A bounded operator $T$ acting on a Banach space $\cal B$ is said to be supercyclic if there is a vector $x\,{\in}\,\cal B$ such that the projective orbit$\{\lambda T^n x\,{:}\,n\,{\geq}\,0\text{ and } \lambda\,{\in}\,\Bbb C \}$ is dense in $\cal B$. Examples of supercyclic operators are hypercyclic operators, in which the orbit itself is dense without the help of scalar multiples. Supercyclic operators are, in turn, a special case of cyclic operators. An operator is called cyclic if the linear span of the orbit of some vector is dense in the underlying space. This survey describes some recent results on linear subspaces in which all elements, except the zero vector, are supercyclic for a given supercyclic operator.

The projective tensor product $\ell_2\,{\hat{\otimes}}\,X$ of $\ell_2$ with any Banach space $X$ sits inside the space ${\rm Rad}(X)$ of all almost unconditionally summable sequences in $X$. If $X$ is of cotype 2 and $u: X \longrightarrow Y$ is 2-summing, then $u$ takes ${\rm Rad}(X)$ into $\ell_2\,{\hat{\otimes}}\,Y$. Consequently, if $X$ is of cotype 2, then every operator from $X$ to $\ell_2$ is 1-summing if and only if $\ell_1\,{\check{\otimes}}\,X \subseteq \ell_2\,{\hat{\otimes}}\,X$. In this case, each 2-summing operator from $\ell_2$ to $X$ is nuclear, and $X$ does not have non-trivial type provided that ${\rm dim}\,X = \infty$.

Let $2\le p <\infty$, and let $X$ be a complex Banach space. It is shown that $X$ is $p$-uniformly PL-convex if and only if there exists $\lambda >0$ such that $ \|f\|_{H^p(X)}\,{\ge}\, (\|f(0)\|^p+\lambda\int_{\mathbb D} (1-|z|^2)^{p-1}\|f^{\prime}(z)\|^p\,dA(z))^{1/p}$, for all $f\in H^p(X)$. Applications to embeddings between vector-valued BMOA spaces defined via Poisson integral or Carleson measures are provided.

A sharp function estimate is established for multilinear singular integral operators. As applications, some weighted norm inequalities are obtained for such operators.This research was supported by NSF of Henan Province and NNSF (No. 10271015) of China.

Assuming that the integrated density of states of a Schrödinger operator admits a high-energy asymptotic expansion, the authors give explicit formulae for the coefficients of this expansion in terms of the heat invariants.

The problem of classifying finitely generated groups up to virtual isomorphism is strictly harder than that of classifying them up to isomorphism, in the sense that there is a Borel reduction from the isomorphism problem to the virtual isomorphism problem, but not vice versa.Research partially supported by NSF Grants.

The automorphism group of a finitely generated free group is the normal closure of a single element of order 2. If $m<n$, then a homomorphism ${\rm Aut}(F_n)\to {\rm Aut}(F_m)$ can have image of cardinality at most 2. More generally, this is true of homomorphisms from ${\rm Aut}(F_n)$ to any group that does not contain an isomorphic image of the symmetric group $S_{n+1}$. Strong restrictions are also obtained on maps to groups that do not contain a copy of $W_n=({\mathbb Z}/2)^n\rtimes S_{n}$, or of ${\mathbb Z}^{n-1}$. These results place constraints on how ${\rm Aut}(F_n)$ can act. For example, if $n\ge 3$, any action of ${\rm Aut}(F_n)$ on the circle (by homeomorphisms) factors through ${\rm det} : {\rm Aut}(F_n)\to{\mathbb Z}_2$.The research of the first author was supported by an EPSRC Advanced Fellowship. The research of the second author is supported in part by NSF grant DMS-9307313.

In this paper, Conway's group, $\Co_3$, is characterized among simple groups of even type with $e(G)=3$, by a restriction on the $p$-local structure for some odd prime $p$ for which $m_{2,p}(G) = 3$.

Co-Hopfian finitely generated torsion-free nilpotent groups are characterized in this paper in terms of their Lie algebra automorphisms. Many examples of such groups are also constructed.Partially supported by NSF Grant # DMS-0203979.

It is shown that a function whose critical locus is an isolated complete intersection singularity of arbitrary dimension, and that has finite codimension (in the sense of R. Pellikaan, Proc. London Math. Soc. (3) 57 (1998) 357–382) with respect to the ideal defining the isolated complete intersection singularity, can be approximated by a function whose critical locus is a finite number of Morse points together with the Milnor fibre of the isolated complete intersection singularity, having there well-known types of singularities.

Let $E$ be an elliptic curve over a finitely generated infinite field $K$. Let $K^{\rm s}$ denote a separable closure of $K$, $\sigma$ an element of the Galois group $G_{K}\,{=}\,\hbox{Gal}(K^{\rm s}/K)$, and $K^{\rm s}(\sigma)$ the invariant subfield of $K^{\rm s}$. If the characteristic of $K$ is not 2 and $\sigma$ belongs to a suitable open subgroup of $G_K$, then $E(K^{\rm s}(\sigma))$ has infinite rank.Partially supported by the Sloan Foundation and by NSF Grant DMS 97-27553.

This paper provides a method of generating infinitely many integer solutions of the simultaneous equations $x_1^r+x_2^r+x_3 ^r=y_1^r+y_2^r+y_3^r$ where $r$ is equal to $2$, $3$ or $4$. Three numerical solutions of this system of equations have also been given in this paper.

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic. As an application, it is shown that an open subset of the real hyperbolic plane ${\mathbb R}H^2$ cannot be minimally immersed into the Euclidean space. As another application, a proof is given that if an irreducible Kähler manifold is minimally immersed in a Euclidean space, then its restricted holonomy group must be $U(n)$, where $n = \dim_{\mathbb C}M$.Supported by an EPSRC Grant GR/R69174.

Crispin St John Alvah Nash-Williams was born in Cardiff on 19 December 1932. His father was Keeper of Archaeology at the National Museum of Wales and also Senior Lecturer in Archaeology at University College, Cardiff. War broke out when Crispin was six; his father joined the army and his consequent absence changed the family's life.