  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.
