Description

Riemann surfaces can arise, for instance, as zero sets of complex polynomials in ℂ^2 , but they can also arise from more topological techniques such as branched surfaces, or as ``multivalued functions´´ -- in an appropriate sense -- such as the square root of a complex number.

Certain properties of complex analysis carry over to Riemann surfaces. For instance, the maximum principle implies that a compact Riemann surface never admits a non-constant holomorphic function. On the other hand, meromorphic functions on compact Riemann surfaces can only admit finitely many zeros and poles, and the count of these with multiplicities is always zero. For holomorphic differential forms, objects that generalise functions, the number of poles and zeros is related to the Euler characteristic of the Riemann surface.

Riemann surfaces are a fascinating topic where interesting interactions between (complex) analysis and topology occur. Applications include Picard's small theorem, stating that a holomorphic function f:ℂ -> ℂ can miss at most one point in the image.

Prerequisites

Co-requisites

Resources

Relevant books include the following:

• Riemann surfaces, Simon Donaldson, Oxford University Press
• Lectures on Riemann surfaces, Otto Foster, Springer graduate texts in mathematics
• Complex algebraic curves, Frances Kirwan, Cambridge University Press

These are available online in the Library for instance here.