Reconstruct Morphological Phases from Nonlocal Geometric Systems

模式的形成是自组织原理的有序产物。 形态相是在物理和生物系统中形成结构化图案的复杂几何结构。 实例包括嵌段共聚物中的形态相、动物皮毛和细胞发育中的皮肤色素沉着。 这些模式形成系统的共同点是，偏离均匀性会对进一步增加产生强烈的正反馈。 此外，图案形成提供了局部自增强过程的更长范围的限制。 研究图案形成揭示了这些系统的机械，光学，电学，离子，屏障和其他特性。 在这个项目中，研究人员研究具有多种成分的生物和物理系统的形态学阶段。 这些阶段通常是在实验中观察到的，而目前没有理论或模型预测它们。 研究人员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，其在商业上用作热塑性弹性体：酒瓶塞，光纤电缆的外覆盖物，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。研究员研究物理和生物系统中的形态学阶段。 特别感兴趣的体系是Ohta-Kawasaki、FitzHugh-Nagumo、Gierer-Meinhardt以及三嵌段和四嵌段共聚物。 典型地，形态相由通过窄界面分离的微域组成。 通常，窄界面形态相可以通过其尖锐的界面极限来近似，其中组分被没有厚度的界面完全分离。 在尖锐的接口的情况下，该项目调查的集合环面，霍普夫链接，双泡，和三泡。 分析方法的发展，使用尖锐的界面形态相作为骨架重建形态相与窄界面。 通过重整化和无限维约化技术，库仑力被视为长程相互作用，低维结构的自由能推导出来。 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.