Using a combination of techniques geometric, topological, and computational, we have resolved two longstanding open problems in the theory of hyperbolic three-manifolds. To wit,

  • whether or not any orientable one-cusped hyperbolic three-manifolds besides the figure-eight knot complement admit more than eight exceptional fillings (no, they don’t); and
  • whether or not the closed orientable hyperbolic three-manifolds of second and third smallest volume are those of second and third smallest volume in the SnapPea census (yes, they are).