The shocking connection between complex numbers and geometry

Meromorphic functions are complex-differentiable functions f: ℂ → ℂ whose only singularities are poles—no essential singularities allowed.

Riemann’s existence theorem establishes profound correspondence between compact Riemann surfaces (complex analysis) and algebraic curves (algebraic geometry).

The Riemann sphere compactifies the complex plane by adding one “point at infinity,” creating a sphere topologically equivalent to a globe.

Riemann surfaces are surfaces that locally resemble the complex plane at each point, generalizing complex analysis from flat domains to curved spaces.