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Applied and Combinatorial Topology (Dagstuhl Seminar 24092)

Authors: Paweł Dłotko, Dmitry Feichtner-Kozlov, Anastasios Stefanou, Yusu Wang, and Jan F Senge

Published in: Dagstuhl Reports, Volume 14, Issue 2 (2024)

Abstract

This report documents the program and the outcomes of Dagstuhl Seminar 24092 "Applied and Combinatorial Topology". The last twenty years of rapid development of Topological Data Analysis (TDA) have shown the need to analyze the shape of data to better understand the data. Since an explosion of new ideas in 2000’ including those of Persistent Homology and Mapper Algorithms, the community rushed to solve detailed theoretical questions related to the existing invariants. However, topology and geometry still have much to offer to the data science community. New tools and techniques are within reach, waiting to be brought over the fence to enrich our understanding and potential to analyze data. At the same time, the fields of Discrete Morse Theory (DMT) and Combinatorial Topology (CT) are developed in parallel with no strong connection to data-intensive TDA or to other statistical pipelines (e.g. machine learning). This Dagstuhl Seminar brought together a number of experts in Discrete Morse Theory, Combinatorial Topology, Topological Data Analysis, and Statistics to (i) enhance the existing interactions between these fields on the one hand, and (ii) discuss the possibilities of adopting new invariants from algebra, geometry, and topology; in particular inspired by continuous and discrete Morse theory and combinatorial topology; to analyze and better understand the notion of shape of the data. The different talks in the seminar included both introductory talks as well as current research expositions and proved fruitful for the open problem and break-out sessions. The topics that were discussed included 1) algorithmic aspects for efficient computation as well as Morse theoretic approximations 2) topological information gain of multiparameter persistence 3) understanding the magnitude function and its relation to graph problems.

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Paweł Dłotko, Dmitry Feichtner-Kozlov, Anastasios Stefanou, Yusu Wang, and Jan F Senge. Applied and Combinatorial Topology (Dagstuhl Seminar 24092). In Dagstuhl Reports, Volume 14, Issue 2, pp. 206-239, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@Article{dlotko_et_al:DagRep.14.2.206,
  author =	{D{\l}otko, Pawe{\l} and Feichtner-Kozlov, Dmitry and Stefanou, Anastasios and Wang, Yusu and Senge, Jan F},
  title =	{{Applied and Combinatorial Topology (Dagstuhl Seminar 24092)}},
  pages =	{206--239},
  journal =	{Dagstuhl Reports},
  ISSN =	{2192-5283},
  year =	{2024},
  volume =	{14},
  number =	{2},
  editor =	{D{\l}otko, Pawe{\l} and Feichtner-Kozlov, Dmitry and Stefanou, Anastasios and Wang, Yusu and Senge, Jan F},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagRep.14.2.206},
  URN =		{urn:nbn:de:0030-drops-205059},
  doi =		{10.4230/DagRep.14.2.206},
  annote =	{Keywords: Applied Topology, Topological Data Analysis, Discrete Morse Theory, Combinatorial Topology, Statistics}
}

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DOI: 10.4230/LIPIcs.SoCG.2022.31

Persistent Cup-Length

Authors: Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou

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

Abstract

Cohomological ideas have recently been injected into persistent homology and have for example been used for accelerating the calculation of persistence diagrams by the software Ripser. The cup product operation which is available at cohomology level gives rise to a graded ring structure that extends the usual vector space structure and is therefore able to extract and encode additional rich information. The maximum number of cocycles having non-zero cup product yields an invariant, the cup-length, which is useful for discriminating spaces. In this paper, we lift the cup-length into the persistent cup-length function for the purpose of capturing ring-theoretic information about the evolution of the cohomology (ring) structure across a filtration. We show that the persistent cup-length function can be computed from a family of representative cocycles and devise a polynomial time algorithm for its computation. We furthermore show that this invariant is stable under suitable interleaving-type distances.

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Marco Contessoto, Facundo Mémoli, Anastasios Stefanou, and Ling Zhou. Persistent Cup-Length. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{contessoto_et_al:LIPIcs.SoCG.2022.31,
  author =	{Contessoto, Marco and M\'{e}moli, Facundo and Stefanou, Anastasios and Zhou, Ling},
  title =	{{Persistent Cup-Length}},
  booktitle =	{38th International Symposium on Computational Geometry (SoCG 2022)},
  pages =	{31:1--31:17},
  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.2022.31},
  URN =		{urn:nbn:de:0030-drops-160398},
  doi =		{10.4230/LIPIcs.SoCG.2022.31},
  annote =	{Keywords: cohomology, cup product, persistence, cup length, Gromov-Hausdorff distance}
}

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Applied and Combinatorial Topology (Dagstuhl Seminar 24092)

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Persistent Cup-Length

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