CMSP Joint ICTP-SISSA Seminar: The Dirac operator and the dynamics of topological signals
Channel: ICTP Condensed Matter and Statistical Physics
Date: 2023-01-24

Speaker: Ginestra Bianconi (Queen Mary University of London)

ABSTRACT:
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 machine learning and network science. However, little is known about the collective dynamical phenomena involving topological signals. In this talk, I will introduce the Hodge Laplacian  and 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 and simplicial complexes. I will present the topological Dirac equation in which the spinor has a geometrical interpretation and is defined on both nodes and links of the network and the potential implications of this model.   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. The coherent state of these processes are localized on the higher-dimensional cavities of the simplicial complex opening new perspectives to characterize the interplay between topology and dynamics.

2023_01_17-11_30-Bianconi.mp4