Topological waves in passive and active fluids on curved surfaces

Topological methods are well-established in hard condensed matter, and recently similar approaches have been shown to explain phenomena in soft condensed matter. In this talk I will examine two soft matter systems, one passive and one active, and consider how the models and topological modes are connected. I solve for these modes directly without recourse to the bulk-edge correspondence and provide an index theorem linking the number of modes determined by the topology of momentum-space to the real-space topology of the surface on which they are hosted. Our work suggests some simple rules to engineer topological modes on surfaces in passive and active soft matter systems.