Abstract

It is well-known that the kernel of the graph Laplacian captures the topological properties (number of cycles and connected components) of a graph. In a similar fashion, the kernel of a persistent Laplacian captures the information contained in the persistent homology of a given simplicial complex. Our main goal is to understand what we can deduce from the remaining eigenvalues and -vectors in the more general cellular sheaf setting, which theoretically incorporate further information of the faces of a simplicial complex. In this talk, I will discuss work in progress towards this aim and present a recently-established theoretical foundation for this goal, where we show that the eigenvalues are stable under small perturbation of the sheaf and simplicial complex. The upshot of this result is that we can reasonably assume that the additional information encoded by the other eigenvalues and -vectors are a faithful representation of other geometric or topological properties of the underlying simplicial complex, although precisely what this information represents remains to be investigated (current work in progress proceeds with a machine learning approach). Joint work with Shiv Bhatia, Daniel Ruiz Cifuentes, Jiyu Fan and Anthea Monod.