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

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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.