Mathematical Problems from Materials Science

这个项目涉及数学、物理和材料科学之间的几个主题。一个主题是外延生长，特别是连接微观和宏观描述生长的任务。第二个主题是体能和表面能的相互作用，它决定了不同系统（如应力薄膜和介电流体）中缺陷的结构和模式的选择。第三个主题是复合材料微观结构的分析，其中为图像处理开发的方法似乎提供了新的机会。物质在许多长度尺度上都有结构。(a)预测这种结构和(b)理解其后果的任务是现代材料科学的主要挑战之一。数学方法，如均匀化、渐近分析和多分辨率方法可以提供帮助。该项目将在特定材料系统的背景下开发和应用这些方法，从而增强对这些系统的理解，并改进数学方法本身

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