Nonlocal Aggregation Dynamics with Near-Neighbor Weighting, and the Associated Geometry of Measures

In this thesis we introduce a new weighted-averaging variant of the familiar "nonlocal biological aggregation equation" in Euclidean space, with weights dependent on the nearness of neighbors, which are added for more realism and flexible modeling. We discover how the gradient flow structure of the original equation is realized again via the introduction of a new metric tensor, one that penalizes movement in crowded configurations (nonlocally). We interpret this metric tensor and its global metric, examine the formal differential geometry structure, understand its boundedness when infinite spreading can occur, and finally establish the topology for a version of the metric defined in a bounded set. Numerical simulations follow to illustrate the behavior of the aggregation dynamics and the metric's geodesics.