The stellar and sub-stellar IMF of simple and composite populations
  The current knowledge on the stellar IMF is documented. It appears to become
top-heavy when the star-formation rate density surpasses about 0.1Msun/(yr
pc^3) on a pc scale and it may become increasingly bottom-heavy with increasing
metallicity and in increasingly massive early-type galaxies. It declines quite
steeply below about 0.07Msun with brown dwarfs (BDs) and very low mass stars
having their own IMF. The most massive star of mass mmax formed in an embedded
cluster with stellar mass Mecl correlates strongly with Mecl being a result of
gravitation-driven but resource-limited growth and fragmentation induced
starvation. There is no convincing evidence whatsoever that massive stars do
form in isolation. Various methods of discretising a stellar population are
introduced: optimal sampling leads to a mass distribution that perfectly
represents the exact form of the desired IMF and the mmax-to-Mecl relation,
while random sampling results in statistical variations of the shape of the
IMF. The observed mmax-to-Mecl correlation and the small spread of IMF
power-law indices together suggest that optimally sampling the IMF may be the
more realistic description of star formation than random sampling from a
universal IMF with a constant upper mass limit. Composite populations on galaxy
scales, which are formed from many pc scale star formation events, need to be
described by the integrated galactic IMF. This IGIMF varies systematically from
top-light to top-heavy in dependence of galaxy type and star formation rate,
with dramatic implications for theories of galaxy formation and evolution.
