The initial mass distribution of stars in a galaxy (the initial mass
function or IMF) is usually described as being invariant. The universality
of the IMF is one of the main assumptions for most of the
chemical/chemodynamical models in the literature. However, evidence is
emerging that this is not the case: the IMF appears to become top-heavy in
galaxies characterized by high star formation rates. The integrated
galactic IMF (IGIMF) theory determines the correlation between the star
formation rate and the IMF in a galaxy. In this seminar I will show how
the global chemical evolution of galaxies change if the hypothesis of an
invariant IMF is relaxed and the IGIMF theory is adopted instead. In
particular, I will show: (i) how the [alpha/Fe] abundance ratios in
early-type galaxies are affected by the steepening of the IMF with
decreasing star formation rate, predicted by the IGIMF theory, (ii) what
are the consequences of the IGIMF for the chemical evolution of the Milky
Way, (iii) how the [alpha/Fe] ratios in the most metal-poor stars of a
galaxy vary as a function of its IGIMF.