Riemann surfaces
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
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
- 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 |
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.
- R. Godement. Topologie algébrique et théorie des faisceaux.
- C. A. Weibel. An introduction to homological algebra.
Ressources en bibliothèque
- Riemann surfaces / Donaldson
- Compact Riemann Surfaces / Jost
- An introduction to homological algebra / Weibel
- Principles of algebraic geometry / Griffiths & Harris
- Topologie algébrique et théorie des faisceaux / Godement
- Introduction to Compact Riemann Surfaces, Jacobians, and Abelian Varieties / Bost
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