This project aims to understand the propulsion mechanism and interactions of vertically oscillating objects that float on a fluid interface. These newly discovered “surfers” have been shown to self-propel along a fluid bath, and to self-organize through their mutual wave field into moving clusters. However, the mechanisms by which they self-propel and interact have remained elusive. The insights gained from this research have the potential to impact the biology and engineering communities, as scientific attention has been given to the self-propulsion of both inanimate objects and living organisms at the liquid-gas interface. These observations have inspired the development of biomimetic robots that self-propel using similar mechanisms. Capillary-scale robots have many engineering applications including environmental monitoring, water remediation and cargo transport. The project will train undergraduate and graduate students in mathematical modeling and physical applied mathematics, giving them tools that can be broadly applied to problems arising in the natural sciences and engineering. From a mathematical perspective, the main challenge is that surfers generate interfacial waves, unlike so-called “dry” active matter systems whose constituents propel and self-organize solely through steric interactions. A consequence of the inertial wave-mediated coupling between constituents is that multiple interaction modes coexist for the same experimental parameters. To address this