Research and Education in Physical Mathematics

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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.