Chad Topaz (Williams College)

Title: Topological data analysis of collective dynamics

Abstract:

From nanoparticle assembly to synchronized neurons to locust swarms, collective behaviors abound anywhere in nature that objects or agents interact. Investigators modeling collective behavior face a variety of challenges involving data from simulation and/or experiment. These challenges include exploring large, complex data sets to understand and characterize the system, inferring the model parameters that most accurately reflect a given data set, and assessing the goodness-of-fit between experimental data sets and proposed models. Topological data analysis provides a lens through which these challenges may be addressed. This talk consists of three parts. In the first part, I apply topological data analysis to the seminal aggregation model of Vicsek et al. (1995) in order to identify dynamical events that traditional methods do not. In the second part, I use topological data analysis to choose between unbiased correlated random walk models that potentially describe motion tracking experiments on pea aphids. Finally, moving towards a theory of reduced topological descriptions of complex behavior, I present open questions on the topology of random data, complementing research in random geometric graph theory. Throughout the talk, the key approach is to characterize a system's dynamics via the time-evolution of topological invariants called Betti numbers, accounting for persistence of topological features across multiple scales. No prior knowledge of topology is necessary.