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

DMS奖摘要奖#：0135078PI：Venkataramani，Shankar研究所：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人员是分析师、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，发展思考多尺度问题的新方法。为此，本项目利用泛函分析、几何学、拓扑学、匹配渐近性和标度变元的思想，结合数值模拟，重点研究了五个跨学科问题。这些问题是：(1)广义皱缩：结合泛函分析方法和几何/拓扑思想来研究奇点和微观结构；(2)抛物型偏微分方程中的爆破：将Morse理论应用于爆破解的动力学；(3)微观结构动力学：研究模型系统中涉及微观结构变化的全局分叉；(4)图案形成和非平衡相变：研究存在噪声的非线性系统中的多重尺度行为；以及(5)拓扑相变：使用偏微分方程的工具来研究2个流体系统中的拓扑相变。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。对于活的有机体、地质和地球物理系统、技术上重要的复合材料，甚至社会结构和等级制度来说，情况无疑都是如此。因此，许多学科的研究人员努力解决以下两个问题，这两个问题是多尺度分析的本质：(1)大尺度(宏观)行为是如何从小尺度(微观)单位的集体行为中产生的？(2)大尺度行为的规则是什么，这对小尺度单位的行为有什么影响？这个项目的研究部分通过材料科学和物理中出现的问题，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，以开发能够成功处理复杂现实世界多尺度问题的工具。这与一种综合的教育学方法相结合，这种方法的特点是大力参与本科生和研究生的研究，为在数学和物理科学中代表性不足的群体开发研究机会，在研究生和本科生水平上开发课程，并开发面向公众的科学宣传材料。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