Topology and dynamics in active and living matter

Topology underlies much of our understanding of equilibrium matter in terms of defects in ordered media and topologically protected states. In active systems composed of individually powered units, such as bacterial suspensions or epithelial cell layers, topological phenomena can take on new and surprising roles. In this talk I will highlight topological phenomena in active fluids with polar and nematic liquid crystalline order. I will show how broken time-reversal symmetry due to the active drive allows polar flocks on a curved surface to support topologically protected sound modes. In active nematics, activity instead causes topological disclinations to become spontaneously motile, driving a non-equilibrium variant of the Berezinskii-Kosterlitz-Thouless transition and novel phases of defect order and chaos. Such topological phenomena offer intriguing possibilities for biology.