Riemann's Existence Theorem

If you search for statements of Riemann’s Existence Theorem, you will find many seemingly different statements which all might deserve their own name. These statements, which go under the name RET, range from the existence of nonconstant meromorphic functions on compact Riemann surfaces to the classification of finite branched covers via fundamental group monodromy. Also related is the fact that all compact Riemann surfaces are algebraic. In this talk, I will explain how these different concepts are closely related (while using as little analysis as possible).