Research and Education in Physical Mathematics

本项目涉及流体力学、材料科学和生物学等当前重要的科学和技术问题。具体来说，第一个项目包括对液滴溅落在固体或液体基片上之前发生的事件进行数学研究。这是一个具有巨大技术重要性的问题，其应用范围从喷墨打印机的设计到海洋表面二氧化碳的夹带。尽管如此，这个过程的基本特征仍然不被理解，我们的数学模型旨在解决这个关键问题。第二个主题是影子光刻的数学研究，影子光刻是一种用于创建小纳米结构阵列和新型光学材料的新兴方法。我们的目标是开发一个完整的数学描述模式，可以形成使用这种方法，导致一个结构的逻辑字典，可以作出。第三个项目旨在更好地了解生物膜的发展，细菌菌落牢固地附着在表面上。这些都被认为是医学和技术领域感染和衰败的原因。导致生物膜发育的生物转变尚不清楚。我们将建立一个模型生物中这些转变的数学模型。这有可能导致对生物膜发展和如何控制它的新的基本见解。更广泛的影响集中在研究生和博士后的人才培养，以及通过烹饪教授科学和数学的教材的开发。我们在科学和烹饪方面的教学计划已经通过在线课程覆盖了近10万人，我们的目标是将范围和内容扩展到中学教育。液滴溅射的研究将从描述液体中的不可压缩势流开始，并结合周围空气中的可压缩动力学。用润滑近似法很好地捕捉了空气中的动力学。以前的工作是用势流来模拟液相的初始动力学。然而，理论估计和实验都表明，黏性力在液体的后期变得重要；我们将扩展数学描述来包括这一点，使用液体中粘性效应的边界层描述。阴影光刻的研究使用几何方法来完全表征由球体阵列产生的阴影。结构的数学表征将用球体的二维掩模来完成，也将把目前的实验方法扩展到透明矩阵中不透明球体的三维掩模。后者可能导致制造复杂器件的新方法。生物膜的研究采用一种新的实验方法对枯草芽孢杆菌的基因表达进行全膜成像，这使人们对生物膜中发生变化的时空动态有了前所未有的了解。该项目将在实验中建立营养场的数学模型，然后研究营养场本身是否足以解释这些转变，或者是否还需要信号分子。预测将在实验中得到验证，并可能导致对芽孢杆菌如何发展的理解。

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.