A vector $x$ in a Banach space $\cal B$ is called hypercyclic for a bounded linear operator $T: \cal B \rightarrow \cal B $ if the orbit $\{T^n x : n \geq 1 \}$ is dense in $\cal B$. We prove that if $T$ is hereditarily hypercyclic and its essential spectrum meets the closed unit disk, then there is an infinite-dimensional closed subspace consisting, except for zero, entirely of hypercyclic vectors for $T$. The converse is also true even if $T$ is not hereditarily hypercyclic. In this way, we improve and extend to Banach spaces a recent result for Hilbert spaces. We also investigate the corresponding problem for supercyclic operators. In this case we obtain a description of the norm closure of the class of all supercyclic operators that have an infinite-dimensional closed subspace of hypercyclic vectors. Moreover, for certain kinds of supercyclic operators we can characterize when they have an infinite-dimensional closed subspace of supercyclic vectors. 1991 Mathematics Subject Classification: 47B38, 47B99.

For $K$ a connected finite complex and $G$ a compact connected Lie group, a finiteness result is proved for gauge groups ${\mathcal G}(P)$ of principal $G$-bundles $P$ over $K$: as $P$ ranges over all principal $G$-bundles with base $K$, the number of homotopy types of ${\mathcal G}(P)$ is finite; indeed this remains true when these gauge groups are classified by $H$-equivalence, that is, homotopy equivalences which respect multiplication up to homotopy. A case study is given for $K = S^4$, $G = \text{SU}(2)$: there are eighteen $H$-equivalence classes of gauge group in this case. These questions are studied via fibre homotopy theory of bundles of groups; the calculations in the case study involve $K$-theories and $e$-invariants. 1991 Mathematics Subject Classification: 54C35, 55P15, 55R10.

We study finite and infinite entangled graphsin the bond percolation process in three dimensionswith density $p$.After a discussion of the relevant definitions,the entanglement critical probabilities are defined.The size of the maximal entangled graph at the origin is studied for small $p$, and it is shown that this graph has radius whose tail decays at least as fast as $\exp(-\alpha n/\log n)$; indeed, the logarithm may be replaced by any iterate of logarithmfor an appropriate positive constant $\alpha$. We explore the question of almost sure uniqueness of the infinite maximal open entangled graph when $p$ is large, and we establish two relevant theorems. We make several conjectures concerning the properties of entangled graphs in percolation. http://www.statslab.cam.ac.uk/$\sim$grg/ 1991 Mathematics Subject Classification: primary 60K35; secondary 05C10, 57M25, 82B41, 82B43, 82D60.

We classify the finite-dimensional quotient Hopf algebras of the deformed algebra of functions of the general linear group (over an algebraically closed field of zero characteristic). This gives an interesting class of Hopf algebras if the deformation parameter is a root of unity (of odd order). We investigate the properties of these Hopf algebras and construct a new counterexample to Kaplansky's tenth conjecture. E-mail: [email protected] 1991 Mathematics Subject Classification: 81R50, 16W30, 17B37.

The variation of the Hodge structure of a Calabi--Yau 3-fold induces a canonical K\"ahler metric on its Kuranishi moduli space, known as the Weil--Petersson metric. Similarly, {\it special} pseudo-K\"ahler manifolds correspond to certain (abstract) variations of Hodge structure which generalize the above example. We give the classification of homogeneous special pseudo-K\"ahler manifolds of semisimple groups with compact stabilizer. 1991 Mathematics Subject Classification: 53C30, 53C25.

A finite Borel measure $\mu$ on $\mathbb R^d$ is called R-O varying with index $F$ if there exist a $\operatorname{GL}(\mathbb R^d)$-valued function $f$ varying regularly with index $(-F)$, an increasing function $k : (0,\infty)\to (0,\infty)$ with $k(t)\to\infty$ and $k(t+1)/k(t)\to c\geq 1$ as $t\to\infty$, and a $\sigma$-finite measure $\phi$ on $\mathbb R^d\setminus\{0\}$ such that \[k(t)\cdot(f(k(t))\mu)\to\phi\quad \text{as $t\to\infty$.}\] R-O varying measures generalize regularly varying measures introduced by Meerschaert (see M.~M.~Meerschaert, `Regular variation in $\mathbb R^k$', {\it Proc.\ Amer.\ Math.\ Soc.} 102 (1988) 341--348) and have numerous applications in limit theorems for probability measures. For an R-O varying measure $\mu$ and $-\infty<\infty$ let \begin{equation*} \begin{split} V_a(t,\theta) &= \int_{|\langle x,\theta\rangle|>t} |\langle x,\theta\rangle|^a\,d\mu(x),\\ U_b(t,\theta) &= \int_{|\langle x,\theta\rangle|\leq t} |\langle x,\theta\rangle|^b\,d\mu(x) \end{split} \end{equation*} denote the tail- and truncated moment functions of $\mu$ in the direction $\|\theta\|=1$. The purpose of this paper is to show that R-O variation of a measure implies sharp bounds on the growth rate of the tail- and truncated moment functions depending on the real parts of the eigenvalues of the index $F$ along a compact set of directions. Furthermore, bounds on the ratio of these functions for certain values of $a$ and $b$ are obtained. 1991 Mathematics Subject Classification: 60B10, 28C15.