  The cohomology of the Hilbert schemes of points on smooth projective surfaces
can be approached both with vertex algebra tools and equivariant tools. Using
the first tool, we study the existence and the structure of universal formulas
for the Chern classes of the tangent bundle over the Hilbert scheme of points
on a projective surface. The second tool leads then to nice generating formulas
in the particular case of the Hilbert scheme of points on the affine plane.
