The investigator focuses on the coupling of the analytical methods of applied mathematics to numerical computation, in the context of both research and education. The topics considered address a range of problems in engineering and science using a combination 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 both improving understanding of the surface patterns produced technique (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 elastic sheets into three-dimensional structures. Here there are important issues both with respect to assembly kinetics and in determining the shape of the sheet that leads to the lowest energy structure, when folded into a given topology; (3) An analysis of the shapes of pollen grains, and the mechanisms 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 how the concepts of applied mathematics should be taught, given the prevalence 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 a phenomenon) from computer graphics (which produces a picture reproducing it)? The investigator is developing new courses (and modifications 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, for complicated systems there are serious issues in understanding when to judge whether a computer simulation is correct. This project addresses this general question both in the context of specific research projects and also through the development of educational courses and tools to help students learn how to answer this question. The research projects include analyses of complex, technologically important phenomena for which both computer simulation and mathematical analysis can be combined. These project include a study of ion beam sputtering, a complex process that occurs when surfaces are irradiated with high energy ions. The investigator and his colleagues study this problem using both computer simulations and analytic methods, coupled with experiments, in an effort to understand whether ion beams can create specific, technologically useful, patterns in a surface. Another project studies the self-assembly of elastic sheets, in an effort to understand whether it is possible to create small objects which self-assemble into useful components in three dimensions. A third project studies the process when a droplet of liquid splashes on a solid substrate. Despite the ubiquity of this phenomenon it is not known why a droplet creates a splash. On the educational front, the project aims to develop material to teach students both how to carry out computer simulations, and equally importantly how to evaluate whether they are correct.
