This article discusses the dynamics of iterated cubic maps from the real or complex line to itself and will describe the geography of the parameter space for such maps. It is a rough survey with few precise statements or proofs and depends strongly on work by Douady, Hubbard, Branner and Rees.

Using the LLL algorithm and the second author's "ladder'' method, we find (conjectural) $\Z$-linear relations among polylogarithms of order up to 16 evaluated at powers of a single algebraic number. These relations are in accordance with theoretical predictions and are valid to an accuracy of 300 decimal digits, but we cannot prove them rigorously.

The aim of this article is to provoke a discussion concerning the general nature of the topological closure of the set of rational solutions of systems of polynomial equations with rational coefficients.

The infinitesimal Kobayashi metric of an ellipsoid of the form $$ E_m=\{(z_1,z_2)\in \C^2:|z_1|^2+|z_2|^{2m}<1\} $$ is calculated explicitly, modulo a parameter that is determined by solving a transcendental equation. Using this result, we show that the metric is $C^1$ on the tangent bundle away from the zero section. We also describe software that will calculate, using a Monte Carlo method, the infinitesimal Kobayashi metric on a domain of the form $$ \Omega_\rho=\{(z_1,z_2)\in\C^2:\rho(z_1,z_2)<0\}, $$ where $\rho$ is a real-valued polynomial. We compare results of computer calculations with those obtained from the explicit formula for the Kobayashi metric.

Queneau observed that certain 1-additive sequences (defined by Ulam) are regular in the sense that differences between adjacent terms are eventually periodic. This paper extends Queneau's work and my recent work toward characterizing periods and fundamental differences of all regular 1-additive sequences. Relevant computer investigations of associated nonlinear recurring sequences give rise to unexpected evidence suggesting several conjectures.

If a projective plane $\Pi$ of order 15 contains a line-extended (15,3)-Kirkman design $\Sigma^*$, no collineation of $\Pi$ of order 7 fixes (setwise) the point set of $\Sigma^*$.

We consider the pentagram map on the space of plane convex pentagons obtained by drawing a pentagon's diagonals, recovering unpublished results of Conway and proving new ones. We generalize this to a "pentagram map'' on convex polygons of more than five sides, showing that iterated images of any initial polygon converge exponentially fast to a point. We conjecture that the asymptotic behavior of this convergence is the same as under a projective transformation. Finally, we show a connection between the pentagram map and a certain flow defined on parametrized curves.