This workshop will bring together mathematicians and other scientists who are making strides in understanding these defects and their motions in their particular areas of expertise, with the expectation that the exchange of ideas and techniques will benefit all.
Below are some of the specific mathematical areas to be covered.
The Allen-Cahn equation and its various generalization
The Allen-Cahn equation is a well-known model for the mathematical study of bi-phase transition. It has received extensive study, and numerous important results have been obtained. For example, it is found that the interfacial surface moves by its mean curvature (in the limit as the diffusion coefficient approaches zero); the final shape of the interface is some minimal surface; the stationary shape of the interface has to be mainly smooth (i,e, the singular set of the interface must be of lower Hausdorff dimension), etc. However, there are still some very interesting open questions such as the uniqueness of the basic profile near the interface when it is not smooth, the rigidity of the interface under certain constraints, etc. These questions are also related to the De Giorgi conjecture on the symmetry of certain stationary transition solutions to the Allen-Cahn equation. This is still open in higher dimensions even though proofs exist for low dimension and partial results exist for higher dimensions. More importantly, some more sophisticated phase transition problems may be modeled by variations of the Allen-Cahn equations, such as a non-local version where the interaction between material particles has long range. Discrete (e.g. lattice) and continuum models with non-local interaction are important for both material science and neuroscience, the latter situation being the modeling of large populations of neurons interconnected through dendrites and synapses. The vector-valued versions of the Allen-Cahn and Cahn-Hilliard equations are important for modeling transitions in multi-component materials. Some of these variations display interesting new phenomena and difficulties, such as pinning in the discrete model and triple junction motion in the multi-component case. Many problems are open and require expertise in geometry, PDE?s, dynamical system, etc, and more importantly new ideas growing from the existing approaches.
Ginzburg-Landau and Nonlinear Shrödinger Equations
The modeling of vortices in superconductors with complex Ginzburg-Landau equations has been a very active area over the last decade. There are strong groups in France, Japan, PRC, and the US. Separately, they have made tremendous progress in showing the existence of vortex solutions, describing their stability and motion, and demonstrating the onset or nucleation of superconducting regions as the applied field is decreased past a certain critical value. We expect that the interaction of representatives of these groups through the workshop will allow greater progress to be made. We also expect that the interaction of these groups with those working in other areas will be beneficial to all. The motion of defects under the Nonlinear Shrödinger equation is also a very interesting and timely area of study. In this case, there are difficult problems associated with the spectrum of the linearized operator and also, the conserving nature of the flow shows that there is a lack of compactness. Still, many new ideas have been injected over the last five years or so. We hope that techniques for one system will have a bearing on the other.
Gray-Scott and Gierer-Meinhardt Equations
Recently there have been significant advances in the analysis of some well-known models of pattern formation such as the Gray-Scott and Gierer-Meinhardt Equations, arising in chemical reactor and biological systems. The appearance of stable spikes or spots and the onset of instability as spots subdivide are intriguing, both from a mathematical and biological point of view. The motion of several spots as they interact with each other and the boundary of the domain is not well-understood and yet the work on the Allen-Cahn equation and other systems seems to have a bearing here. Again, we bring together experts in both areas at this workshop and expect the interaction between scientists to be both lively and fruitful.