Given a metric space and a positive number $d$, the Vietoris–Rips (VR) complex of scale $d$ is the simplicial complex whose faces are all sets of diameter at most $d$. 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.