MATH-410 / 5 crédits

Enseignant: Mornev Maxim

Langue: Anglais

## 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.

## Content

• Category theory
• Homological algebra
• Sheaves and their cohomology
• Complex manifolds
• Topology of compact Riemann surfaces
• Differential forms
• De Rham cohomology
• Hodge decomposition
• Holomorphic differentials
• Invertible sheaves, divisors and linear systems
• Riemann-Roch theorem
• Serre duality
• Embedding of compact Riemann surfaces into projective spaces

## Keywords

• Riemann surfaces
• holomorphic maps
• differential forms
• meromorphic functions
• cohomology of sheaves

## Required courses

• Complex analysis
• Differential geometry
• Topology

## Recommended courses

• Introduction to differentiable manifolds
• Complex 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 No Others Moodle page

## Bibliography

1. S. K. Donaldson. Riemann surfaces
2. J. Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics
3. J. B. Bost, Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties.
4. P. Griffiths and J. Harris, Principles of algebraic geometry.
5. R. Godement. Topologie algébrique et théorie des faisceaux.
6. C. A. Weibel. An introduction to homological algebra.

## Dans les plans d'études

• Semestre: Automne
• Forme de l'examen: Ecrit (session d'hiver)
• Matière examinée: Riemann surfaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Automne
• Forme de l'examen: Ecrit (session d'hiver)
• Matière examinée: Riemann surfaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Automne
• Forme de l'examen: Ecrit (session d'hiver)
• Matière examinée: Riemann surfaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel
• Semestre: Automne
• Forme de l'examen: Ecrit (session d'hiver)
• Matière examinée: Riemann surfaces
• Cours: 2 Heure(s) hebdo x 14 semaines
• Exercices: 2 Heure(s) hebdo x 14 semaines
• Type: optionnel

## Semaine de référence

Jeudi, 8h - 10h: Cours DIA004

Jeudi, 10h - 12h: Exercice, TP DIA004

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