A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations.
The theory of Riemann surfaces has been a source of inspiration and examples for many fields of mathematics. We will study the most important theorems concerning closed Riemann surfaces: the Riemann-Roch theorem, Abel’s theorem, and the Jacobi inversion theorem.
This course will be completed with ''Advanced topics in Analysis B'' in the next quarter.
In this course, our goal is to understand and hence to prove the Riemann-Roch theorem.
At the end of this course, students are expected to understand the Riemann-Roch theorem.
Riemann surfaces, the Riemann-Roch theorem
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | Riemann surfaces | Details will be provided during each class session. |
Class 2 | Topology of Riemann surfaces | |
Class 3 | Differential forms | |
Class 4 | Harmonic differetials, holomorphic differetials | |
Class 5 | Bilinear relations | |
Class 6 | Divisors | |
Class 7 | The Riemann-Roch theorem I | |
Class 8 | The Riemann-Roch theorem II |
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
None in particular
H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag
Assignments. Details will be announced during the session.
None
None in particular