Higher-order networks  describe the many-body interactions of a large variety of complex systems, ranging from the brain to collaboration networks and social contact networks. Simplicial complexes are generalized network structures which allow us to capture the combinatorial properties, the topology and the geometry of higher-order networks. In this talk we will show that simplicial complexes provide a very general mathematical framework to reveal how higher-order dynamics including synchronization and epidemic spreading depends on simplicial network topology. We will show that higher-order synchronization describing the dynamics of topological signals defined on link, triangles and higher-dimensional simplices is explosive [2-4] and we will show that this rich dynamics can have an important role for understanding brain rhythms. We will also show how epidemic spreading on higher-order networks  can take into account for time-dependent contacts due to co-location in space and how this modelling can help us understand the spreading dynamics of airborne diseases.
G. Bianconi, Higher-order networks: An introduction to simplicial complexes (Cambridge University Press, 2021)
 Millán, A.P., Torres, J.J. and Bianconi, G., 2020. Explosive higher-order Kuramoto dynamics on simplicial complexes. Physical Review Letters, 124(21), p.218301.
 Ghorbanchian, R., Restrepo, J.G., Torres, J.J. and Bianconi, G., 2021. Higher-order simplicial synchronization of coupled topological signals. Communications Physics, 4(1), pp.1-13.
 Calmon, L., Restrepo, J.G., Torres, J.J. and Bianconi, G., 2021. Topological synchronization: explosive transition and rhythmic phase. arXiv preprint arXiv:2107.05107.
 St-Onge, G., Sun, H., Allard, A., Hébert-Dufresne, L. and Bianconi, G., 2021. Universal nonlinear infection kernel from heterogeneous exposure on higher-order networks. Physical Review Letters, 127 (15), 158301