Discrete Morse theory is a wonderful tool from combinatorial algebraic topology which transports Morse theory from smooth manifolds to the realm of cell complexes. Critical cells of discrete Morse functions on a given complex generate a new (and smaller) complex possessing the same homology groups. We apply discrete Morse theory to filtrations of complexes and provide a novel approach to efficient computation of persistent homology.

Here is a link to the associated software project which computes persistent homology very fast.

This is joint work with Konstantin Mischaikow.

Given a compact Riemannian submanifold of Euclidean space, it is possible to recover (with high confidence) the homotopy type of this manifold from a sufficiently dense uniform point sample. In our work, we demonstrate a similar probabilistic result for functions. We show that under mild assumptions it is possible to reconstruct a continuous function between two such manifolds up to homology if we are given: a) dense point samples taken from the domain and the range, and b) the images of the domain samples under the action of the function. The result is robust under perturbations arising from bounded sampling noise.

This project is joint work with Konstantin Mischaikow.

Joint work with Marcio Gameiro, Yasuaki Hiraoka, Miroslav Kramar and Konstantin Mischaikow. Details coming soon!