This paper proves conditional existence results for non-trivial solutions of the equation

\begin{equation}\sum_{i=1}^{n}a_{i}X_{i}^{3}=0\quad (n=4\mbox{ or }5), \tag{$*$}\end{equation}

where the coefficients $a_{i}$ and the unknowns $X_{i}$ are taken to be rational integers.

No such results were previously known for $n\leq 6$.The proofs use elementary facts about the 3-descentprocedure for elliptic curves of the form $E_{A}: X^{3}+Y^{3}=AZ^{3}$.

Thus, when $n=4$, and the $a_{i}$ are each prime, and are all congruent to 2 modulo 3, it is shown that ($*$) will have non-trivial solutions,providing that the Selmer conjecture holds for the curves $E_{A}$.One may replace the Selmer conjecture by an appropriate form of the Generalized Riemann Hypothesis, when $n=5$ and the $a_{i}$ are againtaken to be primes, all congruent to 8 modulo 9. Finally, when $n=5$, one may require only that the $a_{i}$ be square-free and coprime to 3,providing one assumes both the Selmer conjecture and a special case of Schinzel's conjecture (on the representation of primes by cubicpolynomials).

1991 Mathematics Subject Classification: 11D25, 11G05, 14G05.

Let $M_{g,k}$ be the moduli space of flatSU(2)-connections on a compact Riemann surface$\Sigma_g$ of genus $g \ge 2$ punctured in apoint $p$ in a small open disc $D$, withholonomy $(-1)^k$ about $\partial D$.This paper gives an explicit decompositionof the rational intersection homologyof $M_{g,k}$ into irreducible representationsof the oriented mapping class group$\mbox{Map}^+(\Sigma_g)$ of $\Sigma_g$.

1991 Mathematics Subject Classification: primary 14D20; secondary 14H60, 55N33.

Engel-4 groups of exponent 5 are now known to be locally finite. In this paper we determine the orders of the largest finite groups of this kind on $m$ generators.

In the process we show that such a group has a subdirect decomposition into a group which is centre-by-Engel-3 and one which is nilpotent of class at most 10. We work via associated Lie algebras and Lie superalgebrasand make use of computer calculations.

1991 Mathematics Subject Classification: 20F45.

Schwarz's lemma asserts that analytic mappings from the unit disc into itself decrease hyperbolic distances. In this paper, inner functions which decrease hyperbolic distances as much as possible, when one approaches the unit circle, are constructed. Actually, it is shown that a quadratic condition governs the best decay of the hyperbolic derivative of an inner function. This is related to a result of L. Carleson on the existence of singular symmetric measures. As a consequence, some results on composition operators are obtained, bringing out the importance of the Bloch spaces in this connection. Another consequence is a uniform way of producing singular measures which are simultaneously symmetric and Kahane.

1991 Mathematics Subject Classification: primary 30D50; secondary 30D45, 26A30, 47B38.

Let $F: {\Bbb C}^n \to {\Bbb C}^n$ be a holomorphic map,$F^k$ be the $k$th iterate of $F$, and $p \in {\Bbb C}^n$be a periodic point of $F$ of period $k$. That is,$F^k(p) = p$, but for any positive integer $j$ with$j < k$, $F^j(p) \ne p$. If $p$ is hyperbolic, namelyif $DF^k(p)$ has no eigenvalue of modulus $1$, then it iswell known that the dynamical behaviour of $F$ is stablenear the periodic orbit $\Gamma = \{p, F(p),\ldots, F^{k-1}(p)\}$.But if $\Gamma$ is not hyperbolic, the dynamical behaviourof $F$ near $\Gamma$ may be very complicated and unstable.In this case, a very interesting bifurcational phenomenon mayoccur even though $\Gamma$ may be the only periodic orbit in someneighbourhood of $\Gamma$: for given $M \in {\Bbb N}\setminus \{1\}$,there may exist a $C^r$-arc $\{F_t: t\in [0,1]\}$(where $r \in {\Bbb N}$ or $r = \infty$)in the space ${\Bbb H}({\Bbb C}^n)$of holomorphic maps from ${\Bbb C}^n$ into ${\Bbb C}^n$,such that $F_0 =F$ and, for $t \in (0,1]$, $F_t$ has an$Mk$-periodic orbit $\Gamma_t$ with $d(\Gamma_t, \Gamma) =\sup_{p \in \Gamma_t}\inf_{q \in \Gamma} \|p - q\| \to 0$as $t \to 0$. The period thus increases by a factor$M$ under a $C^r$-small perturbation! If such an $F_t$ does exist,then $\Gamma$, as well as $p$, is said to be {\em $M$-tuplingbifurcational.} This definition is independent of~$r$.

For the above $F$, there may exist a $C^r$-arc $F^*_t$ in${\Bbb H}({\Bbb C}^n)$, with $t \in [0,1]$, such that$F^*_0 = F$ and, for $t \in (0,1]$, $F^*_t$ has two distinct$k$-periodic orbits $\Gamma_{t,1}$ and $\Gamma_{t,2}$ with$d(\Gamma_{t,i}, \Gamma) \to 0$ as $t \to 0$ for $i = 1,2$.If such an $F^*_t$ does exist, then $\Gamma$, as well as $p$,is said to be {\em $1$-tupling bifurcational.}

In recent decades, there have been many papers andremarkable results which deal withperiod doublingbifurcations of periodic orbits of parametrized maps.L. Block and D. Hart pointed out that period$M$-tuplingbifurcations cannot occur for $M > 2$ in the 1-dimensionalcase. There are examples showing that for any $M\in{\Bbb N}$, period $M$-tupling bifurcations can occurin higher-dimensional cases.

An $M$-tupling bifurcational periodic orbit as defined hereacts as a critical orbit which leads to period $M$-tuplingbifurcations in some parametrized maps. The main resultof this paper is the following.

{\sc Theorem.} {\em Let $k \in {\Bbb N}$ and $M \in {\Bbb N}$,and let $F: {\Bbb C}^2 \to {\Bbb C}^2$ be a holomorphic mapwith $k$-periodic point $p$. Then $p$ is $M$-tuplingbifurcational if and only if $DF^k(p)$ has a non-zeroperiodic point of period $M$.}

1991 Mathematics Subject Classification: 32H50, 58F14.

Consider the differential equation

$$\ddot{x} +n^2 x+h_L (x) =p(t),$$

where $n=1,2,\dots$ is an integer, $p$ is a $2\pi$-periodic function and $h_L$ is the piecewise linear function

$$h_L (x)=\begin{cases}L& \text{if$x\geq 1$},\\

Lx & \text{if$|x|\leq 1$},\\

-L & \text{if$x\leq -1$}.\end{cases}$$

A classical result of Lazer and Leach implies that thisequation has a $2\pi$-periodic solution if and only if

\begin{equation}\label{ll}|\hat{p}_n |<{2L\over \pi},\end{equation}

where

$$\hat{p}_n :={1\over 2\pi}\int_0^{2\pi} p(t)e^{-int}\, dt.$$

In this paper I prove that if $p$ is of class $C^5$ thenthe condition (\ref{ll}) is also necessary and sufficientfor the boundedness of all the solutions of the equation.

The proof of this theorem motivates a new variant ofMoser's Small Twist Theorem. This variant guaranteesthe existence of invariant curves for certain mappingsof the cylinder which have a twist that may depend onthe angle.

1991 Mathematics Subject Classification: 34C11, 58F35.

Bihermitian complex surfaces are oriented conformal four-manifoldsadmitting two independent compatible complex structures. Non-anti-self-dual bihermitian structures on ${\mathbb R}^4$ and the four-dimensional torus $T^4$ have recently been discovered by P. Kobak. We show that an oriented compact 4-manifold, admitting a non-anti-self-dual bihermitian structure, is a torus or K3 surface in the strongly bihermitian case (whenthe two complex structures are independent at each point) or, otherwise,must be obtained from the complex projective plane or a minimal ruledsurface of genus less than 2 by blowing up points along someanti-canonical divisor (but the actual existence of bihermitian structuresin the latter case is still an open question). The paper includes ageneral method for constructing non-anti-self-dual bihermitian structures on tori, K3 surfaces and $S^1\times S^3$. Further properties of compact bihermitian surfaces are also investigated.

1991 Mathematics Subject Classification: 53C12, 53C55, 32J15.

The energy minimization problem associated to uniform, isotropic, linearly elastic rods leads to a geometric variational problem for the rod centreline,whose solutions include closed, knotted curves. We give a complete description of the space of closed and quasiperiodic solutions.

The quasiperiodic curves are parametrized by a two-dimensional disc. The closed curves arise as a countable collection of one-parameter families, connecting the $m$-fold covered circle to the $n$-fold covered circle for any relatively prime $m$and $n$. Each family contains exactly one self-intersecting curve, one elastic curve, and one closed curve of constant torsion.Two torus knot types are represented in each family, and all torus knots are represented by elastic rod centrelines.

1991 Mathematics Subject Classification: primary 53A04, 73C02; secondary 57M25.

Let $\cal V$ be a complete connected hyperbolic 3-manifoldof finite volume, with Liouville measure $m$, geodesic flow$\Gamma_t$ and Brownian motion $Z_t$. Let $\omega$ be a smooth 1-form, closed in the cusps of $\cal V$. Then the limit laws as $t \to \infty$ of $(t\log t)^{-1/2}\int_0^t\omega(\Gamma_s)$ under$m$ and of $(t\log t)^{-1/2}\int_0^t\omega(Z_s)$are calculated, and seen to be Gaussian, and equal. Thegeodesic flow case is studied via the Brownian case.

1991 Mathematics Subject Classification: 60J65, 58F17, 51M10.