- Linear Algebra
- Basic Algebra (Groups, Rings and Fields)
- Basic point set topology, including a first introduction to the fundamental group
- Familiarity with the notion of a manifold
- Complex Analysis: holomorphic and meromorphic functions in one variable, Cauchy's Integral Theorem and the Residue Theorem
Aim of the course
In this course we introduce and study Riemann surfaces, which are 1-dimensional complex manifolds. This is a fascinating area of mathematics which mixes ideas from topology, geometry, algebra and analysis. We will see how to naturally generalise many notions and results from the complex plane to Riemann surfaces, such as holomorphic and meromorphic functions. We will study the theory through many concrete examples.
The main goal of the course is to describe the geometry of compact Riemann surfaces. Among many interesting theorems we will cover, one of the most important in terms of its wide-ranging geometric consequences is the Riemann-Roch Theorem. This will enable us to study projective embeddings, which in turn allows us to relate compact Riemann surfaces to complex algebraic curves.
Victoria Hoskins (RU)