Reconstruct Morphological Phases from Nonlocal Geometric Systems

模式的形成是自组织原则的有序结果。形态阶段是复杂的几何结构，在物理和生物系统中形成结构模式。例子包括嵌段共聚物的形态相、动物皮毛和细胞发育中的皮肤色素沉着。在这些模式形成系统中常见的是，对同质性的偏离对其进一步增加具有强烈的正反馈。此外，模式的形成为局部自增强过程提供了更大范围的限制。研究模式的形成揭示了这些系统的机械、光学、电学、离子、势垒和其他性质。在这个项目中，研究者研究具有多种成分的生物和物理系统的形态阶段。这些阶段通常是在实验中观察到的，而目前还没有理论或模型预测它们。研究者的目标是进行严格的分析，不仅预测这些结构的存在，而且预测它们的特殊性质。形态相在现代材料中很重要，例如嵌段共聚物，它被商业上用作热塑性弹性体：酒瓶塞、光纤电缆的外层、粘合剂、沥青改性剂或人工器官技术。研究生和本科生参与该项目的工作。研究者研究物理和生物系统的形态阶段。特别感兴趣的体系是Ohta-Kawasaki， FitzHugh-Nagumo， Gierer-Meinhardt以及三和四嵌段共聚物。典型的形态相由由窄界面分隔的微畴组成。通常，窄界面形态相可以通过其尖锐的界面限制来近似，其中成分被没有厚度的界面完全分开。在尖锐的界面情况下，本项目研究环面、霍普夫链、双气泡和三重气泡的组合。提出了用尖锐界面形态相作为骨架来重构窄界面形态相的分析方法。通过重整化和无限维降维技术，其中库仑力被视为远程相互作用，导出了低维结构的自由能。研究生和本科生参与该项目的工作。

Pattern formation is an orderly outcome of self-organization principles. Morphological phases are complex geometric structures that form as structured patterns in physical and biological systems. Examples include morphological phases in block copolymers, animal coats, and skin pigmentation in cell development. Common in these pattern-forming systems is that a deviation from homogeneity has a strong positive feedback on its further increase. In addition, pattern formation provides a longer ranging confinement of the locally self-enhancing process. Studying pattern formation reveals the mechanical, optical, electrical, ionic, barrier, and other properties of these systems. In this project, the investigator studies morphological phases in biological and physical systems having multiple constituents. Often these phases have been observed in experiments where no current theory or models predict them. The investigator aims for a rigorous analysis that not only predicts the existence of these structures but also their particular properties. Morphological phases are important in modern materials such as block copolymers, which are commercially used as thermoplastic elastomers: wine bottle stoppers, outer coverings for optical fibre cables, adhesives, bitumen modifiers, or in artificial organ technology. Graduate and undergraduate students participate in the work of the project.The investigator studies morphological phases in physical and biological systems. Systems of particular interest are the Ohta-Kawasaki, FitzHugh-Nagumo, Gierer-Meinhardt, and tri- and tetrablock copolymers. Typically a morphological phase consists of micro-domains separated by narrow interfaces. Often, narrow-interface morphological phases can be approximated by their sharp interface limits, where constituents are fully separated by interfaces with no thickness. In the sharp interface case this project investigates assemblies of tori, Hopf links, double bubbles, and triple bubbles. Analytical methods are developed to use sharp interface morphological phases as skeletons to reconstruct morphological phases with narrow interfaces. Through renormalization and an infinite-dimensional reduction technique, where Coulomb forces are treated as long-range interactions, free energies of lower-dimensional structures are derived. Graduate and undergraduate students participate in the work of the project.