Riemann Surfaces - M1 - 8EC

Prerequisites

  • Linear Algebra
  • Basic Algebra (Groups, Rings and Fields, e.g. Chapters I-III & V in P. Aluffi "Algebra: Chapter 0")
  • 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, 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, which in turn allows us to relate them to complex algebraic curves.

Rules about homework/exams

During the semester, there will be three homework assignments. The average grade of these homework assignments counts for 40% of the final grade (in the case of a resit, the homework also counts for 40% of the final grade). At the end of the semester, there will be a written exam. In order to pass the course, the grade of the final exam has to be 5 or higher, regardless of your grade for the homework assignments.

Lecturer

Victoria Hoskins