  A two-dimensional electron gas in a high magnetic field displays
macroscopically degenerate Landau levels, which can be split into Hofstadter
subbands by means of a weak periodic potential. By carefully engineering such a
potential, one can precisely tune the number, bandwidths, bandgaps and Chern
character of these subbands. This allows a detailed study of the interplay of
disorder, interaction and topology in two dimensional systems. We first explore
the physics of disorder and single-particle localization in subbands derived
from the lowest Landau level, that nevertheless may have a topological nature
different from that of the entire lowest Landau level. By projecting the
Hamiltonian onto subbands of interest, we systematically explore the
localization properties of single-particle eigenstates in the presence of
quenched disorder. We then introduce electron-electron interactions and
investigate the fate of many-body localization in subbands of varying
topological character.
