# Understanding singularities of differential equations

Is there a simple reference which explains how to see geometrically an algebraic differential equation ?

I tried to read "Équation différentielles à points singuliers réguliers" of Pierre Deligne but it was a bit too complicated for me.

I only follow a basic course on Riemann surfaces, is there some reference or books which can helps me to understand it better this paper of Deligne (or more generally what is the geometry hidden behind theses equations and these singularities ?

Thanks in advance and sorry if my question is a bit unclear.

• Have you studied ordinary differential equations and their singular points in an easy book or did you just jump into the differential algebra interpretation of ODE's? An algebraic differential equation is just a differential equation where one interprets the differential operators as derivations right? So your question seems to be 'what is a differential equation geometrically?' Feb 24, 2015 at 7:07
• Yes I already studied the basic theory of ordinary differential equations so my question is what could help me to understand deeper the connection between these equations and all the geometry/topology involved in the paper of Deligne.
– user171326
Feb 25, 2015 at 7:34
• Nice ! Why not posted it as answer ? Thanks.
– user171326
Feb 25, 2015 at 8:16

## 1 Answer

According to this the theory of linear differential equations has been generalized to the geometry of connections on manifolds, and the monodromy of their solutions in terms of representations of fundamental groups (note the theory of special functions arising from linear differential equations is unified via representation theory). Naturally it should be possible to analyze all of this with stokes theorem or more generally cohomology and sheaves, thus as this (following Deligne) shows one should be able to unify all these interpretations to discuss singularities, and relate it all to Riemann-Hilbert problems.

I think one can view this as the inverse Piccard-Vessiot problem, with the direct problem being treated in an introductory way in Abel's theorem & Galois' Dream.