# Riemann Surfaces

## Description

The theory of Riemann surfaces lies between differential geometry, algebraic geometry, and complex analysis. Riemann surfaces can be viewed as the simplest examples of complex manifolds or (when compact) algebraic varieties, as they are one-dimensional versions of these heavily studied higher-dimensional objects. This course will take a modern approach to Riemann surfaces, and will use the setting of Riemann surfaces to develop many important tools which have become essential in higher-dimensional geometry, with an emphasis on techniques that bridge algebraic and differential geometry. Thus the course is aimed at anyone who would like to gain an appreciation for these ideas, which should be useful to anyone with an interest in geometry (differential or algebraic) or analysis.

The course will follow Donaldson's book "Riemann surfaces" quite closely. In particular, we will aim to cover:

• briefly and somewhat informally, the smooth classification of two (real) dimensional manifolds via Morse theory;
• the basic definitions and examples of Riemann surfaces;
• holomorphic maps between Riemann surfaces and the relation with topology;
• calculus on manifolds as it applies to Riemann surfaces;
• the existence of meromorphic functions on Riemann surfaces, and constructing projective embeddings;
• the uniformisation theorem.

The main results of the course will be the final two bullet points, which demonstrate clearly the links with algebraic geometry. The proofs of these two results will involve some linear PDE theory which will be developed in this part of the course, which also provide a foundation for the many related results in geometry (such as Hodge theory) that we will not touch upon.

Depending on time and the interests of the audience, we may cover (probably only one of) the further topics:

• divisors, line bundles, cohomology and Jacobians;
• moduli and deformations;
• valuations and other algebraic approaches to the theory;
• further differential-geometric aspects: curvature, hyperbolic geometry.

The course will be taught in person for students in Glasgow (and streamed online for those elsewhere). The lectures will be 10:00-12:00 on Tuesdays, and the lecture theatre will vary from week to week:

10th October: 12 UNIV GDNS:101

17th October: ST ANDREWS BUILDING:ROOM 345

24th October: 12 UNIV GDNS:101

31st October: 12 UNIV GDNS:101

7th November: ST ANDREWS BUILDING:ROOM 368 SEM

14th November: 12 UNIV GDNS:101

21st November: ST ANDREWS BUILDING:ROOM 368 SEM

28th November: ST ANDREWS BUILDING:ROOM 345