The main objective of this project is to investigate problems at the interface between mathematics, materials science, and biology, by (1) developing and applying state-of-the-art adaptive numerical methods to large-scale computation, and (2) performing analytical, numerical, and modeling studies of important constituent processes. At the interface between mathematics and materials science, we will focus on fundamental studies of nanoscale, crystalline thin films, where we will investigate the dynamics and coarsening of nanometer-scale structures in strongly anisotropic systems. Our study is important not only for gaining a fundamental understanding of thin-film evolution, but also for elucidating mechanisms that can be used to promote spatially ordered nanostructures, and thus has the potential to impact the design of novel electronic devices such as high-efficiency lasers, and quantum cellular automata architectures. At the interface between mathematics, materials science, and biology, we will focus on fundamental studies of inhomogeneous biomembranes, where we will investigate the coupling between shape deformation, fluid flow, and surface phase segregation of the multiple lipid components that comprise the biomembrane. Our fundamental study has the potential to impact the understanding of many cellular and subcellular biological processes, since biomembranes play an active and critical role in cell functions such as cell locomotion, adhesion, signal transduction, etc. In spite of the very different physical origins of the problems we propose to study, the mathematical governing equations have many similarities and thus can be treated using common analytical and computational techniques. For example, a key component of our study is to investigate the central role that the surface tension and bending forces play in the dynamics of both applications. As part of this study, we will develop new mathematical and adaptive numerical techniques that will also have application beyond the present context; examples include alloy microstructures, emulsions and polymer blends, blood flow, tissue and solid tumor growth. Indeed, the ability of mathematical analyses to elucidate, predict, and control fundamental processes in seemingly disparate physical and biological systems, by exploiting the commonalities in their mathematical descriptions, is what makes mathematics so valuable. Finally, a course on crystal and thin-film growth for gifted high-school students will be developed as part of the Calif. State Summer School for Mathematics and Science (COSMOS) at UC Irvine. This course will also help to recruit new math and science majors and enhance the participation of high-school students in research. Two Ph.D students and one postdoctoral researcher will receive interdisciplinary training while performing the proposed work. The proposed work also broadens the participation of underrepresented groups, as one of the students is a woman.
