For day 2, Sara Kalisnik and Andrew Blumberg are giving summaries of the talks we heard:
Gunnar Carlsson: Persistence barcodes are natural invariants of finite metric spaces useful for studying point cloud data. However, they are not well adapted to standard machine learning methods. One approach to this problems is to equip the set of barcodes with a metric, but an even simpler would be to provide coordinates for the set of barcodes. Gunnar Carlsson talked about coordinatizations of barcode spaces and their properties. He also proposed some ways in which they could be used to obtain information from multidimensional persistence profiles.
Vanessa Robins focused on on-going work about applications of discrete Morse theory and persistent homology to analyzing images of porous materials. She explained specific connections between the physical structure of the material and the patterns of persistence diagrams. The results presented were particularly exciting insofar as they represented a very serious and thorough application of computational topology to large quantities of real data.
Radmila Sazdanovic introduced categorification and provided several examples in pure mathematics, especially knot theory. Among others, categorifications of Jones and chromatic polynomials.
Sayan Mukherjee had two distinct themes. In the first part of the talk, he discussed results (with Boyer and Turner) on the "persistent homology transform" and "Euler characteristic transform", which roughly speaking are invariants of an object in or obtained by taking the ensemble of persistent homology or Euler characteristic of slices (relative to some fixed orientation vector). It turns out these are sufficient statistics and moreover seem quite successful in classification problems (e.g., for primate bones). The second part of the talk focused on the problem of manifold learning in the context of mixtures of hyperplanes of different dimensions. The key insight is that a Grassmanian embedding due to Conway, Hardin, and Sloane allows the use of distributions on the sphere to carry out statistical procedures on spaces of hyperplanes.