The Möbius Homology Laplacian for Persistence

Abstract

Persistent homology provides a framework for studying the evolution of topological features across filtered spaces. While persistence diagrams give discrete invariants that record the birth and death of homological features, they do not directly encode analytic or geometric structure associated to persistence modules. This dissertation develops a spectral framework for persistent homology by introducing and studying a Laplace operator on the Möbius chain complex, called the Möbius homology Laplacian. Möbius homology categorifies Möbius inversion for persistence modules by replacing integer-valued data with vector-space-valued chain complexes. By equipping these chain complexes with inner products, we define a Laplace operator on the Möbius chain complex whose kernel provides canonical cycle representatives for the associated Möbius homology space. We analyze the spectral properties of this operator in two primary settings: one-parameter persistence modules and birth-death modules arising from filtrations of simplicial complexes. In both cases, we show that the spectrum reflects the combinatorial structure of the associated persistence diagram.