In recent years there have been many important developments in the mathematical treatment of localization phenomena in reaction-diffusion problems especially those connected with the equilibrium or static theory of elliptic partial differential equations that possess a variational structure. However, many open problems remain, most notably those in the realm of nonlinear dynamics, bifurcation behavior, and the numerical computation of localized structures in reaction-diffusion systems. In addition, there is a strong need for researchers in this area to be exposed to new classes of PDE's involving localization phenomena that arise from other biological and physical situations.
There are two main aims of this workshop. A primary goal is to enhance the interaction between researchers in nonlinear PDE, in dynamical systems theory, and in applied and computational mathematics, who model, compute and analyze various localization phenomena in the mathematical, biological or physical sciences. This interaction will stimulate new mathematical ideas, and also expose the mathematical community to new biological and physical problems that await a mathematical understanding. The second main focus of the workshop is to expose a limited number of Postdoctoral fellows and advanced graduate students to current problems associated with localization behavior in reaction-diffusion systems, and to show different mathematical approaches to study this behavior.
To illustrate the importance and relevance of this research area, over the past three years there have been many conferences devoted to the analysis of localization phenomena in reaction-diffusion systems. A sample of these include a one-week conference in Crete, Greece in June 1999 on nonlinear dynamics for PDE's related to materials science (organized by N.~Alikakos of U.~Tennessee), a one-week conference on reaction-diffusion systems at the Chinese University of Hong Kong in Dec. 1999 (organized by J.~Wei of CUHK and M.~Mimura of U.~Hiroshima), a two-week conference in pattern formation at the Lorentz Institute in Leiden, Holland in March 2001 (organized by D.~Hillhorst of U.~Paris-Sud and H.~Matano of U.~Tokyo), the PIMS conference on point-condensation phenomena in Vancouver, Canada in July 2001 (organized by C.~Gui and N.~Ghoussoub of UBC), a week-long conference in Patterns and Waves in Nonlinear Chemistry at the Lorentz Institute in Leiden in August 2001 (organized by A.~Doelman and Y.~Nishiura). In addition, there were several week-long workshops devoted to localization phenomena during the six-month programme (Jan. 2001-Jun. 2001) in Nonlinear Partial Differential Equations at the Newton Institute in Cambridge, England (organized by N.~Dancer of U.~Sydney and H.~Brezis of Paris 6).
We are not aware of any workshops in the study of localization phenomena that are currently proposed for the summer of 2003. Our intention is to invite many of the main participants of these previous meetings to our workshop, with the view of highlighting the advances made on open problems. A key feature of the workshop is that we intend to also invite certain participants who work on localization phenomena in other areas of science, where there has been to date only a limited involvement by the mathematical community.
FORMAT OF THE WORKSHOP:
There will be four main areas of focus during the workshop
The expertise of each of the participants listed below is well matched to one of these core areas, with roughly equal representation in these four areas. Our intention is to have talks in each of these areas to cover four days of the workshop, with one day for each area. For the remaining day, talks will be given by selected advanced graduate students and Postdocs in the morning, with a free afternoon left for discussions and interaction between the participants. For the first three core areas mentioned above, our intention is to invite a keynote speaker from the participant list given below to give a one-hour lecture. This lectures should provide an overview of the area, with regards to the progress to date and the open problems that remain that are either of a mathematical or a modeling nature. Each keynote lecture will be followed by a series of (30+5)-minute lectures dealing with various specific problems in the field. For the remaining area of emerging applications, the expertise of the participant list ranges from hot-spot behavior in microwave heating (Kriegsmann), the motion of defects in Ginzburg-Landau (Mielke), pattern formation in growing domains (Maini), and moving-mesh methods for the numerical computation of localized solutions to PDE's (Russel, Wang).
Final Report (in PDF format)