Adventures Through Time and Phase Space: Characterizing the Dynamic Radio Sky, From MHz to GHz, From Seconds to Years

Time domain radio astronomy is on the cusp of a revolution. This is particularly true at low radio frequencies, where wide-field arrays are conducting deeper surveys than ever before, in pursuit of elusive transient phenomena. This thesis has opened an unprecedented window on the dynamic radio sky, spanning a wide range in frequency -- from MHz to GHz -- and a wide range of phenomena -- from searches for coherent radio bursts indicative of coronal mass ejections from nearby stars, to the detection of incoherent synchrotron emission from an outflow associated with the tidal disruption of a star by a supermassive black hole at the center of a galaxy. A significant part of this work has involved the building of the Owens Valley Radio Observatory Long Wavelength Array (OVRO-LWA), a uniquely powerful radio telescope, with the ability to image the entire viewable sky at a 10 second cadence and across nearly 60 MHz of bandwidth. With the OVRO-LWA, I have developed the methods and techniques necessary for pushing further into unexplored regions of radio transient phase space, in the pursuit of sources of radio emission that provide unique insight into various astrophysical phenomena. This includes the follow-up of gravitational wave events in order to search for the signatures of prompt, coherent radio emission associated with compact object mergers, as well as targeted searches for stellar and exoplanetary radio emission as a means of better understanding planetary habitability.