It is shown that the system of two additive equations a_1 x_1^k + \ldots + a_s x_s^k = b_1 x_1^k + \ldots + b_s x_s^k =0 where $k \ge 2$ and $a_j$, $b_j$ are any given integers, has non-trivial solutions in all $p$-adic fields provided only that $s > 8k^2$. The constant 8 can be reduced when $k$ is not a power of 2. It is expected, in accordance with a classical conjecture of Artin, that the bound $8k^2$ can be replaced by $2k^2$.

2000 Mathematical Subject Classification:11D72.

Let $\varphi:A\to B$ be a homomorphism of finite-dimensional algebras over an algebraically closed field and $\varphi^{(c)}:{\rm mod}_B^c\to{\rm mod}_A^c$ the induced morphism of the associated module schemes for any integer $c\geq 1$. We prove that if the induced functor ${\rm mod\,} B\to{\rm mod\,} A$ is hom-controlled then the restriction of $\varphi^{(c)}$ to any connected component of ${\rm mod}_B^c$ is a composition of a smooth morphism followed by an immersion. Some new results on the types of singularities in the orbit closures of module schemes are also proved.

2000 Mathematical Subject Classification: 14B05, 14L30, 16G10.

Let V be a pencil of curves in ${\bf P}^2$ with one place at infinity, and $X \longrightarrow {\bf P}^2$ the minimal composition of point blow-ups eliminating its base locus. We study the cone of curves and the cones of numerically effective and globally generated line bundles on X. It is proved that all of these cones are regular. In particular, this result provides a new class of rational projective surfaces with a rational polyhedral cone of curves. The surfaces in this class have non-numerically effective anticanonical sheaf if the pencil is neither rational nor elliptic. An application is a global version on X of Zariski's unique factorization theorem for complete ideals. We also define invariants of the semigroup of globally generated line bundles on X depending only on the topology of V at infinity.

2000 Mathematical Subject Classification: primary 14C20; secondary 14E05.

We describe the current state of progress on the maximal subgroup problem for the Monster sporadic simple group. Any unknown maximal subgroup is an almost simple group whose socle is in one of 19 specified isomorphism classes.

2000 Mathematical Subject Classification:20D08.

This paper provides a strengthening of the theorems of small cancellation theory. It is proven that disc diagrams contain 'fans' of consecutive 2-cells along their boundaries. The size of these fans is linked to the strength of the small cancellation conditions satisfied by the diagram. A classification result is proven for disc diagrams satisfying small cancellation conditions. Any disc diagram either contains three fans along its boundary, or it is a ladder, or it is a wheel. Similar statements are proven for annular diagrams.

2000 Mathematical Subject Classification: 20F06.

By studying the braid group action on Milnor's construction of the 1-sphere, we show that the general higher homotopy group of the 3-sphere is the fixed set of the pure braid group action on certain combinatorially described groups. This establishes a relation between the braid groups and the homotopy groups of the sphere.

2000Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10.

A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, that is, subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analysed by representation-theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.

2000 Mathematical Subject Classification: 16G20, 14L30, 17B37.

Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.

2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

We give several sufficient conditions on a pair of Banach spaces X and Y under which each Lipschitz mapping from a domain in X to Y has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability. Most of these conditions are stated in terms of the moduli of asymptotic smoothness and convexity, notions which have appeared in the literature under a variety of names. We prove, for example, that for $\infty > r > p \ge 1$, every Lipschitz mapping from a domain in an $\ell_r$-sum of finite-dimensional spaces into an $\ell_p$-sum of finite-dimensional spaces has, for every $\epsilon > 0$, a point of $\epsilon$-Fréchet differentiability, and that every Lipschitz mapping from an asymptotically uniformly smooth space to a finite-dimensional space has such points. The latter result improves, with a simpler proof, an earlier result of the second and third authors. We also survey some of the known results on the notions of asymptotic smoothness and convexity, prove some new properties, and present some new proofs of existing results.

2000 Mathematical Subject Classification: 46G05, 46T20.

We construct a Banach space X whose algebra of bounded linear operators has K0-group the additive group of rational numbers. We achieve this using a Gowers-Maurey space.

2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10