Visual Demo of Discrete Stratified Morse Theory (Media Exposition)

Abstract

Discrete stratified Morse theory, first introduced by Knudson and Wang, works toward a discrete analogue of Goresky and MacPherson’s stratified Morse theory. It is inspired by the works of Forman on discrete Morse theory by generalizing stratified Morse theory to finite simplicial complexes. The class of discrete stratified Morse functions is much larger than that of discrete Morse functions. Any arbitrary real-valued function defined on a finite simplicial complex can be made into a discrete stratified Morse function with the proper stratification of the underlying complex. An algorithm is given by Knudson and Wang that constructs a discrete stratified Morse function on any finite simplicial complex equipped with an arbitrary real-valued function. Our media contribution is an open-sourced visualization tool that implements such an algorithm for 2-complexes embedded in the plane, and provides an interactive demo for users to explore the algorithmic process and to perform homotopy-preserving simplification of the resulting stratified complex.