Joint with the Workshop Fuchsian Differential Equations.

In the 1860's, Fuchs looked at linear ordinary differential equations with variable coefficients, say, holomorphic functions. The issue was that the highest derivative could have a coefficient which vanished at some point. Just think of the equation $x^2 y' + y = 0$ with singularity at $x = 0$, or the hypergeometric equations.

So he asked, quite naturally, when the solutions would have a moderate behaviour at these singular points: he was willing to allow meromorphic functions, roots of them, and logarithms (examples suggest these are a good class), but not functions with worse behaviour (the precise definition will be given in the talk).

And — surprise — he was able to give an algebraic and effective criterion to characterize these "regular" singularities (the terminology is pityful). His discovery and the proof of it was like an explosion, with many subsequent works (Frobenius, Thomé, Fabry, ...) and many important results, applications and generalizations.

After describing the approach of Fuchs, we will briefly address Grothendieck's conjecture in this context: He proposes to understand the differential equation (when defined over $\mathbb{Q}$) by means of its reduction modulo prime numbers $p$. Fortunately, his conjecture is still wide open, so researchers have something to think about.