Riemann-Hurwitz formula

The Riemann-Hurwitz formula relates the complexity (genus) of two interconnected shapes called Riemann surfaces, when one is mapped onto the other. It accounts for how many times the covering surface 'wraps around' points on the base surface, especially where the map behaves differently, called ramification points. Essentially, it quantifies how the shape's fundamental features change through such mappings by balancing the genus of both surfaces, the degree of the map (how many sheets cover), and the ramification data. This formula is fundamental in complex analysis and algebraic geometry, helping mathematicians understand how complex structures transform under mappings.