Supplemental file(s) description: Interactive visualization of knot electric potential critical sets, Interactive visualization of equipotential surfaces, Python code for numerics of charged knots. ; The author's research program involves several topics which differ at first glance. However, they all share the common theme of exploring how geometry and topology influences dynamical equilibria. The dissertation is broken into three parts: the hyperbolic geometry of higher-dimensional Kuramoto oscillators, electrostatic knot theory, and minimal surfaces with Mobius energy on the boundary. Each part is further divided into chapters adapting the author's preprints and published papers, which have appeared in journals in physics, applied mathematics, and pure mathematics.