MATH-410 / 5 credits

Teacher: Viazovska Maryna

Language: 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.

## Keywords

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

## Required courses

• Complex analysis
• Differential geometry
• Topology

## 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 No Others Moodle page

## 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.

## In the programs

• Semester: Fall
• Exam form: Written (winter session)
• Subject examined: Riemann surfaces
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Fall
• Exam form: Written (winter session)
• Subject examined: Riemann surfaces
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Fall
• Exam form: Written (winter session)
• Subject examined: Riemann surfaces
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks
• Semester: Fall
• Exam form: Written (winter session)
• Subject examined: Riemann surfaces
• Lecture: 2 Hour(s) per week x 14 weeks
• Exercises: 2 Hour(s) per week x 14 weeks

## Reference week

 Mo Tu We Th Fr 8-9 CM1221 9-10 10-11 CM1221 11-12 12-13 13-14 14-15 15-16 16-17 17-18 18-19 19-20 20-21 21-22

Thursday, 8h - 10h: Exercise, TP CM1221

Thursday, 10h - 12h: Lecture CM1221