Lectures are a sequel to ''Advanced topics of Analysis A'' in the previous quarter.
A Riemann surface is a two-real-dimensional manifold with holomorphic coordinate transformations.
We will study the most important theorems concerning closed Riemann surfaces,
in this course, Abel’s theorem and the Jacobi inversion theorem.
Using these and the Riemann-Roch theorem a topic in the previous quarter, we will study the Jacobian varieties and holomorphic maps of closed Riemann surfaces.
At the end of this course, students are expected to understand the main classical results of the theory of closed Riemann surfaces, like the Riemann-Roch Theorem, Abel's Theorem and the Jacobi inversion.
Riemann surfaces, Abel’s theorem, the Jacobi inversion theorem, Jacobian varieties
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard lecture course.
|Course schedule||Required learning|
|Class 1||Applications of the Riemann-Roch Theorem I (Weierstrass points)||Details will be provided during each class session.|
|Class 2||Applications of the Riemann-Roch Theorem II (automorphisms of closed Riemann surfaces)|
|Class 3||Abel’s theorem|
|Class 4||The Jacobi inversion theorem|
|Class 5||The Jacobian varieties I|
|Class 6||The Jacobian varieties II|
|Class 7||Holomorphic maps of closed Riemann surfaces I|
|Class 8||Holomorphic maps of closed Riemann surfaces 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.
Students are expected to have completed Advanced topics in Analysis A (MTH.C401).
None in particular