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

points.