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PERSISTENT MAYER HOMOLOGY AND PERSISTENT MAYER LAPLACIAN.

Li Shen1, Jian Liu2,1, Guo-Wei Wei1,3,4

  • 1Department of Mathematics, Michigan State University, MI 48824, USA.

Foundations of Data Science (Springfield, Mo.)
|September 29, 2025
PubMed
Summary

This study introduces Mayer Laplacians and persistent Mayer homology for generalized N-chain complexes. These methods offer new topological and geometric insights, showing promise for analyzing complex data in topological data analysis.

Keywords:
Mayer LaplacianMayer homologyN-chain complexPrimary: 55N31persistencestability

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Area of Science:

  • Algebraic Topology
  • Topological Data Analysis
  • Computational Geometry

Background:

  • The standard differential in algebraic topology satisfies d^2 = 0.
  • Generalized differentials (d^N = 0) and Mayer homology on N-chain complexes have been studied for over 80 years.
  • Mayer homology provides a framework for studying topological structures beyond the standard d^2 = 0 condition.

Purpose of the Study:

  • Introduce Mayer Laplacians on N-chain complexes.
  • Explore the application potential of Mayer homology and Mayer Laplacians for topological and geometric insights.
  • Develop persistent Mayer homology and persistent Mayer Laplacians for data analysis.
  • Investigate the stability and bottleneck distance of persistence diagrams associated with Mayer homology.

Main Methods:

  • Development of Mayer Laplacians for N-chain complexes.
  • Introduction of persistent Mayer homology and persistent Mayer Laplacians.
  • Analysis of bottleneck distance and stability for persistence diagrams.
  • Computational experiments on large, complex datasets.

Main Results:

  • Mayer homology and Mayer Laplacians provide significant topological and geometric insights.
  • Persistent Mayer homology and Laplacians demonstrate potential for analyzing complex data.
  • The bottleneck distance and stability of persistence diagrams were investigated.

Conclusions:

  • Mayer homology and Mayer Laplacians are valuable tools for understanding topological and geometric properties of spaces.
  • Persistent Mayer homology and Laplacians show promise for topological data analysis, especially for large and complex datasets.
  • This work lays the foundation for integrating Mayer homology and Laplacians into mainstream topological data analysis.