Simplicial complexes naturally describe discrete topological spaces and when their links are assigned a length they describe discrete geometries. As such simplicial complexes have been widely used in quantum gravity approaches that involve a discretization of spacetime. Recently they are becoming increasingly popular to describe complex interacting systems such a brain networks or social networks. In this talk we present non-equilibrium statistical mechanics approaches to model large simplicial complexes. We propose the simplicial complex model of Network Geometry with Flavor and we explore the hyperbolic nature of its emergent geometry. Finally we reveal the rich interplay between Network Geometry with Flavor and synchronization of coupled oscillators. We show that the skeleton of these simplicial complexes display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network. This result can shed light on the recent experimental finding that neuronal cultures have a dynamics that is strongly dependent on the network geometry and in particular on their dimensionality of the neuronal networks.