2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/41363The conventional approach to perturbative evolution is illegal because the expansion in powers of $α_s$ of the the DGLAP splitting matrix ${\bf P(z,α_s)}$ breaks down at small $z$. The small-$x$ data for the proton structure function $F_2(x,Q^2)$ and its charm component $F^c_2(x,Q^2)$ show that a hard pomeron, with intercept close to 1.4, must be added to the familar soft pomeron. Conventional perturbative evolution may be applied to the hard-pomeron component and provides a striking test of perturbative QCD.Lectures at Corfu2001High Energy Physics - Phenomenologytext