PECASE: Self-Sustained Interfacial Motions: Dynamics, Instabilities, and Singularities

该学院早期职业发展(CALEAR)计划奖的目标是调查和了解表面活性物质(表面活性剂)驱动的Marangoni效应背景下复杂流体界面的自持运动、奇点形成和拓扑变化的物理起源和动力学。实现这些目标将需要开发新的理论工具，以确定表面活性剂驱动的不稳定性和由此产生的自诱导动态运动的发生条件，并揭示界面奇异性在维持此类运动中的作用。这项拟议的研究将导致基于动力系统、微分几何、几何流动和奇点理论的协同作用的新的几何启发的理论方法的阐述。如果成功，这项研究计划还将帮助人们理解奇点和不稳定现象之间的相互作用，从而使人们能够在阿诺德将卡坦理论和奇点理论结合起来的问题上取得进展。界面的自我持续运动是生物体中所有马达的基础。从基础和实践的角度来看，揭示和人工创造这些功能都是重要的，例如在人工细胞的发展中。解决这些问题将推动科学和工程领域的新领域。特别是，拟议的研究计划将增强在中观和宏观尺度上理解某些类型的生物马达的机制和人工创造生物功能的理论基础。由于复杂的界面无处不在，这项研究计划的结果应该会为表面活性剂在从工程到生物等各种所需功能的众多应用中进行最有效的设计提供思路。在传播层面，所研究的现象提供了一个很好的机会，向人们介绍复杂程度不同的科学和技术，并鼓励科学培训。研究生、本科生和地区性高中生不仅将从课堂教学中受益，还将获得第一手研究经验。

The objective of this Faculty Early Career Development (CAREER) Program award is to investigate and understand the physical origin and quantify the dynamics of self-sustained motions, singularity formation, and topological changes of complex fluid interfaces in the context of Marangoni effects driven by surface-active substances (surfactants). Achieving these objectives will require developing new theoretical tools in order to identify the conditions for the occurrence of surfactant-driven instabilities and resulting self-induced dynamical motions, and uncover the role of interfacial singularities in sustaining such motions. The proposed research will lead to the elaboration of new geometrically inspired theoretical methods based on the synergy of dynamical systems, differential geometry, geometric flows, and singularity theories. If successful, this research program will also help one to understand the interplay between singularity and instability phenomena and thus allow one to make progress on Arnold's problem of combining Cartan's and singularity theories.Self-sustained motions of interfaces are at the basis of all motors in living organisms. Revealing and artificially creating such functions is important both from the fundamental and practical points of view, e.g. in the development of artificial cells. Addressing these questions will advance new fields in science and engineering. In particular, the proposed research program will enhance the theoretical basis for understanding mechanisms of certain types of biological motors at meso- and macroscopic scales and for creating biological functions artificially. Since complex interfaces are ubiquitous, the outcome of this research program should provide ideas for the most efficient designs in numerous applications of surfactants with desired functions ranging from engineering to biology. At the dissemination level, the studied phenomena provide a great opportunity to introduce people to science and technology at varying degrees of complexity and to encourage scientific training. Graduate, undergraduate, and regional high-school students will benefit not only through classroom instruction, but will also receive firsthand research experience.