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Documents authored by Knudson, Kevin


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Media Exposition
Visual Demo of Discrete Stratified Morse Theory (Media Exposition)

Authors: Youjia Zhou, Kevin Knudson, and Bei Wang

Published in: LIPIcs, Volume 164, 36th International Symposium on Computational Geometry (SoCG 2020)


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.

Cite as

Youjia Zhou, Kevin Knudson, and Bei Wang. Visual Demo of Discrete Stratified Morse Theory (Media Exposition). In 36th International Symposium on Computational Geometry (SoCG 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 164, pp. 82:1-82:4, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{zhou_et_al:LIPIcs.SoCG.2020.82,
  author =	{Zhou, Youjia and Knudson, Kevin and Wang, Bei},
  title =	{{Visual Demo of Discrete Stratified Morse Theory}},
  booktitle =	{36th International Symposium on Computational Geometry (SoCG 2020)},
  pages =	{82:1--82:4},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-143-6},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{164},
  editor =	{Cabello, Sergio and Chen, Danny Z.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2020.82},
  URN =		{urn:nbn:de:0030-drops-122409},
  doi =		{10.4230/LIPIcs.SoCG.2020.82},
  annote =	{Keywords: Discrete Morse theory, stratified Morse theory, discrete stratified Morse theory, topological data analysis, data visualization}
}
Document
Discrete Stratified Morse Theory: A User's Guide

Authors: Kevin Knudson and Bei Wang

Published in: LIPIcs, Volume 99, 34th International Symposium on Computational Geometry (SoCG 2018)


Abstract
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.

Cite as

Kevin Knudson and Bei Wang. Discrete Stratified Morse Theory: A User's Guide. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 54:1-54:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{knudson_et_al:LIPIcs.SoCG.2018.54,
  author =	{Knudson, Kevin and Wang, Bei},
  title =	{{Discrete Stratified Morse Theory: A User's Guide}},
  booktitle =	{34th International Symposium on Computational Geometry (SoCG 2018)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-066-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{99},
  editor =	{Speckmann, Bettina and T\'{o}th, Csaba D.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2018.54},
  URN =		{urn:nbn:de:0030-drops-87671},
  doi =		{10.4230/LIPIcs.SoCG.2018.54},
  annote =	{Keywords: Discrete Morse theory, stratified Morse theory, topological data analysis}
}
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