Abstract Jerrard Jerrard will work on three classes of problems. Two of these grow out of joint work carried out by Jerrard in collaboration with H.M Soner. Jerrad and Soner examined the dynamics of topological defects in solutions of Ginzburg-Landau (GL) systems taking values in the plane. They showed that, under an appropriate scaling limit and for appropriate initial data, the energy density of solutions concentrates around codimension 2 manifolds, and they characterized the evolution of these manifolds. In planar domains, these singular manifolds are simply singular points which evolve via a system of ordinary differential equations; in higher dimensional domains, one obtains in the limit manifolds which evolve via codimension 2 mean curvature flow. Jerrard now proposes to study a number of questions about dynamics of point defects of solutions of evolution equations of GL type, related to his earlier work. In particular, he wants to look at dynamics of point defects in GL Schroedinger equations and in generalized GL systems in higher dimensions. Secondly, he plans to look at problems concerning the evolution of line defects or, more generally, singular manifolds of dimension greater than 1 and codimension at least 2. These would include questions about evolution of line defects in GL Schroedinger equations. A third, completely different class of problems has to do with developing a framework within which one can make sense of level set equations in which certain standard monotonicity assumptions on the coefficients are violated. Heuristically, such an equation would specify that every level set evolves by some geometric rule, but that different level sets evolve by different rules. Such equations are of mixed parabolic/hyperbolic type. A notion of weak solution would need to combine viscosity solution techniques with entropy conditions, and would probably also require some other ingredients. The problems described above are drawn from two areas: physics of superconductors and sup erfluids, and image progessing. Ginzburg-Landau systems are a widely-used mathematical model for certain kinds of superconductors and superfluids, and they seem to agree quite well with experiment. Some classes of superconducting materials, including many high temperature superconductors, exhibit small islands of normal, nonsuperconducting behavior within a larger superconducting matrix. These islands are known as vortices, and they can have a negative impact on the performance of superconductors. A major focus of this propsal is to attempt to understand, from a mathematical viewpoint, the behavior of these Ginzburg-Landau vortices. Similar vortices are seen in superfluids, such as liquid helium. Finally, a major problem in image processing is to eliminate "meaningless" noise, while retaining contrasts that carry some significant information. Many image processing schemes amount to finding a solution of some differential equation, which takes a given image as the input and produces the processed image as a solution. In this framework, it is useful to try to have a well-developed mathematical theory for equations which have the effect of smoothing out small irregularities in the given image, while sharpening major contrasts. This is the other main area in which Jerrard proposes to work.
