Mathematics of Collective Behavior: From Self-Organized Dynamics to Fluid Turbulence

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The theory of collective behavior is a rapidly developing area of mathematics that studies emergence of global phenomena in many biological, social and technological systems. Examples of such systems include flocks of birds, schools of fish, social networking, and exchange of political opinions among people. Although these systems arise from seemingly disconnected contexts, they share many common principles of self-organization. The project puts forward a comprehensive program of research to study those principles. More specifically, it aims to understand how "agents" in collective systems driven by their natural laws of communication can build global formations, such flocks, using only local interactions. This project will systematically classify types of global formations, their structure and stability under presence of an external force, such as gust of wind in the context of bird flocks or media influence in the context of opinion dynamics. On the way, the project will bring a connection between real applications and a new purely theoretical area of mathematics that studies spread of information in diffusive systems. The project will involve one graduate student, who will be involved in theoretical aspects as well as numerical and visual implementations of the proposed research. The technical implementation of the goals of the project involves modeling of emergent dynamics through a novel system of fractional parabolic equations. The system is designed to predict the end-state behavior of a "flock" through a careful long-time analysis of its global solutions. The new feature of proposed model involves inclusion of an adaptive anisotropic diffusion kernel as the main alignment mechanism of global behavior. This feature is not only mathematically motivated but is also observed in many biological and social congregations. The tools used in the analysis will be extended to study regularity problems in a more general class of parabolic and elliptic equations. The project will address validity of the classical laws such as energy conservation and will bring connection with the celebrated Onsager conjecture of 1949 that studies sharp regularity conditions under which such laws hold.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.