Toward a Spectral Theory of Cellular Sheaves

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.

Updated with more background material and discussion of a wider array of applications

References in corpus (4)

• A roadmap for the computation of persistent homology
• User-friendly tail bounds for sums of random matrices
• Spanning forests and the vector bundle Laplacian
• Isoperimetric Inequalities in Simplicial Complexes

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