Geometry and topology in active and driven systems

The key characteristic of active matter is the motion of an emergent collection (such as a flock of birds), which is driven by the consumption of energy by its active components (i.e. individual birds). In this thesis, the central question I consider is: how do topology and geometry affect the motion of active and other driven systems?I answer this question by considering several examples. Starting with active nematics, I show that geometry can eliminate the threshold reported for such systems and demonstrate how this can be realised, which can be useful in designing laminar flows at low activity. I then construct a minimal hydrodynamic theory of a living liquid crystal, in which bacteria are controlled by nematic patterning on a substrate which exhibit non-trivial topology. Finally I investigate topological mechanical chains; after explaining the different phases of the chain, in the process uncovering a new superspinner phase, I go on to analyse how these chains swim at low Reynolds number, and find that this is connected to both the topology and geometry of the chain.