  We numerically investigate the surface states of a strong topological
insulator in the presence of strong electron-electron interactions. We choose a
spherical topological insulator geometry to make the surface amenable to a
finite size analysis. The single-particle problem maps to that of Landau
orbitals on the sphere with a magnetic monopole at the center that has unit
strength and opposite sign for electrons with opposite spin. Assuming
density-density contact interactions, we find superconducting and anomalous
(quantum) Hall phases for attractive and repulsive interactions, respectively,
as well as chiral fermion and chiral Majorana fermion boundary modes between
different phases. Our setup is preeminently adapted to the search for
topologically ordered surface terminations that could be microscopically
stabilized by tailored surface interaction profiles.
