Topology and hydrodynamics in active matter

Collective topological states play a fundamental role in active matter systems, from topological defects in biofilms and flocks to knotted structures in fungal aggregates and worm collectives. How topological rules give rise to stable long-lived structures continues to present theoretical challenges. Here we investigate the interplay between topology, statistical mechanics and hydrodynamics in the collective motion of particles and filaments. We first demonstrate the formation of low-energy statistical edge modes in the Helmholtz-Onsager point vortex system, which has emerged as a candidate model for certain classes of active particles. Turning to active, swimming filaments, we next examine the topological dynamics exhibited by aquatic California blackworms, which form living tangled structures in minutes but can rapidly untangle in milliseconds. Using blackworm locomotion datasets, we construct stochastic trajectory equations that explain how the dynamics of individual active filaments controls their emergent topological state. By identifying how topological principles produce stable yet responsive structures, these results have applications in understanding broad classes of topologically tunable active systems.