The Topological Dirac Operator and the Dynamics of Topological Signals

Explore the fascinating world of topological signals in this comprehensive lecture by Ginestra Bianconi. Delve into the spectral properties of higher-order Laplacians and their impact on higher-order diffusion and synchronization dynamics. Discover the topological Dirac operator and its applications in processing topological signals of different dimensions simultaneously. Examine the operator's spectral properties on networks, simplicial complexes, and multiplex networks, and understand its relationship to higher-order Laplacians. Learn how the Dirac operator enables topological synchronization of locally coupled signals on network nodes and links. Gain insights into simplicial complex data, boundary properties, and various network geometries. Investigate concepts such as Hausdorff and spectral dimensions, higher-order diffusion, and synchronization transitions. Explore the phase diagram of topological synchronization and the rhythmic phase in this in-depth exploration of topological signal dynamics.