This proposal addresses the coupling of the analytic tools of applied mathematics to numerical computation, in the context of both research and education. The research addresses several unsolved scientific problems through a combination of approximate analysis and computation. The specific projects include: (1) The investigator will determine the continuum equations for a sediment at low volume fraction, with special attention to the proper way to understand particle diffusion, and the role of intrinsic noise in the dynamics; (2) The investigator will attempt to uncover the origin of recently observed violations of the no slip boundary condition on hydrophobic surfaces. In particular the investigator will address both the stability and the dynamical consequences of small bubbles adhering to the solid surface, as observed experimentally; (3) The investigator will study the role of elastic stresses in influencing the morphology of growing tissue, and in particular develop mathematical models for the morphology of growing yeast colonies; (4) The investigator will develop accurate short time solutions to the Fokker Plank equation in arbitrary potentials as a means towards a novel algorithm for numerical computation of dynamics in fluctuating environments. The educational programs center around teaching a unified view of ''the art of approximation'' (including both asymptotic and numerical methods) to students at all levels. New courses at the graduate level and the beginning undergraduate level will be developed to teach how to effectively combine analysis with computation, for understanding difficult mathematics problems.
The proposal consists of two parts: the research questions to be investigated address important problems of current interest, whose solution could lead to significant scientific and technological advances, with impact on fields ranging from applied mathematics, to materials science, to geophysics and biology. The specific problems range from developing mathematical models for understanding when fluid sticks to surfaces to developing new ways of simulating complex objects undergoing Brownian motion to developing a proper mathematical description for a set of particles falling in a fluid. These are all fundamental questions whose solution could significantly advance the design and development of materials for nanotechnology. For example, all nano-scale objects in an underlying fluid undergo Brownian motion; the lack of an efficient numerical algorithm for simulating these objects severely hampers computational design efforts. The second part of the proposal is educational, focusing on developing methodologies for teaching the art of approximation (combining computation and approximate analysis in a single framework). By finding ways of training students at all levels to effectively combine analysis and computation, they will be better educated to address the important questions of today. The materials that are developed will be broadly disseminated, both through web-based and traditional texts.
