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

通过将精细结构的组装引导成最佳的、所需的图案，可以创造出新材料。这个项目是关于控制物体的形状的。控制液滴的形状可以产生用于生物技术的新型微流控设备。了解生物膜(例如细胞膜)的形状和曲率可以为细胞如何运动和功能提供新的见解。而人类的理解和意义可以从高维数据中提取出来，如果它被适当地“展开”的话。这个研究项目的目标是创造新的数学方法/算法来指导自组织、材料设计和从高维数据中学习。这项研究将为运动形状和几何的最优控制奠定基础。这个项目的一部分包括与中小学生互动，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。该计划提供了一种非正式的氛围，包括动手活动，以激励学生，特别是少数族裔和代表不足的群体，追求STEM。研究目标是创造新的数学技术和数值方法，用于自组织。一些例子包括自组装、控制液滴形状(微流体)、折叠生物膜以及通过非线性流形简化进行数据分析/可视化。这些新方法将开辟材料设计的新前沿，实现对物理现象的前所未有的控制，在微生物学方面产生新的理解，并控制大数据，使其可以直接可视化。这项研究将为Landau-de Gennes(LDG)模型的Q张量解创建第一个奇异Maier-Saupe势的数值格式。我们还发展和分析了变域上LDG的不适应有限元方法(FEMS)，这些方法包括以下几个方面。我们将创造在物理和工程问题中控制几何结构随时间变化的方法，例如在液晶和液滴形状中。设计新的方法来模拟和模拟生物膜，用曲面有限元方法很好地逼近全曲率信息(即全形状算子)。此外，我们将扩展我们的生物膜表面有限元技术，对高维数据进行非线性降维到低维空间(用于数据分析和可视化)。这项研究的其他方面将为这里开发的方法创建开放源码软件，使用PI自己的包Felicity和AHF，以及其他开放源码选择(例如Firedrake)。此外，该基金会还将利用他的“与科学家坐在一起”项目(如上所述)对中小学生进行教育。这一奖项反映了NSF的法定使命，通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为是值得支持的。

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.