Vectorial representations of topological features: a spectral perspective.

The rich spectral information of the graph Laplacian has been instrumental in graph theory, machine learning, and graph signal processing for a diverse range of applications. Arguably this is due to the fact that the Laplacian spectrum encodes important topological and geometric properties. In this talk we will argue that these ideas can be naturally generalised to discrete Hodge-Laplacian matrices, whose eigenvectors and eigenvalues provide additional information. Specifically, the eigenvectors and eigenvectors of the Hodge-Laplacians provide us with vectorial representations that encode important topological information such as homology. We illustrate how these insights can be used in a range of applications, including clustering or the signal processing of flows on discrete spaces.