# Motivation for Riemann surfaces

Can someone please motivate me for Riemann surface? I am trying to read Foster's book but couldn't get any motivation. What is the general theme of this subject ? What is the origin of Riemann surface ? What are some ground breaking work in this area ? What are the connections with other part of geometry like differential geometry , lower dimensional topology? I shall be much obliged.

• I think the basic motivation is figuring out how to understand the "graph" of a complex multivalues function such as $\ln (z)$. – Ian Sep 2 '16 at 11:38
• There are several.motivations for Riemann surfaces: complex function theory- Riemann surfaces form natural compact spaces where algebraic complex function live - for example Möbius maps live on the projective space. topology : Riemann surfaces are orientable 2 manifolds Geometry: they are simplest examples of non trivial projective varieties and unlike others they are completely classified. Number theory: a special class of Riemann surfaces are elliptic curves..... There are many more that I don't know. – DBS Sep 2 '16 at 12:42