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Active systems of self-propelled agents, e.g., birds, fish, and bacteria, can organize their collective motion into myriad autonomous behaviors. Ubiquitous in nature and across length scales, such phenomena are also amenable to artificial settings, e.g., where brainless self-propelled robots orchestrate their movements into spatial-temporal patterns via the application of external cues or when confined within flexible boundaries. Like their natural counterparts, these approaches typically require many units to initiate collective motion, so controlling the ensuing dynamics is challenging. Here, we demonstrate a simple mechanism that leverages nonlinear elasticity to tame near-diffusive motile particles in forming structures capable of directed motion and other emergent behaviors. Our elasto-active system comprises two centimeter-sized self-propelled microbots connected with elastic beams. These microbots exert forces that suffice to buckle the beam and set the structure in motion. We first rationalize the physics of the interaction between the beam and the microbots. Then we use reduced-order models to predict the interactions of our elasto-active structures with boundaries, e.g., walls and constrictions, and demonstrate how they can exhibit remarkable emergent behaviors such as maze navigation. These findings demonstrate that allowing and understanding changes in body morphology can enhance the capabilities of active matter systems and enable the design of robotic materials capable of space exploration, adaptation, and complex interactions with their surrounding environment. 

We develop a new method for analyzing moduli problems related to the stack of pure coherent sheaves on a polarized family of projective schemes. It is an infinite-dimensional analogue of Geometric Invariant Theory. We apply this to two familiar moduli problems: the stack of Λ-modules and the stack of pairs. In both examples, we construct a Θ-stratification of the stack, defined in terms of a polynomial numerical invariant, and we construct good moduli spaces for the open substacks of semistable points. One of the essential ingredients is the construction of higher-dimensional analogues of the affine Grassmannian for the moduli problems considered.