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Mathematics > Algebraic Topology

arXiv:1808.01513 (math)
[Submitted on 4 Aug 2018 (v1), last revised 16 Aug 2019 (this version, v2)]

Title:Toward a Spectral Theory of Cellular Sheaves

Authors:Jakob Hansen, Robert Ghrist
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Abstract:This paper outlines a program in what one might call spectral sheaf theory --- an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes results on eigenvalue interlacing, sparsification, effective resistance, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.
Comments: Updated with more background material and discussion of a wider array of applications
Subjects: Algebraic Topology (math.AT); Combinatorics (math.CO)
MSC classes: 55N30, 05C50
Cite as: arXiv:1808.01513 [math.AT]
  (or arXiv:1808.01513v2 [math.AT] for this version)
  https://doi.org/10.48550/arXiv.1808.01513
Related DOI: https://doi.org/10.1007/s41468-019-00038-7

Submission history

From: Jakob Hansen [view email]
[v1] Sat, 4 Aug 2018 17:49:21 UTC (48 KB)
[v2] Fri, 16 Aug 2019 19:25:30 UTC (71 KB)
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