2022 Advanced courses in Analysis B

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Tanabe Masaharu 
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Course description and aims

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.

Student learning outcomes

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

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

Standard lecture course.

Course schedule/Required learning

  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

Out-of-Class Study Time (Preparation and Review)

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

Reference books, course materials, etc.

H. M. Farkas and I. Kra, Riemann surfaces, GTM 71, Springer-Verlag

Assessment criteria and methods

Assignments. Details will be announced during the session.

Related courses

  • ZUA.C301 : Complex Analysis I
  • MTH.C301 : Complex Analysis I
  • MTH.C302 : Complex Analysis II
  • MTH.C401 : Advanced topics in Analysis A

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have completed Advanced topics in Analysis A (MTH.C401).


None in particular

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