Before revision:  By extending a limit theorem for persistent homology on samples from point processes, we provide theoretical validation for our FWER and FDR control methods .
After revision:  A key idea in our paper is that one can compute an empirical distribution under a null hypothesis model that can be used for hypothesis testing in the same way that a normal or t-distribution is used in classical statistics. A argument for the use of this empirical null is based on simulations and limit theorems for persistent homology on samples from point processes, we provide theoretical validation for our FWER and FDR control methods .