RUI: Variational and Topological Approaches to Complex Dynamical Systems

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This research studies two pattern-forming systems in the natural world, aiming to create mathematical frameworks that capture natural laws and to characterize the behavior of the systems. The first system is vegetation in semi-arid environments, which often grow in large-scale, swirling striped patterns visible in satellite imagery. The investigator and his collaborators will create a class of models that are grounded in fundamental ecological principles and are informed by geographic information systems (GIS) observations. They will analyze these models and compare them to data. This work will provide insight into the working of our natural world in the area of vegetation ecology. The second system involves collective behaviors, like those that arise when particles, objects, or agents interact, ranging from nanoparticle assembly to pedestrian crowds to insect swarms. The behavior of these groups can be vexing to characterize as they are neither highly organized nor totally random. The investigator and his collaborators will study collective behavior through the lens of a mathematical area called computational topology, which examines the shape of data obtained from simulations of or experiments on these groups. This work will aid the development of strategies that help humanity understand complex data. Other features of this proposal include: extensive undergraduate student involvement; creation of a network of collaborators across four institutions; a pipeline from research to the classroom; enhancement of computational infrastructure; a focus on the participation of underrepresented groups; efforts to join the applied dynamics and topology mathematical communities; and a study of one-of-a-kind, unpublished mathematical manuscripts of John von Neumann.The investigator and his collaborators study two pattern-forming systems: (1) Semi-arid vegetation patterns are commonly modeled as reaction-diffusion PDE that incorporate some subset of geophysical and ecological ingredients selected by the modeler. The investigator proposes a new paradigm for modeling vegetation patterns that is based on the fundamental ecological principle of energy balance and is both motivated by and validated with GIS data. The primary mathematical contribution is the analysis of a class of optimization problems with nonlocal constraints. The primary applied contribution is a unified framework for vegetation pattern modeling. (2) The collective behaviors of groups of agents can be complex. Traditionally, investigators diagnose these behaviors by studying time series of order parameters chosen a priori to be of possible interest. Taking a different view, the Pl develops a topological theory of collective behavior, building the mathematical machinery to pass from models of collective behavior to simpler descriptions capturing their on-average topological behavior. The primary mathematical contribution is a simpler characterization of the complex collective dynamics that is framed in terms of the evolving Betti numbers of the system. The primary applied contribution is a framework for understanding collective behavior models that does not require an a priori choice of order parameters.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.