2022 Advanced courses in Analysis A

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Tanabe Masaharu 
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Course description and aims

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.

Student learning outcomes

At the end of this course, students are expected to understand the Riemann-Roch theorem.


Riemann surfaces, the Riemann-Roch theorem

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 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

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.C402 : Advanced topics in Analysis B

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



None in particular

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