We study hypersurfaces moving under flow that depends on the mean curvature. The approach is based on a numerical technique that embeds the evolving hypersurface as the zero level set of a family of evolving surfaces. In this setting, the resulting partial differential equation for the motion of the level set function may be solved by using numerical techniques borrowed from hyperbolic conservation laws.

This technique is applied to several problems: the evolution of a dumbbell, and related many-armed surfaces, collapsing under mean curvature; the construction of a minimal surface attached to a given one-dimensional wire frame in $\reals^3$, and, more generally, the construction of surfaces whose mean curvature is a prescribed function of position; the motion of curves on two-manifolds under flow that depends on geodesic curvature.

Some experiments involving flow controlled by Gaussian curvature are also included.

Using a formula of Billey, Jockusch and Stanley, Fomin and Kirillov have introduced a new set of diagrams that encode the Schubert polynomials. We call these objects rc-graphs. We define and prove two variants of an algorithm for constructing the set of all rc-graphs for a given permutation. This construction makes many of the identities known for Schubert polynomials more apparent, and yields new ones. In particular, we give a new proof of Monk's rule using an insertion algorithm on rc-graphs. We conjecture two analogs of Pieri's rule for multiplying Schubert polynomials. We also extend the algorithm to generate the double Schubert polynomials.

The information metric is a construction in statistics which can be used to define a (possibly degenerate) metric on various moduli spaces such as those of instantons and harmonic maps. This metric is shown to be nondegenerate for the space of harmonic maps of the two-sphere onto itself of any degree.

We consider the rational maps given by $z \mapsto |z|^{2\alpha-2}z^2+c$, for $z$ and $c$ complex and $\alpha > {1\over 2}$ fixed and real. The case $\alpha=1$ corresponds to quadratic polynomials: some of the well-known results for this conformal case still hold for $\alpha$ near $1$, while others break down. Among the differences between the two cases are the possibility, for $\alpha\ne1$, of periodic attractors that do not attract the critical point, and the fact that for $\alpha >1$ the Julia set is smooth for an open set of values of $c$. Numerical evidence suggests that the analogue of the Mandelbrot set for this family is connected, but not locally connected if $\alpha \ne 1$.

For $\lambda$ real, we consider the pattern given by the value modulo 2 of the integer part of $\lambda(x^2+y^2)$, where $(x,y)\in\Z\times\Z$. We study the periodicity and other geometric properties of this pattern, and show that it can provide, by visual inspection and an elementary computation, a diophantine approximation for $\lambda$. We conclude with similar results for other moduli.