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.

Let $G$ be a finite group. The strong symmetric genus$\sigma ^{0}(G)$ is the minimum genus of any Riemann surface on which $G$ acts, preserving orientation. For any non-negative integer $g$, there is at least one group of strong symmetric genus $g$. For $g \neq 2$, one such group has the form $Z_{k} \times D_{n}$ for some $k$ and $n$.

Algebraic solutions of certain Painlevé VI equations are produced by solving linear systems with monodromy contained in the octahedral subgroup of $\SO(3)$. The method uses the algebraic geometry of special plane curves, and makes contact with some classical geometrical problems.

It is proved that the automorphism group of a countable structure cannot be a free uncountable group. The idea is that instead of proving that every countable set of equations of a certain form has a solution, it is proved that this holds for a co-meagre family of appropriate countable sets of equations.

An extension is obtained to certain maps into full flag manifolds of compact simple Lie groups of the classical Plücker formulae for holomorphic curves in complex projective space.

A proof is given of the following theorem, which characterizes full automorphism groups of ordered abelian groups: a group$H$is the automorphism group of some ordered abelian group if and only if$H$is right-orderable.

An $\varepsilon$ -entropy theoretical solution is given to an analytic function-theoretical version of the 13th problem formulated by D. Hilbert.

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 ${\rm dim}\,X = \infty$.

Holomorphic principal bundles over a compact Riemann surface X that admits a flat connection are considered. A holomorphic G-bundle over X, where G is a connected semisimple linear algebraic group over ${\Bbb C}$, admits a flat connection if and only if the adjoint vector bundle admits one. More generally, for a complex reductive group G, the necessary and sufficient condition on a G-bundle to admit a flat connection is described. This simplifies the criterion obtained by the authors and given in Math. Ann. 322 (2002) 333–346.

Ramanujan presented (without proof) the following double inequality for the gamma function:\begin{equation}\sqrt{\pi}\Bigl(\frac{x}{e}\Bigr)^x\Bigl[8x^3+4x^2+x+\frac{1}{100}\Bigr]^{1/6} < \Gamma(x+1)< \sqrt{\pi}\Bigl(\frac{x}{e}\Bigr)^x\Bigl[8x^3+4x^2+x+\frac{1}{30}\Bigr]^{1/6} \quad{(x\geq 0)}.\end{equation}Recently, Karatsuba established that these inequalities hold for $x\geq 1$. We show that this can be slightly improved: the inequalities hold for all $x\geq 0$, even if we replace $1/100$ by \[\alpha= \min_{0.6\leq x\leq 0.7}f(x)=0.0100450\dots,\]where $f(x)=(1/\pi)^3[\Gamma(x+1)(e/x)^x]^6-8x^3-4x^2-x$. Moreover, $\alpha$ and $1/30$ are the best possible constant terms.

A complete characterization of the categorical quotients of $({\mathbb P}^1)^n$ by the diagonal action of $\SL(2, {\mathbb C})$ with respect to any polarization is given by M. Polito, in ‘$\SL(2, {\mathbb C})$-quotients de $({\mathbb P}^1)^n$’, C. R. Acad. Sci. Paris Sér. I 321 (1995) 1577–1582. In this paper, these categorical quotients are obtained by certain linear systems on ${\mathbb P}^{n-3}$?>, depending on the given polarization.

In this paper, a geometric interpretation is provided of a new rational Landen transformation. The convergence of its iterates is also established.

Let $X$ be a non-singular real algebraic curve in ${\rm C\!\!\!I}P^2$ of even degree. In this paper a refinement is proved of a theorem of Kharlamov about ($M - 2$)-curves that are invariants under the projective involution. In particular, if the ($M - 2$)-symmetric curve $X$ satisfies the Arnold congruence, then either $X$ or its twin is a separating curve.

Let $C$ be a germ at $O \in {\mathbb R}^2$ of a real analytic plane curve, and $C^{\mathbb C}$ its complexification; let $C_t \subset B_{\varepsilon}$ be a fiber of a real smooth deformation of $C$ in the ball $B_{\varepsilon} = B(O, \varepsilon)$. The following inequality is proved between the integrals of real curvature $k$ of $C_t$ and those of Gaussian curvature $K$ of $C_{t}^{\mathbb C}$: $$ 2 \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^\varepsilon} \vert k \vert \leq \lim_{\varepsilon, t \rightarrow 0} \int_{C_t^{\varepsilon {\mathbb C}}} \vert K \vert.$$ The sharpness of this inequality is proved in the case where $C$ is a real irreducible germ. Similar results are proved for an affine algebraic curve $C \subset {\mathbb R}^2$ of degree $d$.

Barbasch and Vogan showed that the Kazhdan–Lusztig cells of a finite Weyl group are compatible with parabolic subgroups. Their proof uses the known bridge between the theory of cells and the theory of primitive ideals. In this paper, an elementary, self-contained proof of this result is provided, which works for arbitrary Coxeter groups and Lusztig's general definition of cells (involving Iwahori–Hecke algebras with unequal parameters). The argument is based on a recent paper by Howlett and Yin.

It is shown that the classical result of Pringsheim on the convergence of complex continued fractions extends to all dimensions, both geometrically and as a result on continuedfractions whose coefficients lie in a Clifford algebra.

It is proved that in groups with all subgroups subnormal every nilpotent subgroup is contained in a normal nilpotent subgroup.

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.

It is shown that the mod $3$ cohomology of a $1$-connected, homotopy associative mod $3$$H$-space that is rationally equivalent to the Lie group $E_6$ is isomorphic to that of $E_6$ as an algebra. Moreover, it is shown that the mod $3$ cohomology of a nilpotent, homotopy-associative mod $3$$H$-space that is rationally equivalent to $E_6$, and whose fundamental group localized at $3$ is non-trivial, is isomorphic to that of the Lie group $\Ad E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra. It is also shown that the mod $3$ cohomology of the universal cover of such an $H$-space is isomorphic to that of $E_6$ as a Hopf algebra over the mod $3$ Steenrod algebra.

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.