Geometric control of pattern formation in soft matter and active fluids

Geometric constraints can profoundly affect pattern selection and topological defect formation in equilibrium and non-equilibrium systems. In this talk, I will summarize recent experimental and theoretical work that aims to understand (i) how substrate curvature controls symmetry breaking and defect statistics in elastic surface crystals, and (ii) how confinement geometry affects spontaneous flows of microbial suspensions. Our results show that minimal higher-order PDE models can accurately capture the experimentally observed pattern formation transitions in these systems. We first describe phenomenological parallels between 2D elastic and colloidal crystals on spherical surfaces that suggest some universality in the underlying nucleation processes. Subsequently, we demonstrate how microbial flow patterns can be controlled by microstructure to realize bacterial spin lattices. Building on these insights, we will consider generalized Navier-Stokes equations for active fluids, which also reveal a new invariant for the triad dynamics of classical turbulence.