Research and Education in Physical Mathematics

在研究和教育的背景下，研究者专注于应用数学的分析方法与数值计算的耦合。本课程将结合近似分析、计算和实验，探讨工程和科学中的一系列问题。研究者和他的同事进行了以下项目：(1)离子束溅射的数学分析，旨在提高对技术产生的表面图案的理解（例如，为什么特定的形态会在给定的条件下形成？），并了解离子束是否可以用来创建和控制预定的表面形态；(2)二维弹性板自组装成三维结构的研究。当折叠成给定的拓扑结构时，在组装动力学和确定导致最低能量结构的薄片形状方面存在重要问题；(3)分析花粉粒的形状和它们干涸的机制；(4)液滴撞击固体基底的数学分析——特别是液滴是否产生飞溅取决于撞击后立即发生的未知事件。教育活动围绕着如何教授应用数学概念这一普遍问题展开，考虑到计算的普及。讨论的两个主要问题是：如何使用计算机来提高学生的数学（而不是计算）能力？其次，如何教授数学建模，以便清楚地区分建模（产生对现象的理解）和计算机图形学（产生再现现象的图片）？研究者正在开发新的课程（和旧课程的修改）来解决这些问题。在过去的十年里，计算机和计算机模拟在科学和科学论证中扮演着越来越重要的角色。很明显，这一趋势将继续下去。另一方面，对于复杂的系统，在理解何时判断计算机模拟是否正确方面存在严重的问题。该项目在具体研究项目的背景下解决了这个普遍问题，并通过开发教育课程和工具来帮助学生学习如何回答这个问题。研究项目包括分析复杂的、技术上重要的现象，计算机模拟和数学分析可以结合起来。这些项目包括对离子束溅射的研究，这是一个复杂的过程，当表面受到高能照射时发生。研究者和他的同事们利用计算机模拟和分析方法，结合实验来研究这个问题，努力了解离子束是否能在表面上产生特定的、技术上有用的图案。另一个项目研究弹性片的自组装，试图了解是否有可能创造出可以在三维空间中自组装成有用组件的小物体。第三个项目是研究液滴溅落在固体基板上的过程。尽管这种现象无处不在，但尚不清楚为什么液滴会产生飞溅。在教育方面，该项目旨在开发材料，教学生如何进行计算机模拟，同样重要的是如何评估他们是否正确。

BrennerDMS-0605031 The investigator focuses on the coupling of the analyticalmethods of applied mathematics to numerical computation, in thecontext of both research and education. The topics consideredaddress a range of problems in engineering and science using acombination of approximate analysis, computation, and experiments. The investigator and his colleagues conduct the following projects:(1) A mathematical analysis of ion beam sputtering, aimed at bothimproving understanding of the surface patterns producedtechnique (e.g. why do particular morphologies form for given sets of conditions?) and at understanding whether ion beams can be used to create and control prespecified surface morphologies;(2) Studies of the self-assembly of two-dimensional elasticsheets into three-dimensional structures. Here there areimportant issues both with respect to assembly kinetics and indetermining the shape of the sheet that leads to the lowestenergy structure, when folded into a given topology;(3) An analysis of the shapes of pollen grains, and themechanisms through which they dry up; and(4) A mathematical analysis of droplet impact on a solid substrate -- in particular, whether or not a droplet produces a splash depends on an as yet unknown event that occurs immediately after the impact. The educational activities center around the general issue of howthe concepts of applied mathematics should be taught, given theprevalence of computation. The two main issues addressed are:How can computers be used to increase the mathematical (as opposed to computational) sophistication of students? And secondly, how should mathematical modeling be taught, in order to clearly differentiate modeling (which produces understanding of aphenomenon) from computer graphics (which produces a picturereproducing it)? The investigator is developing new courses (andmodifications of old courses) to address these issues. Over the past decade, computers, and computer simulations,play an ever increasing role in science and scientific arguments. It is clear this trend will continue. On the other hand, forcomplicated systems there are serious issues in understandingwhen to judge whether a computer simulation is correct. Thisproject addresses this general question both in the context ofspecific research projects and also through the development ofeducational courses and tools to help students learn how toanswer this question. The research projects include analyses ofcomplex, technologically important phenomena for which bothcomputer simulation and mathematical analysis can be combined. These project include a study of ion beam sputtering, a complexprocess that occurs when surfaces are irradiated with high energyions. The investigator and his colleagues study this problemusing both computer simulations and analytic methods, coupledwith experiments, in an effort to understand whether ion beamscan create specific, technologically useful, patterns in asurface. Another project studies the self-assembly of elasticsheets, in an effort to understand whether it is possible tocreate small objects which self-assemble into useful componentsin three dimensions. A third project studies the process when adroplet of liquid splashes on a solid substrate. Despite theubiquity of this phenomenon it is not known why a droplet createsa splash. On the educational front, the project aims to developmaterial to teach students both how to carry out computersimulations, and equally importantly how to evaluate whether theyare correct.