Recorded 26 October 2022. Emily King of Colorado State University presents "PCA is not dead: Vectorized persistent homology and flag medians" at IPAM's Mathematical Advances for Multi-Dimensional Microscopy Workshop. Abstract: In this talk two methods will be presented which both make use of principal component analysis. The first involves applying persistent homology to atmospheric data and then vectorizing the output. Surprisingly, principal component analysis (PCA) enables a very low dimensional representation of this data that enables clustering. A low-level explanation of persistent homology will be presented for those not familiar with topological data analysis. The second is a novel method inspired by the Weiszfeld algorithm to find prototypes of subspaces of possibly different dimensions which are robust to outliers. This algorithm involves solving a particular series of weighted PCA problems. Some applications to computer vision will be presented. Learn more online at: http://www.ipam.ucla.edu/programs/workshops/workshop-ii-mathematical-advances-for-multi-dimensional-microscopy/?tab=schedule