Toward a Spectral Theory of Cellular Sheaves

TL;DR

This paper introduces spectral sheaf theory, extending spectral graph theory to cellular sheaves by relating Laplacian eigenvalues to sheaf cohomology and cell structure, with results on eigenvalue interlacing and sparsification.

Contribution

It develops a spectral framework for cellular sheaves, connecting Laplacian spectra to topological and combinatorial properties, and explores new spectral results and applications.

Findings

Eigenvalue interlacing established

Sparsification techniques developed

Effective resistance in sheaves analyzed

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.