About differential Galois theory

The standard Picart-Vessiot theory for differential
equations works well in the so-called Fuchs case , because Schlesinger
theorem connects algebra and topology . It does not work for such a
simple equation as dy = y.dx . We shall explain how we can solve this
difficulty by replacing a Galois group by a Galois groupoid , thus permitting
to develop a non-linear theory as Malgrange
and Umemura have done.