books
- Forster
- Miranda
- Wilhelm Schlag
- https://mathweb.tifr.res.in/~srinivas/rsfull.pdf (uses ringed spaces definition of Riemann surfaces where the rings are rings of functions on it)
- https://math.berkeley.edu/~teleman/math/Riemann.pdf
- Donaldson
- Jost
- Renzo Cavalieri, Eric Miles (Hurwitz theory)
- …a hundred more books and notes available online
lecture videos
- These lectures assume covering space theory (algebraic topology) and uniformization theory and does (pre-moduli space) classification of Riemann surfaces: https://www.youtube.com/playlist?list=PLbMVogVj5nJSm4256vuITlsovUT1xVkUL
Riemann Surfaces by Jacob Bernstein ( bernstein@math.jhu.edu)
https://www.bilibili.com/video/BV1fW41197nr/?spm_id_from=333.337.search-card.all.click For MSRI summer school 2014: Complex geometry and geometric analysis on complex manifolds
- Prerequisites:
- Knowledge of basic complex analysis—at the level of Ahlfors, Complex Analysis, Chapters 1-5—will be assumed. Some basic familiarity with (abstract) surface theory and differential forms will be helpful. However, I will review this material as needed.
- Reading:
- The main text will be Donaldson, Riemann Surfaces; get at http://www2.imperial.ac.uk/~skdona/RSPREF.PDF.
- Syllabus: https://www.slmath.org/ckeditor_assets/attachments/106/bernstein_hein_naber_syllabus.pdf
- Other useful references:
- Farkas and Kra, Riemann Surfaces; a classical text on the subject.
- Miranda, Algebraic Curves and Riemann Surfaces; a more algebraic perspective.
- Week 1: Introduction to Riemann Surfaces
- Surfaces and Topology
- Riemann Surfaces and Holomorphic Maps
- Maps between Riemann Surfaces
- Calculus on Riemann Surfaces
- De Rham Cohomology
- Week 2: Geometric Analysis on Riemann Surfaces
- Elliptic Functions and Integrals
- Meromorphic Functions
- Inverting the Laplacian
- The Uniformization Theorem
- Riemann Surfaces and Minimal Surfaces
- A full course on Riemann Surfaces by M Khalkhali with videos on YouTube