  Heavy particles in vortical fluid flow cluster strongly, forming singular
structures termed caustics for their resemblance to focal surfaces in optics.
We show here that such extreme aggregation onto low-dimensional submanifolds
can arise without inertia for self-propelled particles (SPPs). We establish
that a singular perturbation is at the heart of caustic formation by SPPs
around a single vortex, and our numerical studies of SPPs in two-dimensional
Navier-Stokes turbulence shows intense caustics in the straining regions of the
flow, peaking at intermediate levels of self-propulsion. Our work offers a
route to singularly high local concentrations in a macroscopically dilute
suspension of zero-Reynolds number swimmers, with potentially game-changing
implications for communication and sexual reproduction. An intriguing open
direction is whether the active turbulence of a suspension of swimming microbes
could serve to generate caustics in its own concentration.
