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

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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.