Persistent Betti numbers of random \v{C}ech complexes

Understand

We study the persistent homology of random \v{C}ech complexes.

• Generalizing a method of Penrose for studying random geometric graphs, we first describe an appropriate theoretical framework in which we can state and address our main questions.
• Then we define the kth persistent Betti number of a random \v{C}ech complex and determine its asymptotic order in the subcritical regime.
• This extends a result of Kahle on the asymptotic order of the ordinary kth Betti number of such complexes to the persistent setting.