Biological aggregations such as bird flocks, fish schools, and insect swarms are striking examples of self-organization, and serve as the inspiration for algorithms in robotics, computer science, applied mathematics, and other fields. Aggregations give rise to massive amounts of data, for instance, the position and velocity of each group member at each moment in time during a field observation or numerical simulation. Interpreting this data to characterize the group's dynamics can be a challenge. To this end, we apply computational persistent homology — the workhorse of the field of topological data analysis — to the aggregation models of Vicsek et al (1995) and D’Orsogna et al. (2006). We assign a topological signature to each set of simulation data. This signature identifies dynamical events that traditional methods do not. Finally, we pose open questions related to topological signatures averaged over many simulations of stochastic models. This talk assumes no prior knowledge of topology.
Professor of Mathematics Chad Topaz (A.B. Harvard, Ph.D. Northwestern) is an applied mathematician at Williams College. Chad examines problems in biology, chemistry, physics, and the social sciences through several lenses, including data science, modeling, analysis, topology, geometric dynamical systems, numerical simulation, and experiment... all with an eye towards understanding and predicting complex behavior. Passionate about scientific communication and discourse, Chad has delivered over 100 talks at colleges, universities, and scientific meetings, and has co-organized numerous interdisciplinary minisymposia and workshops on chemical reaction diffusion systems, biological swarming, agent-based models, and related topics. His honors include a New Directions Research Professorship at the Institute for Mathematics and its Applications (the first given to a liberal arts college faculty member), a Kavli Frontiers Fellowship from the National Academy of Sciences, a Board of Trustees Award from Macalester College, and the 2013 Outstanding Paper Award of the Society for Industrial and Applied Mathematics.
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