Theorem of Riemann-Hurwitz

The Riemann-Hurwitz theorem relates the shapes of two connected, curved surfaces called Riemann surfaces when one maps onto the other. It states that the number of "holes" (called genus) in the shapes changes in a predictable way based on how many times the mapping wraps around points (branch points). Specifically, it connects the genus of the original and the mapped surface, accounting for these special points where the map behaves differently, with a formula that adds correction terms for these branch points. This helps understand how complex shapes relate when transformed through functions.