In this talk I will explain an application of the theory of Lie groupoids to the study of differential equations with Fuchsian singularities. These are linear ordinary differential equations with a first-order pole at the origin, and their solutions are often multivalued and singular. The study of these equations is classical, and usually proceeds by solving them via power series according to the Frobenius method.

In their work on the Stokes groupoids, Gualtieri-Li-Pym showed that these equations can be converted to differential equations on a naturally defined Lie groupoid. Importantly, these new differential equations are smooth and their solutions are single-valued. In this way, the Lie groupoid provides the natural domain of definition for Fuchsian differential equations. In this talk, I will explain how this perspective can be leveraged to classify and solve these differential equations in a way which illuminates their underlying geometry and the phenomenon of resonance. In particular, the classification of these equations can be reduced to the classification of C* actions.

If time permits, I will also explain how Lie groupoids may be used to give a functorial classification of flat connections with logarithmic singularities on complex manifolds.