CAREER: Singularities and Microstructure - Multiple Scale Analysis for Nonlinear Partial Differential Equations (PDE), Geometric Problems, and the Physical Sciences

摘要/ abstract摘要：奖项编号：0135078项目负责人：Venkataramani， Shankar机构：芝加哥大学项目：应用数学项目经理：Catherine mavriplis标题：职业：奇点和微观结构——非线性偏微分方程、几何问题和物理科学的多尺度分析存在各种各样的工具来研究非线性系统中的多尺度行为，对于特定问题的选择方法通常取决于研究者是分析师、应用数学家还是物理学家。一个重要的目标是有效地融合各种方法，并开发新的思考多尺度问题的方法。为此，本项目将重点研究五个跨学科问题，使用泛函分析、几何、拓扑、匹配渐近和缩放论证的思想，并结合数值模拟。这些问题是：(1)广义皱缩：将泛函分析方法与几何/拓扑思想相结合来研究奇异性和微观结构；(2)抛物型偏微分方程的爆破：将莫尔斯理论应用于爆破解动力学；(3)微观结构动力学：研究涉及模型系统微观结构变化的全局分岔；(4)模式形成和非平衡相变：研究噪声存在下非线性系统的多尺度行为；(5)拓扑转换流体界面：使用PDE的工具研究两种流体系统的拓扑转换。许多现实世界的系统之所以有趣，正是因为它们在不同的尺度上表现出不同的行为。对于生物体、地质和地球物理系统、具有重要技术意义的复合材料，甚至是社会结构和等级来说，都是如此。因此，许多学科的研究人员都在努力解决以下两个问题，这也是多尺度分析的本质：(1)大规模（宏观）行为是如何从小规模（微观）单位的集体行为中产生的？(2)控制大规模行为的规则是什么，这些规则如何影响小规模单位的行为？本项目的研究部分通过材料科学和物理学中出现的问题在数学环境中研究这些问题。总体目标是将各种技术融合在一起，开发出能够成功处理复杂的现实世界多尺度问题的工具。这与综合教学法相结合，其特点是大力参与本科生和研究生的研究，为数学和物理科学领域代表性不足的群体提供研究机会，在研究生和本科生阶段开发课程，以及为公众提供科学宣传材料。日期：2001年12月17日

DMS Award AbstractAward #: 0135078PI: Venkataramani, Shankar Institution: University of ChicagoProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: CAREER: Singularities and Microstructure - Multiple Scale Analysis for Nonlinear PDE, Geometric Problems, and the Physical SciencesThere exist a variety of tools for studying multiple scale behaviorsin nonlinear systems, and the method of choice for a particularproblem often depends on whether the investigator is an analyst, anapplied mathematician or a physicist. An important goal is toeffectively merge the various approaches, and develop new ways ofthinking about multi-scale problems. To this end, this project focuseson investigating five interdisciplinary problems, using ideas fromfunctional analysis, geometry, topology, matched asymptotics andscaling arguments, in conjunction with numerical simulations. Theseproblems are (1) Generalized crumpling: Combining functional analyticmethods with geometric/topological ideas to study singularities andmicrostructure; (2) Blowup in Parabolic PDEs: Applying Morse theory tothe dynamics of blowup solutions; (3) Dynamics of microstructure:Studying global bifurcations involving the change of microstructure ina model system; (4) Pattern formation and non-equilibrium phasetransitions: Investigating multiple scale behaviors in nonlinearsystems in the presence of noise; and (5) Topological transitions influid interfaces: Using tools from PDE to investigate topologicaltransitions in 2 fluid systems.Many real world systems are interesting precisely because they exhibitdifferent behaviors on different scales. This is certainly true forliving organisms, geological and geophysical systems, technologicallyimportant composite materials and even social structures andhierarchies. Thus researchers across many disciplines grapple with thefollowing two questions, which are the essence of multiple scaleanalysis: (1) How does the large scale (macroscopic) behavior emergeout of the collective behavior of the small scale (microscopic)units?, and (2) What are the rules governing the large scale behavior,and how does this influence the behavior of the small scale units? Theresearch component of this project studies these questions in amathematical setting through problems that arise in material scienceand in physics. The overall goal is to meld together a variety oftechniques to develop tools that can successfully handle complexreal-world multiple scale problems. This is combined with anintegrated approach to pedagogy, that features a strong involvement inundergraduate and graduate research, development of researchopportunities for groups that are under-represented in mathematics andthe physical sciences, curriculum development both at the graduate andthe undergraduate level, and development of materials for scientificoutreach to the general public.Date: December 17, 2001