial, which has many of the useful properties of the Tutte polynomial. Although the cyclic graph polynomial generalizes the Tutte polynomial, its definition is very different, and it depends on the embedding in an essential way. 2000 Mathematical Subject Classification: 05C10.

The controlled wild algebra is introduced, a covering criterion for a finite-dimensional algebra to be controlled wild is given, and this criterion is applied to the algebras with radical square zero, algebras with zero relations, local algebras and finite p-group algebras.

2000 Mathematical Subject Classification: 16G20, 16G60.

Suppose $r=p^b$, where $p$ is a prime. Let $V$ be an $n$-dimensional ${\rm GF}(r)$-space and $G$ a subgroup of ${\rm A\Gamma L}(V) \cong {\rm A\Gamma L}(n,r)$ containing all translations and acting primitively on the set of vectors in $V$. Denote by $G_0$ the stabilizer in $G$ of the zero vector, so that $G_0 \le {\rm \Gamma L}(V) \cong {\rm \Gamma L}(n,r)$ and $G$ is the semidirect product of $V$ and $G_0$. Suppose that the generalized Fitting subgroup $F^*(G_0)$ of $G_0$ is an exceptional (twisted or untwisted, quasisimple) Chevalley group and that $\Gamma$ is a graph structure on $V$ on which $G$ acts primitively and distance transitively. The content of this paper is that then $G$ and $\Gamma$ are known. This result solves an open case in the outstanding problem of classifying all finite primitive distance-transitive groups. 2000 Mathematics Subject Classification: primary 20B25; secondary 05C25, 20Gxx, 05E30.

The authors sharpen a result of Baker and Harman (1995), showing that [x, x + x0.525] contains prime numbers for large x. An important step in the proof is the application of a theorem of Watt (1995) on a mean value containing the fourth power of the zeta function. 2000 Mathematical Subject Classification: 11N05.

The main part of the paper finds necessary conditions for solubility of a pencil of curves of genus 1, each of which is a 2-covering of an elliptic curve with at least one 2-division point. As in previous work, these are proved subject to Schinzel's Hypothesis and to the finiteness of the Tate-\u{S}afarevi\u{c} group of elliptic curves defined over a number field. It thus generalizes earlier work of Colliot-Thélène, Skorobogatov and the second author.

The final section gives necessary conditions (though of a rather ugly nature) for the solubility of a Del Pezzo surface of degree 4.

2000 Mathematical Subject Classification: 11D25.

Let $V$ be a vector space over the division ring $D$ of infinite dimension. We study locally finite, primitive groups $G$ of finitary linear automorphisms of $V$. We show that the derived group $G'$ of $G$ is infinite, simple, and lies in every non-trivial normal subgroup of $G$, and that $G' \le G \le {\rm Aut\,}G'$. Moreover if ${\rm char\,}D = 0$, then $G$ is either the finitary symmetric group or the alternating group on some infinite set. If $D$ is commutative, that is, if $D$ is a field, then all these results are known and are the consequence of the collective work of a number of people: in particular (in alphabetical order) V.~V.~Belyaev, J.~I.~Hall, F.~Leinen, U.~Meierfrankenfeld, R.~E.~Phillips, O.~Puglisi, A.~Radford and quite probably others. 2000 Mathematics Subject Classification: 20H25, 20H20, 20F50.

Hurwitz numbers, which count certain covers of the projective line (or, equivalently, factorizations of permutations into transpositions), have been extensively studied for over a century. The Gromov-Witten potential F of a point, the generating series for descendent integrals on the moduli space of curves, is a central object of study in Gromov-Witten theory. We define a slightly enriched Gromov-Witten potential G (including integrals involving one ‘$\lambda$-class’), and show that, after a non-trivial change of variables, G = H in positive genus, where H is a generating series for Hurwitz numbers. We prove a conjecture of Goulden and Jackson on higher genus Hurwitz numbers, which turns out to be an analogue of a genus expansion ansatz of Itzykson and Zuber. As consequences, we have new combinatorial constraints on F, and a much more direct proof of the ansatz of Itzykson and Zuber. We can produce recursions and explicit formulas for Hurwitz numbers; the algorithm presented proves all such recursions. As examples we present surprisingly simple new recursions in genus 0 to 3. Similar recursions should exist for all genera. As we expect this paper also to be of interest to combinatorialists, we have tried to make it as self-contained as possible, including reviewing some results and definitions well known in algebraic and symplectic geometry, and mathematical physics. 2000 Mathematical Subject Classification: primary 14H10, 81T40; secondary 05C30, 58D29.

Let $\Phi=(f,g):({\mathbb C}^{n+1},{\bf 0}) \to({\mathbb C}^2,{\bf 0})$ be a pair of holomorphic germs with no blowing up in codimension 0. (Two examples are the following: $\Phi$ defines an isolated complete intersection singularity; $g=\ell^N$ where $\ell$ is a generic linear form with respect to $f$ and $N>0$.) We study how the Milnor fibrations of the germs $\varphi_{(\alpha:\beta)}=\alpha g-\beta f$ are related to each other when $(\alpha:\beta)$ varies in ${\mathbb P}^1$. More precisely, we construct isotopic subfibrations or subfibres of the Milnor fibrations of any two such germs. The proofs are based on the precise study of the subdiscs of complex lines meeting a fixed complex plane curve germ transversally, generalizing Lê's work on the Cerf diagram.

2000 Mathematical Subject Classification: 32S55, 32S15, 32S30.

In an earlier work, the second author proved a general formula for the equivariant Poincaré polynomial of a linear transformation g which normalises a unitary reflection group G, acting on the cohomology of the corresponding hyperplane complement. This formula involves a certain function (called a Z-function below) on the centraliser CG(g), which was proved to exist only in certain cases, for example, when g is a reflection, or is G-regular, or when the centraliser is cyclic. In this work we prove the existence of Z-functions in full generality. Applications include reduction and product formulae for the equivariant Poincaré polynomials. The method is to study the poset L(CG(g)) of subspaces which are fixed points of elements of CG(g). We show that this poset has Euler characteristic 1, which is the key property required for the definition of a Z-function. The fact about the Euler characteristic in turn follows from the ‘join-atom’ property of L(CG(g)), which asserts that if {X1,...,Xk} is any set of elements of L(CG(g)) which are maximal (set theoretically) then their setwise intersection $\bigcap_{i=1}^kX_i$ lies in L(CG(g)). 2000 Mathematical Subject Classification: primary 14R20, 55R80; secondary 20C33, 20G40.

Lattices in the 3-dimensional Lie group Sol are shown to be not asynchronously automatic. The proof consists of a combination of geometry and logical complexity arguments. 2000 Mathematics Subject Classification: 20F10, 20F16.

We consider a Hamiltonian setup $(\mathcal M,\omega,H,\mathfrak L,\Gamma,\mathcal P)$, where $(\mathcal M,\omega)$ is a symplectic manifold, $\mathfrak L$ is a distribution of Lagrangian subspaces in $\mathcal M$, $\mathcal P$ is a Lagrangian submanifold of $\mathcal M$, $H$ is a smooth time-dependent Hamiltonian function on $\mathcal M$, and $\Gamma:[a,b]\to\mathcal M$ is an integral curve of the Hamiltonian flow $\vec H$ starting at $\mathcal P$. We do not require any convexity property of the Hamiltonian function $H$. Under the assumption that $\Gamma(b)$ is not $\mathcal P$-focal, we introduce the Maslov index $\mathrm i_{\mathrm{maslov}}(\Gamma)$ of $\Gamma$ given in terms of the first relative homology group of the Lagrangian Grassmannian; under generic circumstances $\mathrm i_{\mathrm{maslov}}(\Gamma)$ is computed as a sort of algebraic count of the $\mathcal P$-focal points along $\Gamma$. We prove the following version of the Index Theorem: under suitable hypotheses, the Morse index of the Lagrangian action functional restricted to suitable variations of $\Gamma$ is equal to the sum of $\mathrm i_{\mathrm{maslov}}(\Gamma)$ and a convexity term of the Hamiltonian $H$ relative to the submanifold $\mathcal P$. When the result is applied to the case of the cotangent bundle $\mathcal M=TM^*$ of a semi-Riemannian manifold $(M,g)$ and to the geodesic Hamiltonian $H(q,p)=\frac12 g^{-1}(p,p)$, we obtain a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics with variable endpoints in Riemannian geometry.

2000 Mathematical Subject Classification: 37J05, 53C22, 53C50, 53D12, 70H05.

We prove that, if $M > 4(1+2\sqrt{3})$ and $\varepsilon > 0$, if $\mathcal{V}$ and $\mathcal{W}$ are complex JBW*-triples (with preduals $\mathcal{V}_{*}$ and $\mathcal{W}_{*}$, respectively), and if $U$ is a separately weak*-continuous bilinear form on $\mathcal{V} \times \mathcal{W}$, then there exist norm-one functionals $\varphi_{1},\varphi_{2}\in \mathcal{V}_{*}$ and $\psi_{1},\psi_{2}\in \mathcal{W}_{*}$ satisfying |U(x,y)| \leq M \,\|U\| ( \|x\|_{\varphi_{2}}^{2} + \varepsilon^{2} \, \|x\|_{\varphi_{1}}^{2} )^{\frac 12} ( \|y\|_{\psi_{2}}^{2} + \varepsilon^{2} \, \|y\|_{\psi_{1}}^{2} )^{\frac 12} for all $(x,y)\in \mathcal{V} \times \mathcal{W}$. Here, for a norm-one functional $\varphi$ on a complex JB*-triple $\mathcal{V}$, $\|\cdot\|_{\varphi}$ stands for the prehilbertian seminorm on $\mathcal{V}$ associated to $\varphi$ given by $\|x\|_{\varphi}^{2} := \varphi \{x,x,z\}$ for all $x\in \mathcal{W}$, where $z\in \mathcal{V}^{**}$ satisfies $\varphi (z) = \|z\|=1$. We arrive at this form of ‘Grothendieck's inequality’ through results of C.-H. Chu, B. Iochum, and G. Loupias, and an amended version of the ‘little Grothendieck's inequality’ for complex JB*-triples due to T. Barton and Y. Friedman. We also obtain extensions of these results to the setting of real JB*-triples. 2000 Mathematical Subject Classification: 17C65, 46K70, 46L05, 46L10, 46L70.

Let $F$ be a non-archimedean local field of residual characteristic different from $2$, and let $G$ be a unitary, symplectic or orthogonal group, considered as the fixed point subgroup in $\tilde G={\rm GL}(N,F)$ of an involution $\sigma$. We generalize the notion of a {\it simple character} for $\tilde G$, which was introduced by Bushnell and Kutzko [Annals of Mathematics Studies 129 (Princeton University Press, 1993)], to define semisimple characters. Given a semisimple character $\theta$ for $\tilde G$ fixed by $\sigma$, we transfer it to a character $\theta_{-}$ for $G$ and calculate its intertwining. If the torus associated to $\theta_{-}$ is maximal compact, we obtain supercuspidal representations of $G$, which are new if the torus is split only over a wildly ramified extension. 2000 Mathematics Subject Classification: 22E50.

{\frenchspacing Nous \'etudions l'homotopie d'une vari\'et\'e quasi-projective dans un espace projectif complexe selon la m\'ethode de Lefschetz, c'est-\`a-dire en consid\'erant ses sections par les hyperplans d'un pinceau (tomographie). En particulier, nous aboutissons \`a un th\'eor\`eme du type de Lefschetz qui g\'en\'eralise dans une certaine direction les meilleurs r\'esultats connus dus \`a Hamm, L\^e, Goresky et MacPherson. Ce th\'eor\`eme est d\'emontr\'e par r\'ecurrence sur la dimension de l'espace projectif ambiant \`a partir d'un th\'eor\`eme sur les pinceaux d'axe g\'en\'erique qui constitue le r\'esultat principal de l'article. Ce dernier compare la topologie de la vari\'et\'e \`a celle de sa section par un hyperplan g\'en\'erique du pinceau sur la base des comparaisons (section hyperplane g\'en\'erique -- section par l'axe du pinceau) et (sections hyperplanes exceptionnelles -- section par l'axe); l'incidence des singularit\'es est mesur\'ee par un invariant appel\'e `profondeur homotopique rectifi\'ee globale' (analogue global de la notion de profondeur homotopique rectifi\'ee de Grothendieck).} \vspace{6mm} \noindent We study the homotopy of a quasi-projective variety in a complex projective space following Lefschetz's method, that is, by considering its sections by the hyperplanes of a pencil (tomography). Specifically, we obtain a theorem of Lefschetz type which generalizes in a certain direction the best-known results due to Hamm, L\^e, Goresky and MacPherson. This theorem is proved by induction on the dimension of the ambient projective space with the help of a theorem on pencils with generic axis which is the main result of the paper. The latter compares the topology of the variety with that of its section by a generic hyperplane of the pencil, on the basis of the following comparisons: section by a generic hyperplane with section by the axis of the pencil; and sections by the exceptional hyperplanes with section by the axis. The effect of the singularities is measured by an invariant called `global rectified homotopical depth' (a global analogue of the notion of rectified homotopical depth of Grothendieck). E-mail: eyral@cmi.univ-mrs.fr 2000 Mathematics Subject Classification: 32S50, 14F35, 14F17.

The Hardy operator $T_a$ on a tree $\Gamma$ is defined by \[ (T_af)(x):=v(x) \int^x_a f(t)u(t)\,dt \quad \mbox{for } a, x\in \Gamma. \] Properties of $T_a$ as a map from $L^p(\Gamma)$ into itself are established for $1\le p \le \infty$. The main result is that, with appropriate assumptions on $u$ and $v$, the approximation numbers $a_n(T_a)$ of $T_a$ satisfy \begin{equation*} \tag{$*$} \lim_{n\rightarrow \infty} na_n(T_a) = \alpha_p\int_{\Gamma}|uv|\,dt \end{equation*} for a specified constant $\alpha_p$ and $1 p<\infty$. This extends results of Naimark, Newman and Solomyak for $p=2$. Hitherto, for $p\neq 2$, $(*)$ was unknown even when $\Gamma$ is an interval. Also, upper and lower estimates for the $l^q$ and weak-$l^q$ norms of $\{a_n(T_a)\}$ are determined.

2000 Mathematical Subject Classification: 47G10, 47B10.

We prove that abelian subgroups of fundamental groups of Haken 3-manifolds and closed hyperbolic n-orbifolds are separable. 2000 Mathematical Subject Classification: 20E26, 51M10, 57M05.

We relate the brace products of a fibration with section to the differentials in its Serre spectral sequence. In the particular case of free loop fibrations, we establish a link between these differentials and Browder operations in the fiber. Applications and several calculations (for the particular cases of spheres and wedges of spheres) are given.

2000 Mathematical Subject Classification: 55P35, 55Q15, 55R05, 55R20.

This paper considers various extensions of results of Johnson and Williams and of Fong on the range inclusion of normal derivations. Let $C_p$, with $p \in [1,\infty)$, be the Schatten ideals of compact operators on a Hilbert space $H$ with norms $|\ |_p$, $C_\infty$ be the ideal of all compact operators, and $C_b$ the algebra $B(H)$ of all bounded operators. Any $S \in B(H)$ defines a bounded derivation $\delta_S$ on all $C_p$: $\delta_S(X) = SX - XS$, for $X \in C_p$. Johnson and Williams proved that, for normal $S$, the inclusion of ranges $\delta_T(C_b) \subseteq \delta_S (C_b)$ implies $T = g(S)$, where $g$ is a Lipschitz, differentiable function on $\sigma(S)$. Fong showed that the condition $T = g(S)$ is equivalent to the range inclusion $\delta_T(C_1) \subseteq \delta_S (C_b)$. This paper studies the range inclusion \begin{equation} \delta_T(C_p) \subseteq \delta_S (C_p) \end{equation} for normal $S$ and $p \in [1,\infty] \cup b$, and the classes of functions for which $T = g(S)$. Set $$ p_- = \min\bigg(p, \frac{p}{p - 1}\bigg),\quad p_+ = \max\bigg(p, \frac{p}{p - 1}\bigg), \quad \text{for } p \in (1, \infty), $$ and $$ p_- = 1,\quad p_+ = b, \quad \text{for } p \in \{1,\infty,b\}. $$ This paper shows that condition (1) implies: \begin{enumerate} \item[(i)] the range inclusions $\delta_T(C_r) \subseteq \delta_S (C_r)$, for $r \in [p_-,p_+]$; \item[(ii)] that there exists $D > 0$ such that $|\delta_T(X)|_p \leq D|\delta_S (X)|_p$, for $X \in B(H)$ ($|X|_p = \infty$ if $X \notin C_p$); \item[(iii)]the range inclusions $\delta_{g(A)}(C_p) \subseteq \delta_A (C_p)$ for any normal operator $A$ with $\sigma(A) \subseteq \sigma(S)$. \end{enumerate} It establishes that (1) implies that the function $g$ (in $T = g(S)$) is $C_p$-Lipschitzian on $\sigma(S)$, that is, there is $D > 0$ such that $|g(A) - g(B)|_p \leq D|A - B|_p$ for all normal $A$ and $B$ with spectra in $\sigma(S)$. Conversely, it is proved that, for any selfadjoint $S$ and $C_p$-Lipschitz function $g$ on $\sigma(S)$, $\delta_{g(S)}(C_p) \subseteq \delta_S(C_p)$. The paper also extends the above results of Johnson and Williams to bounded derivations of C$^*$-algebras. 2000 Mathematics Subject Classification: 46L57, 47B47, 58C07.