This is joint work with Konstantin Mischaikow

We introduce an efficient preprocessing algorithm to reduce the number of cells in a filtered cell complex while preserving its persistent homology groups. The technique is based on an extension of combinatorial Morse theory from complexes to filtrations of cell complexes.

This paper provides the theoretical basis for the Perseus software project designed to compute persistent homology of various types of filtrations.

Status: Submitted to Discrete and Computational Geometry.

This project is joint work with Konstantin Mischaikow.

Building upon a fantastic result of Niyogi, Smale and Weinberger from this paper, we provide bounds on sizes of uniform samples required to reconstruct a Lipschitz-continuous function between compact Riemannian manifolds up to homology with high confidence. We show that this reconstruction is robust to bounded sampling noise.

Status: Submitted to Journal of Computational Dynamics.