Flat structure of meromorphic connections on Riemann surfaces

One of the main open problems in local dynamics of several complex variables is the understanding of the dynamics, in a full neighborhood of the origin, of holomorphic germs tangent to the identity. In dimension one, the Leau-Fatou flower theorem provides exactly such an understanding. A few generalizations to several variables of the Leau-Fatou flower theorem have been proved, but none of them was strong enough to be able to describe the dynamics in a full neighborhood of the origin. Abate and Tovena, in 2011, has been able to show a strong relation between dynamics of time-1 maps of homogeneous vector fields in $\mathbb{C}^n$ and dynamics of geodesics of meromorphic connections of Riemann surfaces. In this talk, we will talk about our resent results on dynamics of geodesics of meromorphic connections on Riemann surfaces.