What you'll learn

30 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 structureDifferentials of the three kindsThe Dirichlet principleCompact surfaces are algebraic curvesJacobi inversionTheta functionsBrill-Noether theoryThe Schottky problemHurwitz numbersThe mapping class groupExtremal lengthThe length spectrumTropical curves

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 day its model predicts you would 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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Related Abstract & Pure subjects

Abstract Algebra Representation Theory Lie Groups Universal Algebra Algebraic Topology Homological Algebra