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.

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.

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.

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.

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: [email protected] 2000 Mathematics Subject Classification: 32S50, 14F35, 14F17.

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.

We introduce the notion of tracial topological rank for ${\rm C}^*$-algebras. In the commutative case, this notion coincides with the covering dimension. Inductive limits of ${\rm C}^*$-algebras of the form $PM_n(C(X))P$, where $X$ is a compact metric space with ${\rm dim\,} X\le k$, and $P$ is a projection in $M_n(C(X))$, have tracial topological rank no more than $k$. Non-nuclear ${\rm C}^*$-algebras can have small tracial topological rank. It is shown that if $A$ is a simple unital ${\rm C}^*$-algebra with tracial topological rank $k$ ($<\infty$), then \begin{enumerate} \item[(i)] $A$ is quasidiagonal, \item[(ii)] $A$ has stable rank $1$, \item[(iii)] $A$ has weakly unperforated $K_0(A)$, \item[(iv)] $A$ has the following Fundamental Comparability of Blackadar: if $p,q\in A$ are two projections with $\tau(p)<\tau(q)$ for all tracial states $\tau$ on $A$, then $p\preceq q$. 2000 Mathematics Subject Classification: 46L05, 46L35.

We determine the relationship between the multiplicities of the zeros of certain rational functions defined on the ${\rm SL}_2(\mathbb C)$ and ${\rm PSL}_2(\mathbb C)$ character varieties of knot exteriors. This leads to a formula for the Culler--Shalen seminorms associated to small Seifert Dehn fillings of such manifolds. 2000 Mathematics Subject Classification: 57M05, 57M27, 57R65.