Research and Education in Physical Mathematics

BrennerDMS-0605031 调查员的重点是耦合的分析方法的应用数学数值计算，在研究和教育的背景下。 本课程将结合近似分析、计算和实验，讨论工程和科学领域的一系列问题。研究人员和他的同事进行了以下项目：（1）离子束溅射的数学分析，旨在提高对表面图案产生的理解和技术（例如，为什么在给定的条件下会形成特定的形态？）以及了解离子束是否可以用来产生和控制预定的表面形貌;（2）二维弹性片自组装成三维结构的研究。 这里有两个重要的问题，一个是关于组装动力学，另一个是当折叠成给定的拓扑结构时，不确定导致低能结构的片的形状：（3）花粉粒形状的分析，以及它们干燥的机制;和（4）液滴撞击固体基底的数学分析--特别地，液滴是否产生飞溅取决于撞击后立即发生的未知事件。教育活动的中心是如何应用数学的概念，应该教的一般问题，考虑到计算的普及。 解决的两个主要问题是：如何使用计算机来增加学生的数学（而不是计算）复杂性？ 第二，数学建模应该如何教授，以明确区分建模（产生对现象的理解）和计算机图形学（产生再现现象的图片）？ 调查员正在制定新的课程（以及对旧课程的修改）来解决这些问题。 在过去的十年中，计算机和计算机模拟在科学和科学论证中发挥着越来越重要的作用。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问题。 这个项目解决了这个一般性的问题，既在具体的研究项目的背景下，也通过教育课程和工具的发展，以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，其中计算机模拟和数学分析可以结合起来。这些计划包括对离子束溅射的研究，这是一种当表面受到高能离子照射时发生的复杂过程。 研究人员和他的同事们研究这个问题，使用计算机模拟和分析方法，结合实验，努力了解离子束扫描是否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自组装成三维有用组件的小物体。 第三个项目是研究小滴液体溅到固体基质上的过程。 尽管这种现象普遍存在，但人们不知道为什么一滴水滴会溅起水花。 在教育方面，该项目旨在开发教材，教学生如何进行计算机模拟，以及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.