Abstract (for both lectures): A rational curve 
contained in the projective plane is rectifiable
if and only if it can be transformed into a line by
a birational automorphism of the plane. Determining
which curves are rectifiable is hard in general. An old
open problem, the Coolidge-Nagata conjecture, states
that all complex curves which are homeomorphic to the
line in the Euclidean topology (cuspidal curves) are
rectifiable. I will recall the results of Coolidge,
Kumar-Murthy, Wakabayashi and other necessary tools
from the theory of open surfaces. Then I will prove
some general bounds, which in particular establish
the conjecture for curves with more than three singular