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 preprint, and 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. Here is a preprint of the result.

This project is joint work with Konstantin Mischaikow.

Joint work with Marcio Gameiro, Yasuaki Hiraoka, Shunsuke Izumi, Miroslav Kramar and Konstantin Mischaikow.

Given X-ray crystallography data of a protein molecule from the PDB, we build a van der Waal weighted alpha shape representation of that protein molecule by constructing cells around each atom center. Thus, to each protein we assoicate a set of persistence diagrams (one for each dimension). Using elementary physical principles, we identify certain structural features of molecules that are conjectured to impact compressibility. A simple parameter search through the persistence diagrams isolates these features and provides a robust measure which exhibits remarkable linear correlation with experimentally computed protein compresisbility. Preprint coming soon.