Riemann surfaces
Caution, these contents corresponds to the coursebooks of last year
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
- Complex manifolds
- Topology of compact Riemann surfaces
- Differential forms
- De Rham cohomology
- Hodge decomposition
- Holomorphic differentials
- 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
Learning Prerequisites
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
- weekly submitted homeworks 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 |
| Assistant.e.s | Yes |
| Forum | No |
| Others | Moodle page |
Resources
Bibliography
- S. K. Donaldson. Riemann surfaces
- J. Jost, Compact Riemann Surfaces: An Introduction to Contemporary Mathematics
- J. B. Bost, Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties.
- P. Griffiths and J. Harris, Principles of algebraic geometry.
Ressources en bibliothèque
Moodle Link
In the programs
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Riemann surfaces
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Riemann surfaces
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Riemann surfaces
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional
- Semester: Fall
- Exam form: Written (winter session)
- Subject examined: Riemann surfaces
- Courses: 2 Hour(s) per week x 14 weeks
- Exercises: 2 Hour(s) per week x 14 weeks
- Type: optional