Research and Scientific Training in Applied Mathematics

该奖项将支持卡内基梅隆大学数学科学系非线性分析中心继续和扩大应用数学方面的研究和科学培训。研究将集中在材料科学应用中发生的一系列现象的数学描述和预测上。形状记忆合金、铁电、磁性和磁致伸缩材料、复合材料、液晶和薄膜等新型人造材料的数学模型研究将在该计划中发挥核心作用。将讨论相变、缺陷和缺陷扩展、晶界及其演化、宏观行为的预测及其与微观结构特征的相互作用、超导、弹塑性系统、双曲守恒定律、稳定性、亚稳性以及其他主题。这些问题的研究将推动非线性分析和科学计算的前沿。这项工作的目标是发现和发展数学科学的新应用。主要的焦点将集中在材料的非线性行为以及分析和计算中的相关问题。大规模计算将在我们的所有工作中发挥核心作用。该中心将促进数学家和其他领域科学家之间的密切合作。主要活动将围绕跨学科研究前沿的选定主题举行密集的短期研讨会，重点放在数学和物理科学之间的研究机会的博士后培训计划，以及访客计划。访客方案和研讨会将围绕每年选定的研究主题进行协调。我们的一些资源将用于数学金融的次要研究。

This award will support the continuation and expansion of the research and scientific training in applied mathematics at the Center for Nonlinear Analysis in the Department of Mathematical Sciences, Carnegie Mellon University. Research will focus on the mathematical description and prediction of a range of phenomena that occur in material science applications. The study of mathematical models for novel man-made materials such as shape memory alloys, ferroelectric, magnetic and magnetostrictive materials, composites, liquid crystals, and thin films, will play a central role in the program. Phase transitions, defectiveness and defect propagation, grain boundaries and their evolution, the prediction of macroscopic behavior and its interplay with microstructural features, superconductivity, elasto-plastic systems, hyperbolic conservation laws, stability, metastability, and other topics will be addressed. The study of these questions will advance the frontiers of nonlinear analysis and scientific computation.The goal of this work is to identify and develop new applications of the mathematical sciences. The main focus will be on nonlinear behavior of materials and concomitant issues in analysis and computation. Large scale computing will play a central role in all of our work. The Center will promote close collaboration between mathematicians and scientists from other fields. Main activities will be centered around intense, short workshops on selected topics at the forefront of interdisciplinary research, a postdoctoral training program where emphasis will be placed on research opportunities available at the interface between the mathematical and physical sciences, and a visitors program. The visitors program and the workshops will be coordinated around research themes selected on an annual basis. Some of our resources will be devoted to a secondary research thrust in mathematical finance.