Computer calculations have driven several recent breakthroughs involving the stable homotopy groups of spheres, including the solution due to Lin, Wang, and Xu of the remaining case of the Kervaire invariant problem. I will discuss work in progress, joint with Francis Baer, William Balderrama, and Dan Isaksen, to employ some of the same methods to studying the unstable homotopy groups of spheres via the unstable Adams spectral sequence.