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A computational framework for connection matrix theory

Abstract

The connection matrix is a powerful algebraic topological tool from Conley index theory, a subfield of topological dynamics. Conley index theory is a purely topological generalization of Morse theory in which the connection matrix subsumes the role of the Morse boundary operator. Over the last few decades, Conley’s approach to dynamics has been cast into a purely computational form. In this paper we introduce a computational and categorical framework for connection matrix theory. Broadly speaking, this contribution promotes the computational Conley theory to a computational, homological theory for dynamical systems. More specifically, within this paper we have three specific aims:

  1. 1.

    We cast connection matrix theory in an appropriate categorical, homotopy-theoretic language. We demonstrate the relationship to the previous definitions of connection matrix. Lastly, the homotopy-theoretic language allows us to formulate connection matrix theory categorically.

  2. 2.

    We describe an algorithm for the computation of connection matrices based on algebraic-discrete Morse theory and formalized with the notion of reductions. We advertise an open-source implementation of our algorithm.

  3. 3.

    We show that the connection matrix can be used to compute persistent homology. Ultimately, we believe that connection matrix theory has the potential to be an important tool within topological data analysis.

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Notes

  1. 1.

    See Sect. 3 for formal definitions associated with order theory and algebraic topology.

  2. 2.

    In this example we restrict to filtrations. However, we wish to emphasize that our results hold for multi-parameter persistence; see Sect. 10, in particular Theorem 10.3.

  3. 3.

    We previously introduced the term chain contraction in Sect. 3.3 which agrees with Weibel (1995). This idea should not be confused with reduction.

  4. 4.

    In fact, this is a Corollary of Algorithm 6.2.

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Acknowledgements

The work of SH and KM was partially supported by grants NSF-DMS-1248071, 1521771 and DARPA contracts HR0011-16-2-0033 and FA8750-17-C-0054, and NIH grant R01 GM126555-01. The work of KS was partially supported by the NSF Graduate Research Fellowship Program under grant DGE-1842213 and by EPSRC grant EP/R018472/1. KS would like to thank Chuck Weibel for some very useful discussions regarding homological algebra and connection matrix theory.

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Correspondence to Kelly Spendlove.

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Harker, S., Mischaikow, K. & Spendlove, K. A computational framework for connection matrix theory. J Appl. and Comput. Topology 5, 459–529 (2021). https://doi.org/10.1007/s41468-021-00073-3

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Keywords

  • Connection matrix
  • Conley index
  • Discrete Morse theory
  • Computational topology
  • Computational dynamics
  • Persistent homology

Mathematics Subject Classification

  • 37B30
  • 37B25
  • 55-04
  • 57-04