Abstract: How can geometric ideas help us understand the shapes, patterns and collective behaviours we see around us? Beginning with Higgs bundles and the Hitchin fibration, we will explore how imposing symmetries singles out branes and other distinguished geometric structures, and how varying these constraints opens new geometric possibilities. We will then spend most of the talk following geometric ideas into the applied sciences, from the organisation of viral capsids and the growth of snow crystals to the spread of information and interacting pathogens on networks. Along the way, we will encounter biological transport networks and their inspiration for sustainable optimisation, before turning to recent work on influence in evolving networks, synchronisation and fragmentation in social groups, and the stabilising role of uninformed participants in collective decision making. Throughout, we will ask how structure constrains behaviour, when symmetry helps us understand a system, and what becomes possible when that symmetry is broken.