Laplacians Of Cellular Sheaves: Theory And Applications

Cellular sheaves are a discrete model for the theory of sheaves on cell complexes. They carry a canonical cochain complex computing their cohomology. This thesis develops the theory of the Hodge Laplacians of this complex, as well as avenues for their application to concrete engineering and data analysis problems. The sheaf Laplacians so developed are a vast generalization of the graph Laplacians studied in spectral graph theory. As such, they admit generalizations of many results from spectral graph theory and the spectral theory of discrete Hodge Laplacians. A theory of approximation of cellular sheaves is developed, and algorithms for producing spectrally good approximations are given, as well as a generalization of the notion of expander graphs. Sheaf Laplacians allow development of various dynamical systems associated with sheaves, and their behavior is studied. Finally, applications to opinion dynamics, extracting network structure from data, linear control systems, and distributed optimization are outlined.