Riemann surfaces classification

Riemann surfaces are complex geometric shapes that generalize the idea of multi-valued functions, like square roots or logarithms, making them single-valued. Their classification involves understanding their topological type (related to the number of “holes” or handles) and their geometric structure (such as whether they are spherical, toroidal, or have more complex forms). Mathematically, they are categorized by properties like genus (number of holes), symmetry, and curvature. This classification helps mathematicians understand the fundamental shapes and behaviors of complex functions, revealing how different surfaces can be constructed and related within the realm of complex analysis.