Geometric approach to linear ODEs

There is a classical correspondence between systems of n linear ordinary differential equations (ODEs) of order one and linear ODEs of order n; the correspondence may be viewed as a kind of canonical normal form for systems of ODEs. The correspondence can be restated geometrically: given a Riemann surface C, a vector bundle E on C, and a connection \nabla on E, it is possible to find a rational basis of E such that \nabla is in the canonical normal form.

All of the above objects have a version for arbitrary semisimple Lie group G (with the case of systems of ODEs corresponding to G=GL(n)): we can consider differential operators whose `matrices' are in the Lie algebra of G, and then try to `change the basis' so that the `matrix' is in the `canonical normal form'. However, the statement turns out to be significantly harder. In my talk, I will show how the geometric approach can be used to prove the claim for any G. The claim plays an important role in the geometric Langlands program.