Research and Education in Physical Mathematics

该项目解决了当前科学和技术重要性的几个问题，从流体力学，材料科学和生物学。具体来说，第一个项目包括对液滴在固体或液体基底上飞溅之前发生的事件的数学研究。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的二氧化碳夹带。尽管如此，这个过程的基本特征仍然不清楚，我们的数学模型旨在解决关键问题。第二个主题是阴影光刻的数学研究，这是一种新兴的方法，用于创建小型纳米结构和新光学材料的阵列。我们的目标是开发一个完整的数学描述的模式，可以形成使用这种方法，导致一个逻辑字典的结构，可以。第三个项目旨在更好地了解生物膜的发展，即强烈粘附在表面上的细菌菌落。这些都被认为是医学和技术中感染和腐烂的原因。导致生物膜发展的生物学转变尚未得到很好的理解。我们将在模式生物中建立这些转变的数学模型。这有可能导致新的基本见解生物膜的发展和如何控制它。更广泛的影响集中在研究生和博士后的人员发展，以及通过烹饪教学科学和数学的教育材料的开发。我们的科学和烹饪教学计划通过在线课程接触了近10万人，我们的目标是将范围和内容扩大到中学教育。对液滴飞溅的研究将从描述液体中的不可压缩势流开始，并与周围空气中的可压缩动力学耦合。空气中的动力学很好地被润滑近似所捕获。以前的工作已经模拟了在液相中使用势流的初始动力学。然而，理论估计和实验都表明，粘性力在液体中的后期阶段变得重要;我们将扩展数学描述，包括这一点，使用边界层描述的液体中的粘性效应。阴影光刻技术的研究是利用几何学的方法来全面地描述球体阵列所产生的阴影。结构的数学表征将与两个二维掩模的领域，也将扩展目前的实验方法，以三维掩模的不透明的球体在一个透明的矩阵。后者可能导致制造复杂器件的新方法。生物膜的研究使用了一种新的实验方法，对枯草芽孢杆菌中的基因表达进行全膜成像，这对生物膜中如何发生变化的时空动态进行了前所未有的研究。该项目将在实验中构建营养场的数学模型，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。预测将在实验中进行测试，并可能导致了解如何发展展开芽孢杆菌。

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.