  The $p$-widths of a closed Riemannian manifold are a nonlinear analogue of
the spectrum of its Laplace--Beltrami operator, which corresponds to areas of a
certain min-max sequence of possibly singular minimal submanifolds. We show
that the $p$-widths of any closed Riemannian two-manifold correspond to a union
of closed immersed geodesics, rather than simply geodesic nets.
  We then prove optimality of the sweepouts of the round two-sphere constructed
from the zero set of homogeneous polynomials, showing that the $p$-widths of
the round sphere are attained by $\lfloor \sqrt{p}\rfloor$ great circles. As a
result, we find the universal constant in the Liokumovich--Marques--Neves--Weyl
law for surfaces to be $\sqrt{\pi}$.
  En route to calculating the $p$-widths of the round two-sphere, we prove two
additional new results: a bumpy metrics theorem for stationary geodesic nets
with fixed edge lengths, and that, generically, stationary geodesic nets with
bounded mass and bounded singular set have Lusternik--Schnirelmann category
zero.
