In this post, I will sketch a classification of Riemann surfaces.

For those who haven't heard about the subject before, there is an introduction. For the impatient, look at the bottom of the post, where I have written a very short summary.

Table of contents:

• Definitions
• Examples and counter-examples
• Topological classification, the genus
• Digression about algebraic field extensions
• Biholomorphic classification via universal covering deck transformations
• Brief discussion of Gaussian curvature and conformal geometry
• Summary

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