Riemann surface of genus g

A Riemann surface of genus \( g \) is a shape that locally looks like a flat, two-dimensional plane, but globally can be more complex and twisted, like a doughnut with \( g \) holes. The genus \( g \) counts the number of these holes—so a sphere has genus 0, a torus (doughnut shape) has genus 1, and more complicated surfaces have higher \( g \). These surfaces serve as the natural setting for studying complex functions, extending ideas from complex analysis to more intricate shapes, and are fundamental in understanding the geometry and topology of multi-valued functions.