Random \v{C}ech Complexes on Manifolds with Boundary

Understand

Let $M$ be a compact, unit volume, Riemannian manifold with boundary.

• In this paper we study the homology of a random \v{C}ech-complex generated by a homogeneous Poisson process in $M$.
• Our main results are two asymptotic threshold formulas, an upper threshold above which the \v{C}ech complex recovers the $k$-th homology of $M$ with high probability, and a lower threshold below which it almost certainly does not.
• These thresholds are close together in the sense that they have the same leading term.