The study of multi-legged locomotion provides a rich collection of mathematical challenges. In this talk I will present three vignettes from our work in recent years and highlight the ongoing progress in each: (1) differentiability of of the flows of some classes of non-smooth (hybrid) dynamical systems and ongoing work on efficient algorithms for estimating those differentials; (2) geometric mechanics at the singular limit of friction and ongoing work on data driven geometric gait optimization; (3) motivating examples for a Koopman theory for non-smooth systems and ongoing work discovering a refined understanding of non-resonance conditions necessary and sufficient for eigenfunction uniqueness. In each of these cases the needs arising from practical problems in robotics and biology lead us to theoretical advances with direct practical applications and implications. I will conclude with some general remarks on how to push the mathematical envelope in highly applied problem domains. Speaker(s): Shai Revzen (University of Michigan)