CAREER: Scaling and self-similarity in nonlinear science-education and research

日期：2007年11月21日Proposal：DMS-0748482PI：Menon，Govind Institution：Brown University标题：Career：非线性科学教育和研究中的标度和自相似性在复杂的非线性问题中，经验标度律往往是有序的唯一标志。这样的定律出现在不同的领域，如生物网络、地震和完全湍流的研究。研究人员的研究愿景是将其中一些定律纳入严格的数学分析范围。为此，看似无关的问题在一个统一的框架内处理，该框架结合了动力系统、偏微分方程式和概率论的数学方法。这些区域之间的明确联系被用来提供对(A)材料科学和物理化学中的区域粗化模型、(B)聚合物化学中输运系数的动态标度、(C)湍流模型中“普遍的”标度定律的微观起源的基本理解。与经典概率的类比也被用来指导对动力系统普适性的新研究。这些研究目标与教育和培训的具体改进密切相关。在一个面向本科生的课程改革项目中，标度和自相似被用作一个统一的主题，该项目强调从复杂模型中提取简单的量化答案。这包括开发新的一年级和高级研讨会以及REU计划。这项数学研究是由流体力学、材料科学和物理化学中具有基本技术意义的问题推动的。材料科学中对磁区生长的研究提高了对二元合金稳定性和薄膜衬底上纳米尺度岛粗化的理解。改进的聚合物数值模拟方法推进了生化流动的分析，例如在受限通道中操纵DNA单链。深入了解动力学系统中的普适性，增强了对几种流体流动向混沌转变的认识，是控制这类流动的第一步。本科教育计划早期强调标度分析在生物、物理化学和地球科学等新兴数学应用领域的基本效用。它们还包括在这些领域成功的数学研究的本科生水平上的创新论述的发展。这促进了这些领域对数学复杂性的日益增长的需求，并对持续的影响至关重要。教育计划强调小班、本科生研究和对代表性不足的群体的接触，坚信对学生进行仔细的、个别的指导是建立一支多样化和有才华的科学队伍的基础。

Date: November 21, 2007Proposal: DMS- 0748482PI: Menon, Govind Institution: Brown UniversityTitle: CAREER: Scaling and self-similarity in nonlinear science-education and researchABSTRACTEmpirical scaling laws are often the only signature of order in complex nonlinear problems. Such laws arise in diverse areas such as the study of biological networks, earthquakes and fully turbulent flows. The investigator's research vision is to bring some of these laws within the scope of rigorous mathematical analysis. To this end, seemingly unrelated problems are treated within a unified framework that combines mathematical methods from dynamical systems, partial differential equations and probability theory. Explicit connections between these areas are used to provide a fundamental understanding of the microscopic origin of `universal' scaling laws in (a) models of domain coarsening in materials science and physical chemistry, (b) dynamic scaling of transport coefficients in polymer chemistry, (c) models of turbulence. Analogies with classical probability are also used to guide new investigations of universality in dynamical systems.These research goals are closely tied to concrete improvements in education and training. Scaling and self-similarity are used as a unifying theme in a program for curricular innovation for undergraduates that stresses the extraction of simple quantitative answers from complex models. This includes the development of new freshman and senior seminars and an REU program.This mathematical research is motivated by problems of fundamental technological significance in fluid mechanics, materials science and physical chemistry. Studies of domain growth in materials science improve understanding of the stability of binary alloys and coarsening of nanoscale islands on thin-film substrates. Improved numerical methods for the simulation of polymers advance the analysis of biochemical flows, such as the manipulation of individual strands of DNA in confined channels. A deeper understanding of universality in dynamical systems enhances knowledge of the transition to chaos in several fluid flows, and is the first step in the control of such flows. The plans for undergraduate education place early emphasis on the basic utility of scaling analysis in emerging areas for the application of mathematics such as biology, physical chemistry and geosciences. They also include the development of innovative expositions at the undergraduate level of successful mathematical research in these fields. This contributes to a growing demand for mathematical sophistication in these areas, and is critical for sustained impact. The educational plans stress small classes, undergraduate research, and outreach to under-represented groups, in the firm belief that careful, individual mentoring of students is the foundation for a diverse and talented scientific workforce.