Mathematical Problems from Materials Science

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This project addresses several topics at the interface between mathematics, physics, andmaterials science. One theme is epitaxialgrowth, particularly the task of linkingmicroscopic and macroscopic descriptions ofgrowth. A second theme is the interactionof bulk and surface energies, which determine the structure of defects and the selection ofpatterns in diverse systems such as stressedfilms and dielectric fluids. A third theme isthe analysis of composite microstructure, wheremethods developed for image processing seem tooffer fresh opportunities.Materials have structure on many length scales.The tasks of (a) predicting this structure,and (b) understanding its consequences, are among the major challenges of modern materials science. Mathematical methods such as homogenization,asymptotic analysis, and multiresolution methodscan help. This project will develop and apply suchmethods in the context of specific material systems,leading to enhanced understanding of those systemsand improvement of the mathematical methods themselves