Imperfect Patterns-Existence, Stability, and Essential Spectrum

斑马身上的条纹是怎么来的？细胞膜是如何形成它们的形状并防止破裂的呢？这两个看似不相关的科学主题实际上可以通过本项目概述的一般数学框架来研究。这样的研究不仅适用于上述特定主题，也适用于由类似数学模型产生的任何物理系统，如沙漠中的沙模式、天空中的云模式、细胞裂变/聚变过程、太阳能电池中的铸态离聚体和肥皂水中的网络形态。这个项目的目的是系统地理解图案的不完美，这些图案在自然界中无处不在，在各种科学环境和技术应用中至关重要。研究者研究周期性图案中的缺陷，并应用于理解材料中晶界的发展；他还研究了双分子层中的缺陷，如细胞膜中的缺陷。该项目的一部分内容包括为本科生提供研究体验机会。研究者运用动力系统技术，结合泛函分析、微分几何、渐近分析和大偏差理论来研究图案的形成，重点研究本质谱在周期性图案和双层界面缺陷的形成和稳定性中的作用。最有趣和最具挑战性的情况发生在基本光谱接近原点时；在这里，他寻求条纹图案、两亲结构及其缺陷的形成机制的明确说明。经典的斯威夫特-霍恩伯格方程为周期模式的严格研究提供了一个原型。首先，研究了周期模式对瞬时扰动、恒定扰动和随机扰动的缺陷。其次，在不稳定的情况下，系统的局部偏置和旋转对称性共同产生各种线和点缺陷，如晶界、位错和位错。研究者研究晶界，提供一种新的功能分析机制来构建变形模式。他还研究了功能化Cahn-Hilliard FCH环境中两亲性形态的缺陷，提出了脂筏、端帽和y型连接形成的机制。在这里，线性化算子在双层界面上的基本谱一直到原点是连续的，但不是“简单的”。它可以作为理解基本光谱作用的基准。该项目的一部分内容包括为本科生提供研究体验机会。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。

How do zebras get their stripes? How do cell membranes get their shapes and keep from bursting into pieces? These two seemingly disconnected scientific topics can actually be investigated via a general mathematical framework outlined within this project. Such a study not only applies to the specific topics above, but also to any physical systems arising from similar mathematical models, such as sand patterns in deserts, cloud patterns in the sky, cell fission/fusion processes, cast ionomers in solar cells, and network morphology in soapy water. The aim of this project is to systematically understand imperfections of patterns, which are ubiquitous in nature and pivotal in various scientific settings and technical applications. The investigator studies defects in periodic patterns, with an application to understanding the development of grain boundaries in materials; he also studies defects in bilayers such as arise in cell membranes. Part of the project includes research experience opportunities for undergraduate students.The investigator applies dynamical systems techniques, combined with functional analysis, differential geometry, asymptotic analysis, and large deviation theory, to study pattern formation, with an emphasis on the role of the essential spectrum in the formation and stability of defects of periodic patterns and bilayer interfaces. The most interesting and challenging case occurs when the essential spectrum touches the origin; here he seeks explicit illustrations of formation mechanisms of stripe patterns, amphiphilic structures, and their imperfections. The classical Swift-Hohenberg equation provides a prototype for rigorous studies of periodic patterns. Firstly, the imperfections of periodic patterns to instantaneous, constant, and random perturbations are investigated. Secondly, in the case of instability, the local biases and the rotational symmetry of the system together give rise to various line and point defects, such as grain boundaries, dislocations, and disclinations. The investigator studies grain boundaries, providing a novel functional analytical machinery to construct deformed patterns. He also investigates defects of amphiphilic morphology in the functionalized Cahn-Hilliard FCH) setting, suggesting mechanisms of the formation of lipid rafts, end caps and Y-junctions. Here the essential spectrum of the linearized operator at bilayer interfaces is continuous up to the origin but is not "simple." It serves as a benchmark for understanding the role of essential spectra. Part of the project includes research experience opportunities for undergraduate students.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.