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

DMS Award AbstractAward #： 0135078 PI： Venkataramani，Shankar机构： 芝加哥大学课程： 应用数学项目经理：Catherine Mavriplis职务：职业：奇异性和微观结构-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着各种研究非线性系统多尺度行为的工具，而对特定问题的选择方法往往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合并各种方法，并开发新的方法来思考多尺度问题。为此，本项目重点研究五个跨学科的问题，利用泛函分析、几何、拓扑、匹配渐近和标度参数的思想，结合数值模拟。这些问题是：（1）广义褶皱：将泛函分析方法与几何/拓扑思想相结合，研究奇异性和微结构;（2）抛物偏微分方程中的爆破：将莫尔斯理论应用于爆破解的动力学;（3）微结构动力学：研究模型系统中涉及微结构变化的全局分叉;（4）图案形成和非平衡相变：研究噪声存在下非线性系统的多尺度行为;（5）流体界面的拓扑转变：利用偏微分方程的工具研究二维流体系统的拓扑转变。对于生物体、地质和地球物理系统、重要的技术复合材料，甚至社会结构和等级制度来说，这当然是正确的。因此，许多学科的研究人员都在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）行为如何从小尺度（微观）单元的集体行为中出现？（2）大尺度行为的规则是什么，它如何影响小尺度单元的行为？该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，以开发能够成功处理复杂现实世界多尺度问题的工具。这是与一个综合的方法相结合的教学法，其特点是强烈参与本科和研究生的研究，为那些在数学和物理科学，在研究生和本科水平的课程开发，并为科学推广到广大公众的材料开发代表性不足的群体的研究机会的发展。

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