The Riemann mapping theorem

The Riemann mapping theorem states that any non-empty, simply connected area in the plane (excluding a single point) can be transformed into a unit disk (a perfect circle) in a way that preserves angles. This transformation, called a conformal map, re-shapes the region without distorting its internal angles, allowing complex and irregular shapes to be studied through the well-understood geometry of a circle. It’s a fundamental result in complex analysis, demonstrating that all such simple shapes have a common, standard form when viewed through these special, angle-preserving maps.