Given a metric space and a positive number , the Vietoris–Rips (VR) complex of scale is the simplicial complex whose faces are all sets of diameter at most . Recently, there’s been a push to understand the VR
How many holes (non-trivial homologies) are there?
How big is the largest facet?
How small is the smallest facet?
How many differently-sized facets are there?
Based on joint work with Joe Briggs and Ziqin Feng.