Research and Education in Physical Mathematics

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BrennerDMS-00907985 This award is funded under the American Recovery andReinvestment Act of 2009 (Public Law 111-5). The investigatorand his colleagues study a range of problems in appliedmathematics, focusing on the coupling of the analytical methodsto numerical computation, in the context of both research andeducation. The proposed research projects address a range ofproblems in engineering and science using a combination ofapproximate analysis, computation, and experiments. The specificprojects include (1) a search for potentially unstable singularsolutions of the Euler equation, by focusing on initial dataconsisting of vortex filaments; (2) questions in microfluidmechanics, including developing novel methodologies for creatingthin polymeric fibers, and creating a biomimetic analogue of afungal spore shooter; (3) biological questions, primarily aimedat using theory and data analysis to understand morphologicalvariation in Darwin's Finches. Educational activities centeraround the general issue of how the concepts of appliedmathematics should be taught, given the prevalence ofcomputation. Two specific initiatives are undertaken: The firstinvolves a wiki-based method for teaching applied mathematics anddeveloping applied mathematics texts; the second involves thedevelopment of materials for explaining to broad audiences theways in which mathematics is used and has been used to impact ourworld. The activities of this project use mathematics andmathematical ideas to study, and hopefully solve, problems withbroad impact on the world in which we live. One set of topicsinvolves basic unsolved questions about the nature of fluidflows. Fluid flows occur throughout engineering and thesciences, and understanding how to predict their properties iscritical for critical societal problems and needs, such asenergy, environment materials, and engineering. The projectdevelops methods for using fluid flows to create novel materialsand material processes. Another part of the project aims tounderstand a fundamental question in biology, namely, what is thelogic behind which biological processes lead complex organisms,such as ourselves, to develop and grow? This part of the projectis based on a discovery of the investigator and his colleaguesthat quantitatively explains the morphological diversity ofDarwin's Finches, the classical example of adaptive radiation --the process in which a single species gives rise to multiplespecies that exploit different ecological niches. Theinvestigator extends this work to other bird species and uses theinsights gained to create a mathematical framework forunderstanding development more broadly. Finally, central to theproject is the educational mission of teaching undergraduates,graduate students and postdoctoral fellows how to effectivelyapply mathematics so that it can make a substantive difference inour world.