This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The investigator and his colleagues study a range of problems in applied mathematics, focusing on the coupling of the analytical methods to numerical computation, in the context of both research and education. The proposed research projects address a range of problems in engineering and science using a combination of approximate analysis, computation, and experiments. The specific projects include (1) a search for potentially unstable singular solutions of the Euler equation, by focusing on initial data consisting of vortex filaments; (2) questions in microfluid mechanics, including developing novel methodologies for creating thin polymeric fibers, and creating a biomimetic analogue of a fungal spore shooter; (3) biological questions, primarily aimed at using theory and data analysis to understand morphological variation in Darwin's Finches. Educational activities center around the general issue of how the concepts of applied mathematics should be taught, given the prevalence of computation. Two specific initiatives are undertaken: The first involves a wiki-based method for teaching applied mathematics and developing applied mathematics texts; the second involves the development of materials for explaining to broad audiences the ways in which mathematics is used and has been used to impact our world.
The activities of this project use mathematics and mathematical ideas to study, and hopefully solve, problems with broad impact on the world in which we live. One set of topics involves basic unsolved questions about the nature of fluid flows. Fluid flows occur throughout engineering and the sciences, and understanding how to predict their properties is critical for critical societal problems and needs, such as energy, environment materials, and engineering. The project develops methods for using fluid flows to create novel materials and material processes. Another part of the project aims to understand a fundamental question in biology, namely, what is the logic behind which biological processes lead complex organisms, such as ourselves, to develop and grow? This part of the project is based on a discovery of the investigator and his colleagues that quantitatively explains the morphological diversity of Darwin's Finches, the classical example of adaptive radiation -- the process in which a single species gives rise to multiple species that exploit different ecological niches. The investigator extends this work to other bird species and uses the insights gained to create a mathematical framework for understanding development more broadly. Finally, central to the project is the educational mission of teaching undergraduates, graduate students and postdoctoral fellows how to effectively apply mathematics so that it can make a substantive difference in our world.
