14:00
Topological signals associated not only to nodes but also to links and to the higher dimensional simplices of simplicial complexes are attracting increasing interest in signal processing, machine learning and network science. However, little is known about the collective dynamical phenomena involving topological signals. Typically, topological signals of a given dimension are investigated and filtered using the corresponding Hodge Laplacians. In this talk, I will introduce the topological Dirac operator that can be used to process simultaneously topological signals of different dimensions. I will discuss the main spectral properties of the Dirac operator defined on networks, simplicial complexes and multiplex networks, and their relation to Hodge Laplacians. I will show that topological signals treated with the Hodge Laplacians or with the Dirac operator can undergo collective synchronization phenomena displaying different types of critical phenomena. Finally, I will show how the Dirac operator allows to couple the dynamics of topological signals of different dimension leading to the Dirac signal processing of signals defined on nodes, links and triangles of simplicial complexes.