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Documents authored by Lipiński, Michał

Document
DOI: 10.4230/LIPIcs.SoCG.2026.41
The Depth Poset Under Transpositions in the Filter

Authors: Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The depth poset of a filtered Lefschetz complex reflects the dependencies between the cancellations of different shallow birth-death pairs. Using the fast algorithms for computing the depth poset in [Edelsbrunner et al., 2026] and for updating the persistence diagram under transpositions in [Cohen-Steiner et al., 2006], we give a complete case analysis of how transpositions of cells in the filter affect the depth poset. In addition, we present statistics on the depth poset for random point data and its sensitivity to the transpositions that occur in random straight-line homotopies.
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Herbert Edelsbrunner, Michał Lipiński, Marian Mrozek, Manuel Soriano-Trigueros, and Fedor Zimin. The Depth Poset Under Transpositions in the Filter. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 41:1-41:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{edelsbrunner_et_al:LIPIcs.SoCG.2026.41,
  author =	{Edelsbrunner, Herbert and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Soriano-Trigueros, Manuel and Zimin, Fedor},
  title =	{{The Depth Poset Under Transpositions in the Filter}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{41:1--41:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.41},
  URN =		{urn:nbn:de:0030-drops-258479},
  doi =		{10.4230/LIPIcs.SoCG.2026.41},
  annote =	{Keywords: Algebraic topology, Lefschetz complexes, persistent homology, vines and vineyards, birth-death pairs, shallow pairs, relations, partial orders, transpositions}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.72
Topological Simplification Guided by Forbidden Regions

Authors: Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
Topological simplification is the process of reducing complexity of a function while maintaining its essential features. Its goal is to find a new filter function, which reorders cells of the input complex in a way which eliminates some persistent homological features, without affecting the rest. We present a new approach to simplification based on the concept of forbidden regions and combinatorial dynamics. It allows us to reorder and cancel critical values, whose cancellation is not possible using existing methods because they are not consecutive in the total order. Each such cancellation takes O(c⋅n) time in the worst case, where c is the number of birth-death pairs and n is the size of the input complex.
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Jakub Leśkiewicz, Bartosz Furmanek, Michał Lipiński, and Dmitriy Morozov. Topological Simplification Guided by Forbidden Regions. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 72:1-72:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{leskiewicz_et_al:LIPIcs.SoCG.2026.72,
  author =	{Le\'{s}kiewicz, Jakub and Furmanek, Bartosz and Lipi\'{n}ski, Micha{\l} and Morozov, Dmitriy},
  title =	{{Topological Simplification Guided by Forbidden Regions}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{72:1--72:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.72},
  URN =		{urn:nbn:de:0030-drops-258797},
  doi =		{10.4230/LIPIcs.SoCG.2026.72},
  annote =	{Keywords: persistent homology, topological simplification, depth posets}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2022.35
Tracking Dynamical Features via Continuation and Persistence

Authors: Tamal K. Dey, Michał Lipiński, Marian Mrozek, and Ryan Slechta

Published in: LIPIcs, Volume 224, 38th International Symposium on Computational Geometry (SoCG 2022)

Abstract
Multivector fields and combinatorial dynamical systems have recently become a subject of interest due to their potential for use in computational methods. In this paper, we develop a method to track an isolated invariant set - a salient feature of a combinatorial dynamical system - across a sequence of multivector fields. This goal is attained by placing the classical notion of the "continuation" of an isolated invariant set in the combinatorial setting. In particular, we give a "Tracking Protocol" that, when given a seed isolated invariant set, finds a canonical continuation of the seed across a sequence of multivector fields. In cases where it is not possible to continue, we show how to use zigzag persistence to track homological features associated with the isolated invariant sets. This construction permits viewing continuation as a special case of persistence.
Cite as

Tamal K. Dey, Michał Lipiński, Marian Mrozek, and Ryan Slechta. Tracking Dynamical Features via Continuation and Persistence. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 35:1-35:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{dey_et_al:LIPIcs.SoCG.2022.35,
  author =	{Dey, Tamal K. and Lipi\'{n}ski, Micha{\l} and Mrozek, Marian and Slechta, Ryan},
  title =	{{Tracking Dynamical Features via Continuation and Persistence}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{35:1--35:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-227-3},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{224},
  editor =	{Goaoc, Xavier and Kerber, Michael},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.35},
  URN =		{urn:nbn:de:0030-drops-160439},
  doi =		{10.4230/LIPIcs.SoCG.2022.35},
  annote =	{Keywords: combinatorial dynamical systems, continuation, index pair, Conley index, persistent homology}
}
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