  We study the structure of the absolute differential Galois group of a
rational function field over an algebraically closed field of characteristic
zero. In particular, we relate the behavior of differential embedding problems
to the condition that the absolute differential Galois group is free as a
proalgebraic group. Building on this, we prove Matzat's freeness conjecture in
the case that the field of constants is algebraically closed of countably
infinite transcendence degree over the rationals. This is the first known case
of the twenty year old conjecture.
