Research and Education in Physical Mathematics

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This project addresses several problems of current scientific and technological importance, from fluid mechanics, materials science, and biology. Specifically, the first project includes a mathematical study of the events that occur before a liquid droplet splashes on either solid or liquid substrates. This is a problem of immense technological importance, with applications ranging from the design of ink jet printers to the entrainment of carbon dioxide at the ocean surface. Nonetheless, basic features of the process are still not understood, and our mathematical models aim to resolve the critical issue. The second topic is a mathematical study of shadow lithography, an emerging method for creating arrays of small nanoscopic structures and new optical materials. We aim to develop a complete mathematical description of patterns that can form using this method, leading to a logical dictionary of structures that can be made. The third project aims to better understand the development of biofilms, bacterial colonies which strongly adhere to surfaces. These have been implicated as the cause of infection and decay both in medicine and technology. The biological transitions that lead to biofilm development are not well understood. We will develop a mathematical model of these transitions in a model organism. This has the potential to lead to new fundamental insights into biofilm development and how to control it. The broader impact centers on both personnel development of graduate students and postdocs, and the development of educational materials for teaching science and mathematics through cooking. Our teaching initiatives on science and cooking have reached nearly 100,000 people through an online class, and we aim to extend the reach and content to secondary education. The study of the splashing of droplets will start from a description of incompressible potential flow in the liquid, coupled to the compressible dynamics in the surrounding air. The dynamics in the air is well captured by a lubrication approximation. Previous work has modeled the initial dynamics in the liquid phase using potential flow. However, both theoretical estimates and experiments indicate that viscous forces become important in the liquid in the later stages; we will extend the mathematical description to include this, using a boundary layer description of the viscous effects in the liquid. The study of shadow lithography uses methods in geometry to completely characterize the shadows made by the array of spheres. Mathematical characterization of structures will be made with both two-dimensional masks of spheres and will also extend current experimental methods to three dimensional masks of opaque spheres in a transparent matrix. The latter could lead to new method for fabrication complex devices. The study of biofilms use a new experimental method for whole film imaging of gene expression in Bacillus subtilis, which gives an unprecedented look at the spatiotemporal dynamics of how changes occur in a biofilm. The project will construct mathematical models of the nutrient field in the experiment and then investigate whether the nutrient field itself is sufficient to explain the transitions, or whether signaling molecules are also required. Predictions will be tested in experiments and may lead to understanding of how development unfolds in Bacillus.