Topology, one of the oldest branches of mathematics, captures the concept of shape for spaces of arbitrary type and dimension. This allows to adopt some of its concepts to characterize and compare how complex systems evolve and restructure themselves. In the talk, I will introduce the most common topological techniques, persistent homology and Mapper, to illustrate what novel insights these new descriptive paradigms yield. In particular, I will focus on the impact of topological observables in the analysis of how the brain works at the functional, structural and genetic level, across a range of physiological and pathological conditions. I will then discuss recent advances in our understanding of the effects of higher order interactions on the evolution of dynamical processes, such as contagion and synchronization. Finally, I will discuss the challenges of inferring such higher order interactions in cases where they are not explicit, e.g. starting from timeseries data.
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