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.

Intro
Simplicial complex data
The boundary of a boundary is null
Square d-dimensional lattice
Apollonian and pseudo-fractal hyperbolic simplicial complexes
Network Geometry with Flavor GROWTH
Hausdorff and spectral dimension
Spectral dimension of Apollonian networks and effect of randomness
Higher-order spectral dimension
Spectral dimension of the up-higher-order Laplacian
Higher-order Diffusion
Topological Dirac operator on a network
Sketch of the derivation
Topological Dirac equation on simplicial complexes
Directional Dirac operator on lattices
3-dimensional lattice
Directional Dirac operators on
Synchronization is a fundamental dynamical process
Synchronization transition
Order parameters using the n-dimensional phases
Synchronization of uncoupled topological signals • The uncoupled dynamics of nodes and links of a network
Phase diagram of topological synchronisation
Rhythmic phase
Conclusions
References and collaborators