Abstract & Pure

Learn Riemann Surfaces

The analytic theory of compact Riemann surfaces: degree and ramification, Riemann-Hurwitz, holomorphic differentials and periods, Abel's theorem, uniformization, and the moduli of curves.

Free to start · adaptive placement finds your level · reviews timed so it stays learned.

What you'll learn

18 lessons in Riemann Surfaces

Charts and holomorphic atlasesHolomorphic maps and degreeRamification and the local normal formThe Riemann-Hurwitz formulaHyperelliptic curvesThe field of meromorphic functionsHolomorphic differentialsPeriods and the period matrixAbel's theorem and the JacobianThe Abel-Jacobi map and TorelliWeierstrass points and gapsThe uniformization theoremFuchsian groups and hyperbolic surfacesAutomorphisms and the Hurwitz boundTeichmuller space and moduliQuasiconformal mapsBelyi's theorem and dessinsPuiseux series and local structure
How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the moment just before you'd forget it. A short placement check finds what you already know, so you start Riemann Surfaces exactly where it's useful.

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