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Discrete Stratified Morse Theory: A User's Guide Knudson, Kevin Wang, Bei Discrete Morse theory stratified Morse theory topological data analysis Inspired by the works of Forman on discrete Morse theory, which is a combinatorial adaptation to cell complexes of classical Morse theory on manifolds, we introduce a discrete analogue of the stratified Morse theory of Goresky and MacPherson. We describe the basics of this theory and prove fundamental theorems relating the topology of a general simplicial complex with the critical simplices of a discrete stratified Morse function on the complex. We also provide an algorithm that constructs a discrete stratified Morse function out of an arbitrary function defined on a finite simplicial complex; this is different from simply constructing a discrete Morse function on such a complex. We borrow Forman's idea of a "user's guide," where we give simple examples to convey the utility of our theory. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Kevin Knudson and Bei Wang 2018 Is Part Of LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2018.54 urn:nbn:de:0030-drops-87671 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.54 eng https://creativecommons.org/licenses/by/3.0/legalcode