Rectifications of Homotopy Interleaving Distances

I gave this talk in the Applied Topology in Albany Seminar.

Topological data analysis employs an `interleaving distance’ between persistence spaces as a means of measuring when two such objects are approximately isomorphic. By generalizing this notion to homotopy interleavings, one can instead measure when persistent spaces are approximately weakly equivalent. In 2023, Andrew Blumberg and Michael Lesnick developed a universal homotopy interleaving distance. They additionally conjectured that this homotopy interleaving distance differed from the interleaving distance on a homotopy category by at most a multiplicative constant, which was later proved by Edoardo Lanari and Luis Scoccola. We covered the proof of this conjecture and detailed the failure of this phenomenon for multi-persistent spaces.