Prerequisites

• Basic Algebra (Linear Algebra, Groups, Rings and Fields, e.g. Chapters I-IV and XIII in S. Lang's book 'Algebra', Springer GTM 211)
• Basic point set topology, ideally 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. The study of Riemann Surfaces is a fascinating area of mathematics which mixes ideas from geometry, algebra, analysis and topology. 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, such as hyperelliptic Riemann surfaces.

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 of compact Riemann surfaces and remarkably allows us to relate them to complex algebraic curves.

Lecturers
Ariyan Javanpeykar