Controlling Geometry: Applications in Physics, Biology, and Manifold Learning

通过将精细结构组装成最佳的、理想的模式，可以创建新材料。这个项目是关于控制物体形状的。控制液滴的形状可以产生用于生物技术的新型微流体装置。了解生物膜（如细胞膜）的形状和曲率可以为细胞如何运动和功能提供新的见解。如果适当地“展开”，人类的理解和意义可以从高维数据中提取出来。该研究项目的目标是创建新的数学方法/算法来指导自组织、材料设计和从高维数据中学习。该研究将为运动形状和几何形状的最优控制奠定基础。这个项目的一部分包括与小学生和中学生互动，通过PI的“与科学家坐在一起”计划来强调几何在应用中的重要性。该计划提供了一个非正式的氛围，动手活动，以激励学生，特别是少数民族和代表性不足的群体，追求STEM。研究目标是建立新的自组织数学技术和数值方法。一些例子是自组装，控制液滴形状（微流体），折叠生物膜，以及通过非线性流形约简进行数据分析/可视化。这些新方法将开辟材料设计的新领域，使对物理现象的前所未有的控制成为可能，在微生物学方面产生新的认识，并在“大数据”的统治下使其可以直接可视化。该研究将为Landau-de Gennes （LdG）模型的q张量值解创建第一个奇异Maier-Saupe势的数值格式。我们还开发和分析了LdG在可变域上的非拟合有限元方法（fem），这些方法与以下项目相关。我们将创建控制物理和工程问题中几何结构随时间演变的方法，例如液晶和液滴形状。设计新的方法来建模和模拟生物膜，很好地近似全曲率信息（即全形状算子）使用表面有限元方法。此外，我们将扩展我们的生物膜表面有限元技术，将高维数据非线性降维到低维空间（用于数据分析和可视化）。研究的其他方面将为这里开发的方法创建开源软件，使用PI自己的软件包，FELICITY和AHF，以及其他开源选项（例如Firedrake）。此外，PI还将利用“与科学家坐在一起”的项目（上文提到的）对中小学生进行教育。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

New materials can be created by directing the assembly of fine-scale structures into optimal, desired patterns. This project is about controlling the shapes of things. Controlling the shapes of droplets can yield new micro-fluidic devices for bio-technology. Understanding the shape and curvature of bio-membranes (e.g. cell membranes) can give new insight into how cells move and function. And human understanding and meaning can be extracted from high dimensional data if it is properly "unfolded." The goal of this research project is to create new mathematical methods/algorithms to guide self-organization, material design, and learn from high dimensional data. This research will lay the groundwork for the optimal control of moving shapes and geometries. Part of this project involves interacting with elementary and middle school students to highlight the importance of geometry in applications through the PI's "sit-with-a-scientist" program. The program provides an informal atmosphere, with hands-on activities, to motivate students, especially minorities and under-represented groups, to pursue STEM.The research objective is to create new mathematical techniques and numerical methods for self-organization. Some examples are self-assembly, controlling droplet shape (micro-fluidics), folding biomembranes, and data analysis/visualization through non-linear manifold reduction. These new methods will open new frontiers of material design, enable unprecedented control of physical phenomena, yield new understanding in micro-biology, and reign in "big data" so it can be directly visualized. The research will create the first numerical scheme for the singular Maier-Saupe potential for the Q-tensor-valued solution of the Landau-de Gennes (LdG) model. We also develop and analyze, unfitted finite element methods (FEMs) for LdG, on variable domains, that connect with the following items. We will create methods for controlling the time-dependent evolution of geometric structures in physics and engineering problems, e.g. in liquid crystals and liquid droplet shape. Design new methods for modeling and simulating bio-membranes that well approximate full curvature information (i.e. the full shape operator) using a surface finite element method. Moreover, we will extend our bio-membrane surface FEM techniques to do non-linear dimension reduction of high dimensional data to a low-dimensional space (for data analytics and visualization). Other aspects of the research will create open source software for the methods developed here, using both the PI's own packages, FELICITY and AHF, and other open-source options (e.g. Firedrake). In addition, the PI will educate elementary and middle school students using his "sit-with-a-scientist" program (mentioned above).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.