Collaborative Research: Dynamics and pattern formation of nonlocal collective motion and assembly

该项目的研究人员将研究在非局部集体运动规律下演化的粒子或作用剂的动力学和图案形成。具体地说，PI和共同PI将研究集体行为表现出非平凡的协维度的系统。这种粒子系统的数学横跨许多学科，从物理、化学和生物学到控制理论和工程。这些领域的现代应用包括蛋白质折叠、胶体稳定性和纳米颗粒自组装成超分子结构。在生物学上，类似的数学模型有助于解释在病毒衣壳、蝗虫群和细菌群体中观察到的复杂现象。这个项目的第一阶段将应用和扩展最近开发的数学工具来识别各种物理和化学相互作用，这些相互作用将自然地产生共维一结构。这直接应用于过程的研究，例如多金属氧酸盐(POM)分子簇自组装成球形超分子结构，在这些过程中，实验证据是压倒性的，但对潜在的形成机制缺乏理论上的理解。当相互作用是各向同性时，PI、co-PI和他们的合作者在这类机制的数学理论上取得了许多重要的发展。该项目的这一部分旨在进一步发展这一理论，以证明对其他学科的广泛研究人员有用。拟议的研究项目基本上是跨学科的，因为它从化学、生物和工程等不同领域的未解释现象中得出数学问题。PI和co-PI将应用从动力学系统、偏微分方程组、数学建模和计算方法中提取的广泛的数学工具来解决这些领域中活跃研究的几个问题，包括所谓的纳米自组装中的“设计者势问题”。通过阐明是什么物理力量驱动POM自组装和其他相关现象，如病毒衣壳的形成，我们将为如何设计形成这些重要结构的亚单位的问题提供严格的解决方案。被证明的职业目标之一是促进数学领域中代表人数不足的群体的加入。PI拥有作为旧金山大学多元化学生群体导师所需的个人经验和批判性观点。让这些来自代表性不足的群体的学生接触尖端的数学研究，再加上教师的指导，将激励他们中的许多人攻读数学科学的研究生学位。鉴于美国联邦大学的低收入学生在攻读全日制学位课程期间被迫从事兼职工作的比例很高，这笔拨款支持的学生研究将被证明是该项目招生工作不可或缺的一部分。国际数学联合会在从代表性不足的群体中招募有才华的学生进入数学领域的记录得到了证实，这笔赠款将使这些活动得以继续和扩大。

The investigators of this project will study the dynamics and pattern formation of particles or agents that evolve under non-local collective motion laws. Specifically, the PI and co-PI will study systems in which collective behavior manifests non-trivial co-dimension. The mathematics of such particle systems pervades many disciplines, ranging from physics, chemistry and biology to control theory and engineering. Modern applications in these areas include protein folding, colloid stability and the self-assembly of nanoparticles into supramolecular structures. In biology, similar mathematical models help explain the complex phenomena observed in viral capsids, locust swarms and colonies of bacteria. The first phase of this project will apply and expand recently developed mathematical tools to identifiy various physical and chemical interactions that will naturally produce co-dimension one structures. This has a direct application to the study of processes, such as the self-assembly of Polyoxometalate (POM) molecular clusters into spherical supramolecular structures, in which experimental evidence is overwhelming but a theoretical understanding of the underlying formation mechanisms is lacking. The PI, co-PI and their collaborators have made numerous important developments in the mathematical theory of such mechanisms when the interaction is isotropic. This part of the project aims to further develop this theory in a manner that will prove useful to a broad set of researchers in other disciplines. The proposed research project is fundamentally interdisciplinary in that it derives mathematical problems from unexplained phenomena in diverse fields such as chemistry, biology and engineering. The PI and co-PI will apply a broad set of mathematical tools drawing from dynamical systems, partial differential equations, mathematical modeling and computational methods to solve several problems that are subject to active research in these fields, including the so-called 'designer potential problem' in nano self-assembly. By clarifying which physical forces drive POM self-assembly and other related phenomena, such as viral capsid formation, we will provide rigorous solutions to the question of how to design sub-units that form into these importantstructures. One of the demonstrated career goals of the PI is to further the inclusion of underrepresented groups in the field of mathematics. The PI has the personal experience and critical perspective necessary to serve as a mentor for the diverse student body at University of San Francisco. Exposing these students from underrepresented groups to cuttingedge mathematical research, coupled with faculty mentoring, will inspire many of them to pursue graduate degrees in the mathematical sciences. The student research supported by this grant will prove indispensible to the project's recruitment efforts given USF's high proportion of low-income students who are compelled to work part-time while they pursue full-time academic degree programs. The PI has a proven track record of recruiting talented students from underrepresented groups into mathematics and this grant would allow these activities to continue and expand.