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 extend discrete Morse theory to filtrations of complexes and provide a novel approach to efficient computation of persistent homology. Here is a preprint.
The Perseus software project implements these ideas to compute persistent homology very quickly.
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 homotopy if we are given: a) dense point samples taken from the domain and the co-domain, 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.
Joint work with Konstantin Mischaikow and Steve Ferry.
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. Here is a preprint.
I am a post-doctoral researcher at UPenn working with Rob Ghrist. I received my PhD in Mathematics from Rutgers in October 2012 under the direction of Konstantin Mischaikow. Before coming to Rutgers, I earned a Bachelor's degree in Computer Engineering as well as a Master's degree in Mathematics at Georgia Tech in Atlanta, Georgia. Here is a brief CV. You can also find me on MathOverflow.
I develop topological tools for analyzing large and potentially high-dimensional datasets which arise from measurements made in a variety of contexts.