Toward a spectral theory of cellular sheaves

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...

Full description

Saved in:
Bibliographic Details
Published in:Journal of applied and computational topology 2019-12, Vol.3 (4), p.315-358
Main Authors: Hansen, Jakob, Ghrist, Robert
Format: Article
Language:English
Subjects:
Algebraic Topology
Computational Science and Engineering
Mathematical and Computational Biology
Mathematics
Mathematics and Statistics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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 discussion of eigenvalue interlacing, sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians.
ISSN:2367-1726
2367-1734
DOI:10.1007/s41468-019-00038-7