Find an analytic parameterization of the compact Riemann surfaces in a fixed homeomorphism class. The Ahlfors-Bers theorem proved that Riemann's moduli space gives the solution.
Riemann's Moduli Problem
See also
Ahlfors-Bers Theorem, Riemann's Moduli SpaceExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Riemann's Moduli Problem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannsModuliProblem.html