# Fiches de cours

## Riemann surfaces

Viazovska Maryna

English

#### Summary

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex domains under discontinuous group actions, as algebraic curves. We will cover the following topics:

#### Content

• Topology of Riemann surfaces. Fundamental group. Homology groups
• Maps between Riemann surfaces. Degree of a map. Riemann'Hurwitz formula
• Differential forms
• De Rham cohomology
• Hodge decomposition
• Uniformization of Riemann surfaces
• Holomorphic differentials
• Periods of holomorphic differentials. Jacobian variety
• Abel theorem
• Riemann-Roch theorem
• Embedding of Riemann surfaces into projective space
• Riemann surfaces as algebraic curves
• Jacobians, abelian varieties, and theta functions
• Belyi maps

#### Keywords

• Riemann surfaces
• holomorphic maps
• differential forms
• meromorphic functions
• Jacobian variety

#### Learning Prerequisites

##### Required courses

• Complex analysis
• Vector analysis
• Basic topology and geometry

##### Recommended courses

• Introduction to differentiable manifolds
• Harmonic analysis

##### Important concepts to start the course

• Topological spaces
• Manifolds
• Coordinate charts. Change of coordinates
• Differential forms. Integration of differential forms. Stokes theorem
• Holomorphic functions. Cauchy integration formula
• Meromorphic functions. Residue theorem

#### Learning Outcomes

By the end of the course, the student must be able to:
• Define main mathematical notions introduced in the course
• State main theorems
• Apply main theorems to concrete examples
• Prove main theorems
• Solve problems similar to those discussed on tutorials
• Compute degree of a map, genus of a surface, intersection pairing, period matrix, basis of holomorphic differential forms, image under Abel map, etc.
• Construct examples and counterexamles
• Sketch proves of main results

#### Transversal skills

• Access and evaluate appropriate sources of information.
• Write a scientific or technical report.
• Demonstrate a capacity for creativity.
• Take feedback (critique) and respond in an appropriate manner.

#### Teaching methods

• lectures
• tutorials
• feedback on submitted homework solutions

#### Expected student activities

• attending lectures
• attending tutorials
• submitting written homeworks
• presenting solutions of the exercises

#### Assessment methods

• midterm home exam 40%
• final exam 60%

Dans le cas de l'art. 3 al. 5 du Règlement de section, l'enseignant décide de la forme de l'examen qu'il communique aux étudiants concernés.

#### Supervision

 Office hours Yes Assistants Yes Forum Yes

#### Resources

##### Bibliography

1. P. Griffiths and J. Harris, Principles of algebraic geometry
2. J. Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics
3. J. B. Bost, Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties.

### Dans les plans d'études

• Ingénierie mathématique, 2018-2019, Master semestre 1
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Riemann surfaces
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Ingénierie mathématique, 2018-2019, Master semestre 3
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Riemann surfaces
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Mathématiques - master, 2018-2019, Master semestre 1
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Riemann surfaces
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines
• Mathématiques - master, 2018-2019, Master semestre 3
• Semestre
Automne
• Forme de l'examen
Ecrit
• Crédits
5
• Matière examinée
Riemann surfaces
• Cours
2 Heure(s) hebdo x 14 semaines
• Exercices
2 Heure(s) hebdo x 14 semaines

LuMaMeJeVe
8-9 MAA110
9-10
10-11 MAA110
11-12
12-13
13-14
14-15
15-16
16-17
17-18
18-19
19-20
20-21
21-22
Cours
Exercice, TP
Projet, autre

### légende

• Semestre d'automne
• Session d'hiver
• Semestre de printemps
• Session d'été
• Cours en français
• Cours en anglais
• Cours en allemand