This seminar concerns a class of flow equations for immersed hypersurfaces, modelled on the well-known mean curvature flow. The flows in this class share much of the qualitative behaviour of the mean curvature flow, but are in general fully nonlinear; this compiicates some parts of their analysis. Other calculations are clarified by the general setting. I will present some results on the behaviour of convex hypersurfaces under these flows, which extend work on specific flows by Huisken (Hul), Tso (Tl) and Chow (Cl-2). Also new is a Harnack inequality for solutions of very general flows; this generalises results of Hamilton (Hal) and Chow (C3). Flows of this kind have some applications in geometry; for such purposes the mean curvature flow is not always the best candidate - I will describe an example which applies to manifolds of non-negative sectional curvature.

Shading, or brightness variation, exhibited within an image is a well-known cue for what is variously termed surface shape, depth, or form. As examples, artists commonly use tonal values in drawings to convey a realistic impression of 3-dimensionality, and make-up is sometimes used to give facial shape a flattering appearance. Just how shape might be recovered from shading has long been studied. However, it is only with the advent of computer vision in the last 25 years that substantial progress has been made. It transpires that the problem takes the form of a first-order partial differential equation. The pioneering automated method for determining surface shape from image shading was based on the use of the equations of characteristic strips. This suffered from several drawbacks which were partly overcome with the introduction of parallel schemes, based on minimisation techniques, for use on a 2-dimensional rectangular grid. Typically, these schemes are directed at recovering components of surface normals defined over a region of the image. Recovery of actual depth is left to a subsequent stage of integration. The variational calculus has now become the standard mathematical device by which new schemes are developed. It has also served as a useful means by which their shortcomings may be analysed and rectified. This paper presents the major iterative techniques in the field, and discusses various open problems and difficulties that remain.

A smooth object depicted in a monochrome image will often exhibit brightness variation, or shading. Of interest in computer vision is the problem of how object shape may be recovered from such an image. When the imaging conditions are such that an overhead point-source illuminates a smooth Lambertian surface, the problem may be formulated as that of finding a solution to an eikonal equation. This article will focus on the existence and uniqueness of such solutions, reporting recent results obtained. With regard to existence, shading patterns are exhibited for which there is no corresponding object shape. Specifically, a necessary and sufficient condition is presented for a circularly-symmetric eikonal equation to admit exclusively unbounded solutions; additionally, a sufficient condition is given for an eikonal equation to have no solution. In connection with uniqueness, we consider eikonal equations, defined over a disc, such that the Euclidean norm of the gradient of any solution is circularly-symmetric, vanishes exactly at the disc centre, and diverges to infinity as the circumference of the disc is approached. Contrary to earlier expectations in the area, a class of such eikonal equations is shown to possess simultaneously circularly-symmetric and non-circularly-symmetric bounded smooth solutions.

We summarize recent results on singular variational problems which arise in connection with the n-dimensional analogue of the catenary problem. In addition we give some historical comments.

In recent years major progress was made on the global behaviour of geometric evolution equations, in particular the harmonic map heatflow, the Ricci flow and the mean curvature flow. Longtime existence and regularity could be shown in a number of important cases. On the other hand, it became clear that in general singularities do occur in finite time, and an understanding of their structure should be crucial both for further development in the theory of these equations and for possible applications. In this article we will point out some of the strong analogies in the equations mentioned above and show in the case of the mean curvature flow how rescaling techniques can be used to understand the asymptotic behaviour of many singularities. We emphasize techniques applicable in all the equations under consideration and mention some open problems.

We discuss some recent work on numerical generation of conformal maps and minimal surfaces. Much of this work is joint with Gerd Dziuk. Some of the material on unstable minimal surfaces was developed subsequent to the Workshop.

Smooth interpolation of arbitrary curves and surfaces is a major problem in computer graphics. There are very successful variational formulations of similar problems in smooth interpolation of arbitrary functions; these have given rise to the (linear) theory of splines. However, there appears to be as yet no equivalent useful formulation of the general problem, so present computer graphics algorithms for cunres and surfaces use somewhat ad hoc extensions of the linear results based on parametric representations of splines and surface patches. The paper briefly describes the present state of affairs in the hope that variational geometers will pick up on some of these unsolved problems and develop a coherent theory of smooth interpolation of arbitrary geometrical objects. If such a theory can be developed, it may possibly revolutionize computer graphics.

Local spline approximants offer a means for constructing finite difference formulae for numerical solution of PDEs. These formulae seem particularly well suited to situations in which the use of conventional formulae leads to non-linear computational instability of the time integration. This is explained in terms of frequency responses of the FDF.

We prove existence of new complete embedded minimal surfaces in H3 having the symmetry of a regular tesselation by Coxeter orthoschemes. Each tetrahedron bounds a fundamental piece along four convex symmetry arcs. Its existence is proved by a conjugate surface construction.

In this paper we describe some of the hierarchical methods that have been used to produce computationally efficient algorithms for the calculation of the mutual interactions of a collection of n particles. The work required for such calculations grows quadratically as the number of particles. In this paper we survey the so called "tree codes" or "hierarchical" methods which provide accurate approximations of the required interactions, but for which the computational work is $O(n$ log $n)$ for "simple" hierarchical methods, and $O$(n) for the more sophisticated Fast Multipo!e method. This survey paper follows closely the article by Greengard ["Numerical Solution of the N-Body Problem", Computers in Physics, 1990].