Talk by Timoteo Carletti

Synchronisation is a widespread phenomenon at the root of several biological rhythms or human made technological systems. Synchronisation refers to the spontaneous ability of coupled, regular or non-regular, oscillators to operate at unison and thus exhibit a coherent collective behaviour. Global synchronisation is the resulting phenomenon where all oscillators behave in the same way. Synchronisation has been studied under the assumption of pairwise interactions, i.e., the oscillators are anchored to nodes of a networks, or many-body interactions but still the basic units are assumed to lie on the nodes. In this talk me make one step forward and we consider synchronisation of topological signals, namely dynamical variables defined on nodes, links, triangles, etc. of higher-order networks. Despite topological signals are attracting increasing attention the investigation of their collective phenomena is only at its infancy. In this talk we show how to combine topology and nonlinear dynamics to determine the conditions for global synchronisation to emerge for topological signals defined on simplicial and cell complexes. In the former we show that topological obstruction impedes odd dimensional signals to globally synchronise, while cell complexes can overcome topological obstruction. After having considered the well studied case of diffusive-like coupling, i.e., realised by using the (Hodge) Laplace, we will briefly present some recent results involving synchronisation of topological signals coupled with the Dirac operator.