Three Vignettes in Applied Topology
KEAR 212
Speaker: Chad Giusti
A fundamental tool in applied topology is persistent homology, which measures how the topological structure of a parameterized space evolves with the parameter. This measurement is usually given in the form of a persistence diagram (PD), which encodes the isomorphism type of the resulting algebraic structure. By vectorizing PDs for parameterized complexes built out of observed data, standard statistical methods allow us to use these estimates to infer organization and structure of observed systems. However, simply comparing PDs is often insufficient to solve interesting problems. In this talk, we will survey three of our recent projects in applied topology that go beyond computation of static PDs: 1. Time-varying systems naturally produce paths of PDs. While vectorization schemes for PDs can be naively extended to this setting, these extensions lack important theoretical properties. Leveraging Chen's classical iterated integral construction, we develop a robust feature set for… Read more.