Topology is the study of properties of a space that do not change as the space is continuously deformed. While this provides a powerful approach to classify analytic spaces, it is only recently that topological techniques have been developed that apply to statistically-defined spaces. This is motivated in part by the existence of applications in the physical sciences, where measurements are inherently noisy and topological techniques provide an approach to data analysis that is often robust to experimental error. This talk will introduce the audience to statistical topology by considering three different statistically-defined spaces, along with some of the results and techniques that they have inspired. First, the Erdős-Rényi random graph will motivate the idea of a graph and the study of homology groups. Second, the branched polymer model will motivate the idea of a simplicial complex and the use of persistent homology to calculate fractal dimension. Third, the microstructure of metals will motivate the idea of a cell complex and the use of the cloth to identify the steady-state of a dynamical system.
This talk does not assume any prior knowledge of topology, graph theory, or materials science; where necessary, the relevant concepts will be introduced and explained at a level suitable for a general scientific audience.