We announce the classification of complete, almost embedded surfaces of constant mean curvature, with three ends and genus zero: they are classified by triples of points on the sphere whose distances are the asymptotic necksizes of the three ends.

[From the introduction] Recently there has been much interest in the theory of discrete surfaces in 3- space and its connection with the discretization of soliton equations (see e.g. [4, 11] and references therein). In this article we study a discrete geometry which is the simplest example for both theories. Following [1, 3] we will define a discrete conformal map (DCM) to be a map $z:Z^2\rightarrow P^1$ with the property that the cross-ratio of each fundamental quadrilateral is the same.

We derive macroscopic governing laws of growth velocity, surface tension, mobility, critical nucleus size, and morphological evolution of clusters, from microscopic scale master equations for a prototype surface reaction system with long range adsorbate-adsorbate interactions. It is shown that within the bistable regime, the velocity of the moving cluster boundaries depends on their curvature and is recovered from the microscopic models through derived Kubo-Green formulae.

[From the introduction] Over the last decade there has been considerable success in understanding the construction of certain harmonic 2-tori in symmetric spaces (in particular, the non-superminimal tori in $CP^n$ and $S^n$) using the methods of integrable systems theory. For example, if $\phi: M\rightarrow S^2$ is a non-conformal harmonic torus one knows that $\phi$ is determined by its (real hyperelliptic) spectral curve $X$ equipped with a degree two function $\labmda$. There are already several ways of describing the relationship between the map \$phi$ and its spectral data $(X,\lambda)$: see, for example, [2, 7, 12, 10]. But we lack a direct, geometric picture of how $\phi$ arises from the algebraic geometry of $X$. What I want to present here is such a picture, based on a remarkable property of certain generalised Jacobi varieties. Moreover, it is quite straightforward to extend this picture to produce harmonic (indeed, pluri-harmonic) maps into Grassmannians and $PU_{n+1}$ (i.e. $U_{n+1}/centre) and all of these will be maps of `finite type' in the sense of [3]. From this picture one also sees that some of these harmonic tori are purely algebraic and these present an interesting class for further study.

We use the DPW construction to present three new classes of immersed CMC cylinders, each of which includes surfaces with umbilics. The first class consists of cylinders with one end asymptotic to a Delaunay surface. The second class presents surfaces with a closed planar geodesic. In the third class each surface has a closed curve of points with a common tangent plane.

I construct two families of constant mean curvature genus-zero surfaces with three ends via the DPW construction. One of these families is known and has three unduloid ends; the other is a new family with two unduloid and one nodoid end.

We study the effects of random fluctuations included in microscopic models for phase transitions, to macroscopic interface flows. We first derive asymptotically a stochastic mean curvature evolution law from the stochastic Ginzburg-Landau model and develop a corresponding level set formulation. Secondly we demonstrate numerically, using stochastic Ginzburg-Landau and Ising algorithms, that microscopic random perturbations resolve geometric and numerical instabilities in the corresponding deterministic flow.

A mesoscopic theory for diffusion of molecules interacting with a long range potential is derived for Arrhenius microscopic dynamics. Gradient Monte Carlo simulations are presented on a one- dimensional lattice to assess the validity of the mesoscopic theory. Results are compared for Metropolis and Arrhenius microscopic dynamics. Non Fickian behavior is demonstrated and it is shown that microscopic dynamics dictate the steady-state concentration profiles.

In this paper we present spectral algorithms for the solution of mesoscopic equations describing a broad class of pattern formation mechanisms, focussing on a prototypical system of surface processes. These models are in principle stochastic integrodifferential equations and are derived directly from microscopic lattice models, containing detailed information on particle-particle interactions and particle dynamics. The enhanced computational eficiency and accuracy of spectral methods versus finite difference methods are also described.