Title: Algorithms for computing a minimal homology basis
Speaker: Tianqi Li - The Ohio State University
Abstract: Many applications in science and engineering require computing ``features" in a shape that is finitely represented by a simplicial complex. These features sometimes include topological features such as ``holes" and ``tunnels" present in the shape. A concise definition of these otherwise vague notions can be obtained by considering homology groups and their representative cycles. In particular, a $d$-dimensional homology basis, that is, a maximally independent set of cycles in the $d$-skeleton of the input simplicial complex can be taken as a representative of the ``holes" and ``tunnels" present in the shape. However, instead of any basis, one would like to have a homology basis whose representative cycles are small under some suitable metric, thus bringing the `geometry' into picture along with topology. In this talk, I mainly give the general landscape of algorithms for finding a 1-dimensional minimal homology basis when the input complex is a graph, a manifold, or a general simplicial complex. I then discuss higher dimensional minimal homology bases.
